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APPEARED IN BULLETIN OF THE

arXiv:math/9304209v1 [math.GT] 1 Apr 1993APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 28, Number 2, April 1993, Pages 253-287NEW POINTS OF VIEW IN KNOT THEORYJoan S. BirmanIntroductionIn this article we shall give an account of certain developments in knot theorywhich followed upon the discovery of the Jones polynomial [Jo3] in 1984.Thefocus of our account will be recent glimmerings of understanding of the topologicalmeaning of the new invariants. A second theme will be the central role that braidtheory has played in the subject.

A third will be the unifying principles providedby representations of simple Lie algebras and their universal enveloping algebras.These choices in emphasis are our own. They represent, at best, particular aspectsof the far-reaching ramifications that followed the discovery of the Jones polynomial.We begin in §1 by discussing the topological underpinnings of that most famousof the classical knot invariants—the Alexander polynomial.

It will serve as a modelfor the sort of thing one would like to see for the Jones polynomial. Alexander’s1928 paper ends with a hint of things to come, in the form of a crossing-changeformula for his polynomial, and in §2 we discuss how related formulas made theirappearances in connection with the Jones polynomial and eventually led to thediscovery of other, more general knot and link polynomials.

A more systematicdescription of these “generalized Jones invariants” is given in §3, via the theoryof R-matrices. That is where braids enter the picture, because every generalizedJones invariant is obtained from a trace function on an “R-matrix representation”of the family of braid groups {Bn; n = 1, 2, 3, .

. .

}. The mechanism for finding R-matrix representations of Bn is the theory of quantum groups.

For this reason,the collection of knot and link invariants that they determine have been calledquantum group invariants. We shall refer to them here as either quantum groupinvariants or generalized Jones invariants, interchangeably.

While the theory of R-matrices and their construction via quantum groups gives a coherent and uniformdescription of the entire class of invariants, the underlying ideas will be seen to beessentially combinatorial in nature. Thus, by the end of §3 the reader should beginto understand how it could happen that in 1990 topologists had a fairly coherentframework for constructing vast new families of knot and link invariants, possibly1991 Mathematics Subject Classification.

Primary 57M25; Secondary 57N99.Key words and phrases. Knots, links, knot polynomials, knot groups, Vassiliev invariants,R-matrices, quantum groups.This research was supported in part by NSF Grant DMS-91-06584 and by the US-Israel Bina-tional Science Foundation Grant 89-00302-2This manuscript is an expanded version of an AMS-MAA Invited Address, given on January8, 1992, at the Joint Winter Meeting of the AMS and the MAA in Baltimore, MarylandReceived by the editors September 30, 1992c⃝1993 American Mathematical Society0273-0979/93 $1.00 + $.25 per page1

2JOAN S. BIRMANeven enough to classify unoriented knot and link types, without having the slightestclue to the underlying topology.In §4 we introduce an entirely new collection of invariants, which arose out oftechniques pioneered by Arnold in singularity theory (see the introduction to [Arn1,Arn2]). The new invariants will be seen to have a solid basis in a very interestingnew topology, where one studies not a single knot, but a space of all knots.

Thispoint of view was carried out successfully for the case of knots by Vassiliev [V].(Remark. For simplicity, we restrict our attention in this part of the review toknots.

The theory is, at this writing, less well developed for links.) Vassiliev’sknot invariants are rational numbers.

They lie in vector spaces Vi of dimensiondi, i = 1, 2, 3, . .

., with invariants in Vi having “order” i.On the other hand,quantum group invariants are Laurent polynomials over Z, in a single variable q.The relationship between them is as follows:Theorem 1. Let Jq(K) be a quantum group invariant for a knot type K. LetPx(K) =infXi=0ui(K)xibe the power series over the rational numbers Q obtained from Jq(K) by settingq = ex and expanding the powers of ex in their Taylor series.

Then the coefficientui(K) of xi is 1 if i = 0 and a Vassiliev invariant of order i for each i ≥1.Thus Vassiliev’s topology of the space of all knots suggests the topological under-pinnings we seek for the quantum group knot invariants.A crucial idea in the statement of Theorem 1 is that one must pass to thepower series representation of the knot polynomials before one can understand thesituation. This was first explained to the author and Lin by Bar Natan.

Theorem1 was first proved for special cases in [BL] and then generalized in [Li1]. In §§5–7we describe a set of ideas which will be seen to lead to a new and very simple proofof Theorem 1.

First, in §5, we review how Vassiliev’s invariants, like the Jonespolynomial, the HOMFLY and Kauffman polynomials [FHL, Kau1], and the G2polynomial [Ku] can be described by axioms and initial data. (Actually, all of thequantum group invariants admit such a description, but the axioms can be verycomplicated; so such a description would probably not be enlightening).

In §6 weintroduce braids into the Vassiliev setting, via a new type of object, the monoid ofsingular braids. Remarkably, this monoid will be seen to map homomorphically (weconjecture isomorphically) into the group algebra of the braid group, implying thatit is as fundamental an object in mathematics as the braid group itself.

This allowsus to extend every R-matrix representation of the braid group to the singular braidmonoid. In §7 we use R-matrices and singular braids to prove Theorem 1.

In §8 wereturn to the topology, discussing the beginning of a topological understanding ofthe quantum group invariants. We then discuss, briefly, a central problem: do wenow know enough to classify knots via their algebraic invariants?

We will describesome of the evidence which allows us to sharpen that question.Our goal, throughout this review, is to present the material in the most trans-parent and nontechnical manner possible in order to help readers who work in otherareas to learn as much as possible about the state of the art in knot theory. Thus,when we give “proofs”, they will be, at best, sketches of proofs.

We hope there will

NEW POINTS OF VIEW IN KNOT THEORY3be enough detail so that, say, a diligent graduate student who is motivated to reada little beyond this paper will be able to fill in the gaps.Among the many topics which we decided deliberately to exclude from thisreview for reasons of space, one stands out: it concerns the generalizations of thequantum group invariants and Vassiliev invariants to knots and links in arbitrary 3-manifolds, i.e., the program set forth by Witten in [Wi]. That very general programis inherently more difficult than the special case of knots and links or simply of knotsin the 3-sphere.

It is an active area of research, with new discoveries made everyday.We thought, at first, to discuss, very briefly, the 3-manifold invariants ofReshetikhin and Turaev [RT] and the detailed working out of special cases of thoseinvariants by Kirby and Melvin [KM]; however, we then realized that we could notinclude such a discussion and ignore Jeffrey’s formulas for the Witten invariants ofthe lens spaces [Je]. Reluctantly, we made the decision to restrict our attention toknots in 3-space, but still, we have given at best a restricted picture.

For example,we could not do justice to the topological constructions in [La] and in [Koh1, Koh2]without making this review much longer than we wanted it to be, even though itseems very likely that those constructions are closely related to the central themeof this review. We regret those and other omissions.1.

An introduction to knots and their Alexander invariantsWe will regard a knot K as the embedded image of an oriented circle S1 inoriented 3-space R3 or S3.If, instead, we begin with µ ≥2 copies of S1, theimage (still called K), is a link.Its type K is the topological type of the pair(S3, K), under homeomorphisms which preserve orientations on both S3 and K.Knot types do not change if we replace S3 by R3, because every homeomorphismof S3 is isotopic to one which fixes one point, and that point may be chosen to bethe point at infinity.We may visualize a knot via a diagram, i.e., a projection of K ⊂R3 onto a genericR2 ⊂R3, where the image is decorated to distinguish overpasses from underpasses,for example, as for highways on a map.Examples are given in Figure 1.Theexample in Figure 1(a) is the unknot, represented by a round planar circle. Figure1(b) shows a layered diagram of a knot, i.e., one which has been drawn, using anarbitrary but fixed starting point (which in Figure 1(c) is the tip of the arrow)without the use of an eraser, so that the first passage across a double point in theprojection is always an overpass.

We leave it to the reader to prove that a layereddiagram with µ components always represents µ unknotted, unlinked circles. Fromthis simple fact it follows that any diagram of any link may be systematicallychanged to a diagram for the unlink on the same number of components by finitelymany crossing changes.The diagram in Figure 1(c) was chosen to illustrate the subtleties of knot di-agrams.

It too represents the unknot, but it is not layered. This example wasconstructed by Goeritz [Go] in the mid-1930s.

At that time it was known that afinite number of repetitions of Reidemeister’s three famous moves, depicted (up toobvious symmetries and variations) in Figure 2, suffice to take any one diagram ofa knot to any other. Notice that Reidemeister’s moves are “local” in the sense thatthey are restricted to regions of the diagram which contain at most three crossings.If one defines the complexity of a knot diagram to be its crossing number, a natu-ral question is whether there is a set of moves that preserve or reduce complexity

4JOAN S. BIRMANFigure 1. Diagrams of the unknot.and that, applied repeatedly, suffice to reduce an arbitrary diagram of an arbitraryknot to one of minimum crossing number.

The diagram in Figure 1(c) effectivelyended that approach to the subject, because the eight moves which had by thenbeen proposed as augmented Reidemeister moves did not suffice to simplify thisdiagram.Some insight may be obtained into the question by inspecting the “handle move”of Figure 3. Note that the crossing number in the diagram of Figure 1(c) can bereduced by an appropriate handle move.

The region that is labeled X in Figure3 is arbitrary.The handle move clearly decreases crossing number, but a fewmoments of thought should convince the reader that if one tries to factorize it into

NEW POINTS OF VIEW IN KNOT THEORY5Figure 3. A handle move.Figure 2.

Reidemeister moves.a product of Reidemeister moves, for any sufficiently complicated choice of X, itwill be necessary to use the second Reidemeister move repeatedly in order to createregions in which the third move may be applied. The same sort of reasoning makesit unlikely that any set of local moves suffices to reduce complexity.

Indeed, if weknew such a set of moves, we would have the beginning of an algorithm for solvingthe knot problem, because there are only finitely many knot diagrams with fixedcrossing number.In Figure 4 we have given additional examples, taken from the beginning of atable compiled at the end of the nineteenth century by the physicist Peter GuthrieTait and coworkers, as part of a systematic effort to classify knots.The knotsin the table are listed in order of their crossing number, so that, for example,76 is the 6th knot that was discovered with 7 crossings. The part of the tablewhich we have shown includes all prime knots with at most 7 crossings, up tothe symmetries defined by reversing the orientation on either K or S3.

(We haveshown the two trefoils, for reasons which will become clear shortly.) The tables,which eventually included all prime knots defined by diagrams with at most 13crossings, were an ambitious undertaking.

(Aside: Yes, it is true.Knots, likeintegers, have a decomposition into an appropriately defined product of prime knots,and this decomposition is unique up to order.) Their clearly stated goal [Ta] wasto uncover the underlying principles of knotting, but to the great disappointmentof all concerned they did not even succeed in revealing a single knot type invariantwhich could be computed from a diagram.

Their importance, to this day, is dueto the fact that they provide a rich source of examples and convincing evidence ofboth the beauty and subtlety of the subject.Beneath each of the examples in Figure 4 we show two famous invariants, namely,the Alexander polynomial Aq(K) and the one-variable Jones polynomial Jq(K).

6JOAN S. BIRMANFigure 4.Both are to be regarded as Laurent polynomials in q, the series of numbers rep-resenting the sequence of coefficients, the bracketed one being the constant term.Thus the lower sequence (1,-1,1,-2, [2], -1,1) beneath the knot 61 shows that itsJones polynomial is q−4 −q−3 + q−2 −2q−1 + 2 −q + q2. The richness of struc-ture of both invariants is immediately clear from the sixteen examples in our table.There are no duplications (except for the Alexander polynomials of the two tre-foils, which we put in deliberately to make a point).

The arrays of subscripts andsuperscripts, as well as the roots (which are not shown) and the poles of the Jonesinvariant (about which almost nothing is known), suggest that the polynomials

NEW POINTS OF VIEW IN KNOT THEORY7could encode interesting properties of knots. Notice that Aq(K) is symmetric, i.e.,Aq(K) = Aq−1(K), or, equivalently, the array of coefficients is palendromic.

On theother hand, Jq(K) is not. Both polynomials take the value 1 at q = 1.

(Remark.The Alexander polynomial is actually only determined up to ±multiplicative pow-ers of q, and we have chosen to normalize it to stress the symmetry and so that itsvalue at 1 is +1 rather than −1. )The knots in our table are all invertible; i.e., there is an isotopy of 3-space whichtakes the oriented knot to itself with reversed orientation.

The first knot in thetables that fails to have that property is 817. Neither the Alexander nor the Jonespolynomials changes when the orientation on the knot is changed.

We shall havemore to say about noninvertible knots in §8.While we understand the underlying topological meaning of Aq(K) very well,we are only beginning to understand the topological underpinnings of Jq(K). Tobegin to explain the first assertion, let us go back to one of the earliest problemsin knot theory: to what extent does the topological type X of the complementaryspace X = S3 −K and/or the isomorphism class G of its fundamental groupG(K) = π1(X, x0) suffice to classify knots?

The trefoil knot is almost everybody’scandidate for the simplest example of a nontrivial knot, so it seems remarkablethat, not long after the discovery of the fundamental group of a topological space,Max Dehn [De] succeeded in proving that the trefoil knot and its mirror image hadisomorphic groups, but that their knot types were distinct. Dehn’s proof is veryingenious!

This was the beginning of a long story, with many contributions (e.g.,see [Sei, Si, CGL]) which reduced repeatedly the number of distinct knot typeswhich could have homeomorphic complements and/or isomorphic groups, until itwas finally proved, very recently, that (i) X determines K (see [GL]) and (ii) if Kis prime, then G determines K up to unoriented equivalence [Wh]. Thus there areat most four distinct oriented prime knot types which have the same knot group[Wh].

This fact will be important to us shortly.The knot group G is finitely presented; however, it is infinite, torsion-free, and(if K is not the unknot) nonabelian. Its isomorphism class is in general not easilyunderstood via a direct attack on the problem.

In such circumstances, the obviousthing to do is to pass to the abelianized group, but unfortunately G/[G, G] ∼=H1(X; Z) is infinite cyclic for all knots, so it is of no use in distinguishing knots.Passing to the covering space eX which belongs to [G, G], we note that there is anatural action of the cyclic group G/[G, G] on eX via covering translations. Theaction makes the homology group H1( eX; Z) into a Z[q, q−1]-module, where q is thegenerator of G/[G, G].

This module turns out to be finitely generated. It’s thefamous Alexander module.

While the ring Z[q, q−1] is not a PID, relevant aspectsof the theory of modules over a PID apply to H1( eX; Z). In particular, it splits as adirect sum of cyclic modules, the first nontrivial one being Z[q, q−1]/Aq(K).

ThusAq(K) is the generator of the “order ideal”, and the smallest nontrivial torsioncoefficient in the module H1( eX). In particular, Aq(K) is very clearly an invariantof the knot group.We regard the above description of Aq(K) as an excellent model for what wemight wish in a topological invariant of knots.

We know precisely what it detects,and so we also know precisely what it fails to detect. For example, it turns out thatπ1( eX) is finitely generated if and only if X has the structure of a surface bundleover S1, but there is no way to tell definitively from Aq(K) whether π1( eX) is or

8JOAN S. BIRMANis not finitely generated. On the other hand, if a surface bundle structure exists,the genus of the surface is determined by Aq(K).The polynomial Aq(K) alsogeneralizes in many ways.

For example, there are Alexander invariants of links,also additional Alexander invariants when the Alexander module has more thanone torsion coefficient. Moreover, the entire theory generalizes naturally to higherdimensional knots, the generalized invariants playing a central role in that subject.Returning to the table in Figure 4, we remark that when a knot is replacedby its mirror image (i.e., the orientation on S3 is reversed), the Alexander andJones polynomials Aq(K) and Jq(K) go over to Aq−1(K) and Jq−1(K) respectively.As noted earlier, Aq(K) is invariant under such a change, but from the simplestpossible example, the trefoil knot, we see that Jq(K) is not.

Now recall that G doesnot change under changes in the orientation of S3. This simple argument showsthat Jq(K) cannot be a group invariant!Since there are at most four distinctknot types that share the same knot group G, a first wild guess would be thatJq(K), which does detect changes in the ambient space orientation (but not in knotorientation), classifies unoriented knot types; but this cannot be true because [Kan]constructs examples of infinitely many distinct prime knot types with the sameJones polynomial.

Thus it seems interesting indeed to ask about the underlyingtopology behind the Jones polynomial. If it is not a knot group invariant, whatcan it be?

We will begin to approach that problem by a circuitous route, takingas a hint the central and very surprising role of “crossing-change formulas” in thesubject.2. Crossing changesA reader who is interested in the history of mathematics will find, on browsingthrough several of Alexander’s Collected Works, that many of his papers end withan intriguing or puzzling comment or remark which, as it turned out with thewisdom of hindsight, hinted at future developments of the subject.

For example,in his famous paper on braids [Al1], which we will discuss in detail in §6, he provesthat every knot or link may be represented as a closed braid. He then remarks(at the end of the paper) that this yields a construction for describing 3-manifoldsvia their fibered knots; however, he did it long before anyone had considered theconcept of a fibered knot!

Another example that is of direct interest to us nowoccurs in [Al2], where he reports on the discovery of the Alexander polynomial. Inequation (12.2) of that paper we find observations on the relationship between theAlexander polynomials of three links: Kp+, Kp−, and Kp0, which are defined bydiagrams that are identical outside a neighborhood of a particular double point p,where they differ in the manner indicated in Figure 5.The formula which Alexander gives is:(1a)Aq(Kp+) −Aq(Kp−) = (√q −1/√q)Aq(Kp0).This formula passed unnoticed for forty years.

(We first learned about Alexander’sversion of it in 1970 from Mark Kidwell.) Then, in 1968 it was rediscovered, inde-pendently, by John Conway [C], who added a new observation: If you require, inaddition to (1a), that:(1b)Aq(O) = 1,

NEW POINTS OF VIEW IN KNOT THEORY9Figure 5. Related link diagrams.Figure 6.

Three related diagrams for the unlink.where O is the unknot, then (1a) and (1b) determine Aq(K) on all knots, by arecursive procedure. To see this, the first thing to observe is that if Oµ is theµ-component unlink, then we may find related diagrams in which Kp+, Kp−, andKp0 represent Oµ, Oµ, and Oµ+1 respectively, as in Figure 6.This fact, in conjunction with (1a), implies that Aq(O) = 0 if µ ≥2.

Next,recall (Figure 1b) that any diagram K for any knot type K may be changed toa layered diagram which represents the unknot or unlink on the same number ofcomponents, by appropriate crossing changes. Induction on the number of crossingchanges to the unlink then completes the proof of Conway’s result.

This led himto far-reaching investigations of the combinatorics of knot diagrams.While the Jones polynomial was discovered via braid theory (and we shall havemore to say about that shortly), Jones noticed, very early in the game, that hispolynomial also satisfied a crossing-change formula, vis:(2a)q−1Jq(Kp+) −qJq(Kp−) = (√q −1/√q)Jq(Kp0).which, via Figure 6 and layered diagrams, may be used, in conjunction with theinitial data(2b)Jq(O) = 1,to compute Jq(K) for all knots and links. Motivated by the similarity between(1a)–(1b) and (2a)–(2b), several authors [LM, Ho, PT] were led to consider a moregeneral crossing-change formula, which for our purposes may be described as aninfinite sequence of crossing-change formulas:(3a)q−nHq,n(Kp+) −qnHq,n(Kp−) = (√q −1/√q)Hq,n(Kp0),(3b)Hq,n(O) = 1,where n ∈Z.

It turns out that (3a)–(3b) determine an infinite sequence of one-variable polynomials which in turn extend uniquely to give a two-variable invariantwhich has since become known as the HOMFLY polynomial [FHL]. Later, (1a)–(3b)were replaced by a more complicated family of crossing-change formulas, yielding

10JOAN S. BIRMANthe Kauffman polynomial invariant of knots and links [Kau1]. A unifying principlewas discovered (see [Re2]) which yielded still further invariants, for example, theG2 invariant of embedded knots, links, and graphs [Ku].

Later, crossing-changeformulas were used to determine other polynomial invariants of knotted graphs[Y1] as well.At this time we know many other polynomial invariants of knots, links, andgraphs. In principle, all of them can be defined via generalized crossing-changes,together with initial data.

In general, a particular polynomial will be defined by afamily of equations which are like (2a) and (3a). It is to be expected that each suchequation will relate the invariants of knots which are defined by diagrams whichdiffer in some specified way, in a region which has a fixed number of incoming andoutgoing arcs.

The ways in which they differ will be more complicated than a simplechange in a crossing and “surgery” of the crossing. Thus we have a conundrum:on the one hand, knot and link diagrams and crossing-change formulas clearly havemuch to do with the subject; on the other hand, their role is in many ways puzzling,because we do not seem to be learning as much as we might expect to learn aboutdiagram-related invariants from the polynomials.We make this last assertion explicit.

First, let us define three diagram-relatedknot invariants:(i) the minimum crossing number c(K) of a knot,(ii) the minimum number of crossing changes u(K), i.e., to the unknot, and(iii) the minimum number of Seifert circles s(K),where each of these invariants is to be minimized over all possible knot diagrams.(Aside. For the reader who is unfamiliar with the concept of Seifert circles, wenote that by [Y2] the integer s(K) may also be defined to be the braid index of aknot or link, i.e., the smallest integer s such that K may be represented as a closeds-braid.

If the reader is also not familiar with closed braids, he or she might wishto peek ahead to §3.) All of them satisfy inequalities which are detected by knotpolynomials.

For example, the Morton-Franks-Williams inequality places upper andlower bounds on s(K) [Mo2, FW]. Also, the Bennequin inequality, recently provedby Menasco [Me], gives a lower bound for u(K) which can be detected by the one-variable Jones polynomial.

As another example, the one-variable Jones polynomialwas a major tool in the proof of the Tait conjecture (see [Kau2, Mu]), which relatesto the definitive determination of c(K) for the special case of alternating knotswhich are defined by alternating diagrams. On the other hand, there are exampleswhich show that none of the inequalities mentioned above can be equalities for allknots.

Indeed, at this writing we do not have a definitive method for computingany of these very intuitive diagram-related knot and link-type invariants.3. R-Matrix representations of the braid groupOne of the very striking successes of the past eight years is that, after a periodduring which new polynomial invariants of knots and links were being discovered atan alarming rate (e.g., see [WAD]), order came out of chaos and a unifying principleemerged which gave a fairly complete description of the new invariants; we studyit in this section.

We remark that this description may be given in at least twomutually equivalent ways: the first is via the algebras of [Jo4, BW, Kal] and thecabling construction of [Mu, We2]; the second is via the theory of R-matrices, aswe shall do here.

NEW POINTS OF VIEW IN KNOT THEORY11Figure 7. Braids.Our story begins with the by-now familiar notion of an n-braid [Art].

See Figure7(a) for a picture, when n = 3. Our n-braid is to be regarded as living in a slabof 3-space R2 × I ⊂R3.

It consists of n interwoven oriented strings which joinn points, labeled 1, 2, . .

. , n, in the plane R2 × {0} with corresponding points inR2 × {1}, intersecting each intermediate plane R2 × {t} in exactly n points.

Twobraids are equivalent if there is an isotopy of one to the other which preserves theinitial and end point of each string, fixes each plane R2 × {t} setwise, and neverallows two strings to intersect. Multiplication is by juxtaposition, erasure of themiddle plane, and rescaling.Closed braids are obtained from (open) braids byjoining the initial point of each strand, in R3, to the corresponding end point, inthe manner illustrated in Figure 7(b), so that if one thinks of the closed braid aswrapping around the z-axis, it meets each plane θ =constant in exactly n points.A famous theorem of Alexander [Al1] asserts that every knot or link may be sorepresented, for some n. (Remark.

Lemma 1 of §6 sketches a generalization ofAlexander’s original proof.) In fact, if K is our oriented link, we may choose anyoriented R1 ⊂R3 which is disjoint from K and modify K by isotopy so that thiscopy of R1 is the braid axis.The braid group Bn is generated by the elementary braids σ1, .

. .

, σn−1 illus-trated in Figure 8. For example, the braid in Figure 7(a) may be described, usingthe generators given in Figure 8, by the word σ−21 σ2σ−11 .

Defining relations in Bnare:(4a)σiσj = σjσiif | i −j |≥2,(4b)σiσjσi = σjσiσjif | i −j |= 1,We will refer to these as the braid relations.Let B∞be the disjoint union of the braid groups B1, B2, . .

. .

Define β, β∗∈B∞to be Markov-equivalent if the closed braids ˆβ, ˆβ∗which they define represent the

12JOAN S. BIRMANFigure 8. Elementary braids, singular braids, and tangles.same link type.

Markov’s theorem, announced in [Mar] and proved forty years laterin [Bi], asserts that Markov equivalence is generated by conjugacy in each Bn andthe map Bn →Bn+1 which takes a word W(σ1, . .

. σn−1) to W(σ1, .

. .

σn−1)σ±1n .We call the latter Markov’s second move.The examples in Figures 7(a), 7(b)may be used to illustrate Markov equivalence. The 3-braid σ−21 σ2σ−11shown thereis conjugate in B3 to σ−31 σ2, which may be modified to the 2-braid σ−31usingMarkov’s second move, so that σ−31∈B2 is Markov equivalent to σ−21 σ2σ−11∈B3.Let {ρn : Bn →GLmn(E); n = 1, 2, 3, .

. .} be a family of matrix representationsof Bn over a ring E with 1.

Let in : Bn →Bn+1 be the natural inclusion map whichtakes Bn to the subgroup of Bn+1 of braids on the first n strings. A linear functiontr : ρn(Bn) →E is said to be a Markov trace if:(i) tr(1) = 1,(ii) tr(ρn(αβ)) = tr(ρn(βα))∀α, β ∈Bn,(iii) ∃z ∈E such that if β ∈in(Bn), then tr(ρn+1(βσ±1n )) = (z)(tr(ρn(β)).In particular, by setting β = 1 in (iii) we see that z = tr(ρn(σ±1n−1)).

In view ofMarkov’s theorem, it is immediate that every family of representations of Bn whichsupports a Markov trace determines a link type invariant F(Kβ) of the link typeKβ of the closed braid ˆβ, defined by(5)F(Kβ) = z1−n tr(ρn(β)).Later, we will need to ask what happens to (5) if we rescale the representation, sowe set the stage for the modifications now. Suppose that ρ = ρn is a representationof Bn in GLm(E) which supports a function tr which satisfies (i)–(iii).

Let γ be anyinvertible element in E. Then we may define a new representation ρ′ by the ruleρ′(σi) = γρ(σi) for each i = 1, . .

. , n. It is easy to see that ρ′ is a representation ifand only if ρ is, because (4a) and (4b) are satisfied for one if and only if they aresatisfied for the other.

Properties (i) and (ii) will continue to hold if we replace ρby ρ′, but (iii) will be modified because the traces of σi and σ−1iwill not be thesame in the new representation. To determine the effect of the change on equation(5), let z′ = tr(ρ′(σi)) = γ tr(ρ(σi)) = γz, so z = γ−1z′.

Choose any β ∈Bn.Then β may be expressed as a word σε1µ1σε2µ2 · · · σεrµr in the generators, where eachεj = ±1. Let ε(β) = Pri εi.

Then equation (5) will be replaced by(5′)F(Kβ) = γ−ε(β)(z′)1−n tr(ρ′n(β)).

NEW POINTS OF VIEW IN KNOT THEORY13Thus the existence of a link type invariant will not be changed if we rescale therepresentation by introducing the invertible element γ. Similar considerations applyif we rescale the trace function.

For example, instead of requiring that tr(1) = 1,we could require that if 1n denotes the identity element in Bn, then tr(1n) = zn−1,in which case the factor z1−n in equation (5) would vanish.The invariant which is defined by equation (5) or (5′) depends directly on thetraces of matrices in a finite-dimensional matrix representation ρn of the braidgroup Bn. Any such representation is determined by its values on the generatorsσi, and since these are finite-dimensional matrices they satisfy their characteristicpolynomials, yielding polynomial identities.

From the point of view of this section,such identities are one source of the crossing-change formulas we mentioned earlier,in §2. Another source will be trace identities, which always exist in matrix groups.

(See, e.g., [Pr].) In general one needs many such identities (i.e., polynomial equa-tions satisfied by the images of various special braid words) to obtain axioms whichsuffice to determine a link type invariant.In the summer of 1984, almost eight years ago to the day from this writing,this author met with Vaughan Jones to discuss possible ramifications of a discoveryJones had made in [Jo2] of a new family of matrix representations of Bn, in conjunc-tion with his earlier studies of type II1 factors and their subfactors in von Neumannalgebras.

Before that time, there was essentially only one matrix representation ofthe braid group which had a chance of being faithful and had been studied in anydetail—the Burau representation (see [Bi]). The knot invariant which was associ-ated to that representation was the Alexander polynomial.

Jones had shown thathis representations contained the Burau representation as a proper summand. Thushis representations were new and interesting.

Also, they supported an interestingtrace function. As it developed [Jo3], his trace functions were Markov traces.

If ρnis taken to be the Jones’s representation of Bn, the invariant we called F(Kβ) in(5) is the one-variable Jones polynomial Jq(K).Wandering along a bypath. The reader who is pressed for time may wish toomit this detour.

We interrupt our main argument to discuss how the Jones rep-resentations were discovered, because it is very interesting to see how an almostchance discovery of an unexpected relationship between two widely separated areasof mathematics had ramifications which promise to keep mathematicians busy atwork for years to come! Let M denote a von Neumann algebra.

Thus M is analgebra of bounded operators acting on a Hilbert space H. The algebra M is calleda factor if its center consists only of scalar multiples of the identity. The factor istype II1 if it admits a linear functional tr : M →C, which satisfies:(i) tr(1) = 1,(ii) tr(xy) = tr(yx)∀x, y ∈M,and a positivity condition which shall not concern us here.

It is known that the traceis unique, in the sense that it is the only linear form satisfying (i) and (ii). An olddiscovery of Murray and von Neumann was that factors of type II1 provide a typeof “scale” by which one can measure the dimension dimMH of the Hilbert space H.The notion of dimension which occurs here generalizes the familiar notion of integer-valued dimensions, because for appropriate M and H it can be any nonnegative realnumber or infinity.

The starting point of Jones’s work was the following question:if M1 is a type II1 factor and if M0 ⊂M1 is a subfactor, is there any restriction onthe real numbers λ which occur as the ratio λ = dimM0H/dimM1H?

14JOAN S. BIRMANThe question has the flavor of questions one studies in Galois theory. On theface of it, there was no reason to think that λ could not take on any value in [1,∞],so Jones’s answer came as a complete surprise.

He called λ the index [M1 : M0] ofM0 in M1 and proved a type of rigidity theorem about it:The Jones Index Theorem. If M1 is a II1 factor and M0 a subfactor, then thepossible values of the index λ are restricted to [4, ∞] ∪[4 cos2(π/p)], where p ≥3 isa natural number.

Moreover, each such real number occurs for some pair M0, M1.We now sketch the idea of Jones’s proof, which is to be found in [Jo1]. Jonesbegins with the type II1 factor M1 and a subfactor M0.

There is also a tiny bitof additional structure: In this setting there exists a map e1 : M1 →M0, knownas the conditional expectation of M1 on M0.The map e1 is a projection, i.e.,e21 = e1. His first step is to prove that the ratio λ is independent of the choice ofthe Hilbert space H. This allows him to choose an appropriate H so that the algebraM2 = ⟨M1, e1⟩generated by M1 and e1 makes sense.

He then investigates M2 andproves that it is another type II1 factor, which contains M1 as a subfactor; moreover,|M2 : M1| = |M1 : M0| = λ. Having in hand another type II1 factor, i.e., M2 andits subfactor M1, there is also a trace on M2 which (by the uniqueness of the trace)coincides with the trace on M1 when it is restricted to M1 and another conditionalexpectation e2 : M2 →M1.

This allows Jones to iterate the construction and tobuild algebras M1, M2, . .

. and from them a family of algebras:An = ⟨1, e1, ..., en−1⟩⊂Mn,n = 1, 2, 3, .

. .

.We now replace the ek’s by a new set of generators which are units, defining gk =qek −1 + ek, where (1 −q)(1 −q−1) = 1/λ. The gk’s generate An, because the ek’sdo, and we can solve for the ek’s in terms of the gk’s.

ThusAn = An(q) = ⟨1, g1, . .

. , gn−1⟩,and we have a tower of algebras, ordered by inclusion:A1(q) ⊂A2(q) ⊂A3(q) ⊂· · · .The parameter q, which replaces the index λ, is the quantity now under investiga-tion.

The gi’s turn out to be invertible and to satisfy the braid relations (4a)–(4b),so that there is a homomorphism from Bn to An, defined by mapping the elementarybraid σi to gi. The parameter q is woven into the construction of the tower.

Defin-ing relations in An(q) depend upon q, for example, the relation g2i = (q −1)gi + qholds. Recall that, since Mn is type II1, it supports a unique trace, and, since Anis a subalgebra, it does too, by restriction.

This trace is a Markov trace! Jones’sproof of the Index Theorem is concluded when he shows that the infinite sequenceof algebras An(q), with the given trace, could not exist if q did not satisfy the statedrestrictions.Thus the “independent variable” in the Jones polynomial is essentially the indexof a type II1 subfactor in a type II1 factor!

Its discovery opened a new chapter inknot and link theory.Back to the main road. We now describe a method, discovered by Jones (see thediscussion of vertex models in [Jo5]) but first worked out in full detail by Turaev

NEW POINTS OF VIEW IN KNOT THEORY15in [Tu], which can be applied to give, in a unified setting, every generalized Jonesinvariant via a Markov trace on an appropriate matrix representation of Bn. Asbefore, E is a ring with 1.

Let V be a free E-module of rank m ≥1. For each n ≥1let V ⊗n denote the n-fold tensor product V ⊗E · · · ⊗E V .

Choose a basis v1, . .

. , vmfor V , and choose a corresponding basis {vi1 ⊗· · ·⊗vin; 1 ≤i1, .

. .

, in ≤m} for V ⊗n.An E-linear isomorphism f of V ⊗n may then be represented by an mn-dimensionalmatrix (f j1···jni1···in ) over E, where the ik’s (resp. jk’s) are row (resp.

column) indices.The family of representations of Bn which we now describe have a very specialform. They are completely determined by an E-linear isomorphism R : V ⊗2 →V ⊗2with matrix [Rj1j1i1i2 ] as above.

Let IV denote the identity map on the vector spaceV . The representation ρn,R : Bn →GLmn(E) that we need is defined by(6)ρn,R(σi) = IV ⊗· · · ⊗IV ⊗R ⊗IV ⊗· · · ⊗IVwhere R acts on the ith and (i + 1)st copies of V in V ⊗n.

Thus, if we know how Racts on V ⊗2, we know ρn,R for every natural number n.What properties must R satisfy for ρn,R to be a representation? The first thingto notice is that, if |i −j| ≥2, the nontrivial parts of ρn,R(σi) and ρn,R(σj) willnot interfere with one another, so (4a) is satisfied by construction, independentlyof the choice of R. As for (4b), it is clear that we only need to look at the actionsof R ⊗IV and IV ⊗R on V ⊗3, for if(7)(R ⊗IV )(IV ⊗R)(R ⊗IV ) = (IV ⊗R)(R ⊗IV )(IV ⊗R),then (4b) will be satisfied for every pair σi, σj with |i −j| = 1.

Thus equation (7)is the clue to the construction. It is known as the quantum Yang-Baxter equation(QYBE).

Note that if R satisfies the QYBE equation, then αR will too, for everyinvertible element α ∈E. By our earlier observations (see the discussion of equations(5) and (5′)), if the representation determined by R can be used to define a linkinvariant, then the same will be true if R is replaced by αR.

We will use this factshortly, modifying our choice of R by composing with an invertible scalar, wheneverit simplifies the equations to do so.Caution. One must distinguish between (7), which we call the QYBE, and aclosely related equation (7∗), which is also referred to as the QYBE.

To explain thelatter and to show that both forms have the same geometric meaning, let us returnmomentarily to braids. Our elementary braid generators could have just as wellbeen called σi,i+1 to stress the fact that the nontrivial part of the braid involvesthe adjacent strands i and i + 1.

Relation (4b) in B3 would then beσ12σ23σ12 = σ23σ12σ12.An alternative way to number the braid generators would be to color the strandswith three colors, labeled 1, 2, 3 and to let ˇσi,j denote a positive crossing betweenthe strands of color i and color j, when those two colors are adjacent in the braidprojection. With that convention, we see from the bottom left picture in Figure 14on page 277 (which is the braid version of Reidemeister’s move III) that (4b) wouldbe(4.b∗)ˇσ12ˇσ13ˇσ23 = ˇσ23ˇσ13ˇσ12.

16JOAN S. BIRMANGoing back to operators acting on V ⊗3, let R12 = R ⊗IV and let R23 = IV ⊗R.Then equation (7) may be rewritten asR12R23R12 = R23R12R23.Now let ˇRij be the image of Rij under the automorphism of V ⊗3 induced by thepermutation of {1, 2, 3} which maps 1 to i and 2 to j. The alternative form of theQYBE is(7∗)ˇR12 ˇR13 ˇR23 = ˇR23 ˇR13 ˇR12.In this form, it occurs in nature in many ways, for example, in the theory of ex-actly solvable models in statistical mechanics, where it appears as the star-trianglerelation [Bax].Not long after the discovery that the Jones polynomial generalized to the HOM-FLY and Kauffman polynomials, workers in the area began to discover other, iso-lated cases of generalizations, all relating to existing known solutions to (7*).

Thena coherent theory emerged. It turns out that solutions to (7∗), and hence (7), can beconstructed with the help of known solutions to yet a third equation—the classicalYang-Baxter equation (CYBE).

To explain that equation, let L be a Lie algebraand let r ∈L⊗2. Let r12 be r ⊗1 ∈L⊗3 and let rij be the image of r12 under theautomorphism of L⊗3 induced by the permutation of {1, 2, 3} which sends 1 to iand 2 to j.

Then r is said to be a solution to the CYBE if the following holds:(7∗∗)[r12, r23] + [r13, r23] + [r12, r13] = 0.The theory of the CYBE is well understood, its solutions having been essentiallyclassified. Two good references on the subject, both with extensive bibliographies,are [Sem] and [BD].

The possibility of “quantization”, i.e., passage from solutionsto the CYBE (7∗∗) to those for the QYBE (7∗) was proved by Drinfeld in a seriesof papers, starting with [Dr1]. See [Dr2] for a review of the subject and anothervery extensive bibliography, including the definition of a quantum group and adevelopment of the relevance of the theory of quantum groups to this matter.

Thattheory has much to do with the representations of simple Lie algebras. Explicitsolutions to (7∗) which are associated to the fundamental representations of thenonexceptional classical Lie algebras may be found in the work of Jimbo ([Ji1] andespecially [Ji2]).

See [WX] for a discussion of the problem of finding “classical”solutions to the “quantum” equation (7∗).Now, representations of Bn do not always support a Markov trace. To obtaina Markov trace from the representation defined in (6), where R satisfies (7), oneneeds additional data in the form of an enhancement of R [Re1].

See Theorem 2.3.1of [Tu]. The enhancement is a choice of invertible elements µ1, .

. .

, µm ∈E whichdetermine a matrix µ = diag(µ1, . .

. , µm) such that(8a)µ ⊗µ commutes with R = [Rj1j2i1i2 ],(8b)mXj=1(R±1)k,ji,j · µj = δi,k(Kroneker symbol).

NEW POINTS OF VIEW IN KNOT THEORY17(A remark for the experts. Theorem 2.3.1 of [Tu] contains two factors, α and β.We have chosen them to be 1.

This possibility is discussed in Remark (i), §3.3 of[Tu]. We may do this because, as noted earlier, if R is a solution to the QYBE,then αR is too, for any invertible α ∈E.

)As it turns out, there are invertible elements µ1, . .

. , µm ∈E associated naturallyto every solution to the QYBE which comes from quantum groups and has theproperty that (8a) is true.

This is proved in [Ros, Re 2, RT 1]. It is also knownthat if the R-matrix comes from an irreducible representation of a quantum group,then (8b) is true with δi,k replaced by α−1δi,k for some invertible element α ∈E.Notice that (7) and (8a) still hold if R is replaced by αR.

Thus, possibly replacingR by αR, every solution to the QYBE coming from quantum groups admits anenhancement.Finally, we describe how generalized Jones invariants are constructed out of thedata (R, n, µ1, . .

. , µm).Let ρ = ρn,R.

Let µ⊗n denote the mapping µ ⊗· · · ⊗µ : V ⊗n →V ⊗n. If β ∈Bn, define tr(ρ(β)) to be the ordinary matrix trace ofρ(β) · µ⊗n.

Notice that this implies that, if 1n denotes the identity element in Bn,then tr(1n) = (µ1 + · · · + µm)n is the matrix trace of µ⊗n. With this rescaling ofthe trace the factor z in equation (5) goes away (see the discussion following thestatement of equation (5)).

The link invariant F(Kβ) is given explicitly as(9)F(Kβ) = tr(ρ(β) · µ⊗n) .We leave it to the reader to verify that (8a) and (8b) ensure that F(Kβ) in (9)is invariant under Markov’s two moves. (Alternatively, see Theorem 3.12 of [Tu]for a proof.) The ring E (to the best of our knowledge) will always be a ring ofLaurent polynomials over the integers, the variable being some root of q.

In thespecial case of the HOMFLY polynomial (as defined in (3a) and (3b)), the functionF(Kβ) of (9) turns out to depend only on q±1 when Kβ is a knot.For links,square roots generally occur. The invariant F(Kβ) in (9) is normalized so thatF(O) = µ1 + · · · + µm.The HOMFLY and Kauffman polynomials, which are central to the subject, arerelated to the fundamental representations of the nonexceptional Lie algebras oftype A1n and B1n, C1n, D1n, A2n.

Other knot and link polynomials, for example, theones in [WAD], turn out to be the HOMFLY and Kauffman polynomials of variouscables on simple (noncabled) knots and links. Yet other completely new polynomialsarise out of the representations of the remaining nonexceptional Lie algebras andthe exceptional Lie algebras.

For example, see [Ku] for the case of the exceptionalLie algebra G2 from the point of view of crossing-change formulas and [Kal] for aninterpretation of Kuperberg’s polynomial via traces on representations of Bn. Weunderstand that the representations corresponding to the algebras E6, E7, E8 havealso been constructed, but we do not have a reference.

The important point is thatit is possible to do so.Summarizing our results in this section, we have described a very general con-struction which leads to the construction of vast families of knot and link invari-ants. The proof, via Markov’s theorem, that they do not depend on the choice ofthe representative braid β ∈B∞is straightforward and convincing.

Unfortunately,however, neither the construction (which depends upon the combinatorial prop-erties needed to satisfy (7) or (7*) and (8)) nor the proofs we know of Markov’stheorem ([Be, Bi, Mo1] and most recently [Mak]) give the slightest hint as to the

18JOAN S. BIRMANunderlying topology; moreover, the constructions which actually yield all possiblesolutions to (7) and (8) do not change that situation at all. The crossing-changeformulas in §2 give even less insight.

Thus we can compute the simplest of the in-variants by hand and quickly fill pages with the coefficients and exponents of Jq(K)for not-too-complicated knots without having the slightest idea what they mean.4. A space of all knotsTo proceed further, we need to change our point of view.

The Alexander polyno-mial, as we have seen, contains information about the topology of the complementof a single knot. We now describe yet another method of constructing knot in-variants, discovered in 1989, which yields information about the topology of anappropriately defined space of all knots.Following methods pioneered by Arnold (e.g., see the introduction to [Arn1]) anddeveloped for the case of knots in 3-space by Vassiliev [V], we begin by introducinga change which is very natural in mathematics, shifting our attention from the knotK, which is the image of S1 under an embedding φ : S1 →S3, to the embedding φitself.

A knot type K thus becomes an equivalence class {φ} of embeddings of S1into S3. The space of all such equivalence classes of embeddings is disconnected,with a component for each smooth knot type, and the next step is to enlarge it toa connected space.

With that is mind, we pass from embeddings to smooth maps,thereby admitting maps which have various types of singularities. Let fM be thespace of all smooth maps from S1 to S3.

This space is connected and containsall knot types. Our space will remain connected and will contain all knot typesif we place two mild restrictions on our maps.Let M denote the collection ofall φ ∈fM such that φ(S1) passes through a fixed point ∗and is tangent to afixed direction at ∗.

The space M has some pleasant properties, the main onebeing that it can be approximated by certain affine spaces, and these affine spacescontain representatives of all knot types. The walls between distinct chambers inM constitute the discriminant Σ, i.e.,Σ = {φ ∈M|φ has a multiple point or a place where its derivative vanishes orother singularities} .The space M −Σ is our space of all knots.

(Remark. It is important to include inΣ all maps which have cusp singularities, for if not every knot could be changed tothe unknot by pulling it tight and, at the last moment, popping it!

)The group eH0(M−Σ; Q) contains all rational-valued knot-type invariants. (The“tilde” over H0 indicates that we have normalized by requiring each invariant totake the value 0 on the unknot.) Clearly, eH0(M −Σ; Q) contains enough informa-tion to classify knots.

For example, in principle it is possible to make a list con-taining all knot types, ordered, say, by crossing number, and then to delete repeatsuntil knots which occur in distinct positions on the list have distinct knot types.The position of a knot on the list would then define an element in eH0(M −Σ; Q)which (by definition) classifies knots.Our space M is much too big for direct analysis, but fortunately its very sizemakes it extremely flexible and subject to approximation. Let Γd ⊂M be thesubspace of all maps from R1 to R3 given byt →(φ1(t), φ2(t), φ3(t)),

NEW POINTS OF VIEW IN KNOT THEORY19where each φi is a polynomial of the formtd+1 + ai1td + · · · + aidtwith d even. As t →±∞, the images φ(t) are seen to be asymptotic to the direction±(1, 1, 1), so the images are tangent to a fixed direction at ∗= {∞}.

The space Γdhas three key properties:(i) By the Weierstrass Approximation Theorem, every knot type occurs for someφ ∈Γd for sufficiently large d.(ii) There are ways to embed Γd in Γd′, d < d′. For example [V], map Γd →Γ3d+2by reparametrizing φ(t) as φ(s3 + s).

Thus, one may choose a sequence of positiveintegers d1 < d2 < d3 < · · · so that(10)eH0(M −Σ; Q) = lim←eH0(Γdi −Γdi ∩Σ; Q). (iii) The fact that each φ ∈Γd is determined by 3d real numbers and that each3d-tuple of real numbers determines a map in Γd allows us to identify Γd withEuclidean space R3d.

Alexander duality then applies, and(11)eH0(Γdi −Γdi ∩Σ; Q) ∼= ¯H3d−1(Γdi ∩Σ; Q).where ¯H3d−1 denotes reduced homology. In other words, the object of study isno longer the topology of an individual component K of M −Σ but instead thetopology of the walls Σ which separate the components in our space of all knots.This change has an important consequence: it adds structure in the form of thenatural stratification of Σ which occurs via the hierarchy of its singularities.The discriminant Σ is of course hideously complicated, since maps in Σ may havemultiple points, tangencies, and all sorts of bad singularities; but other studies insingularity theory have shown that certain generic singularities predominate, andthe next goal is to single these out.

Let Mj ⊂Σ be the subspace of maps φ ∈Σwhich have j transverse double points (and possibly other singularities too). Callφ ∈Mj a j-embedding if its only singularities are j transverse double points.

LetΣj = {φ ∈Mj/φ is not a j-embedding}. Then Σ ⊃Σ1 ⊃Σ2 ⊃· · · .

Each mapin Mj −Σj has exactly j generic singularities, where a generic singularity is atransverse double point.The space Mj−Σj is not connected. Its components are singular knot types withexactly j transverse double points.

To illustrate how it might enter our picture,let K and K∗be knot types which are neighbors in M −Σ. Thus a single passageacross Σ suffices to change a representative K of K to a representative K∗ofK∗.At the instant of passage across Σ (assuming that the passage is across a“nice” part of Σ), one will obtain a singular knot K1 in M1.

This singular knotcan be thought of as keeping track of the crossing change.The singular knotK1 will be in M1 −Σ1. The path from K to K∗will in general not be unique,so it may happen that two distinct passages yield singular knots K1, N 1 whichare in distinct components of M1 −Σ1.

To pass from one to another, one mustcross Σ1, obtaining (at the instant of passage) a singular knot K2 ∈M2 −Σ2,and so forth. Vassiliev studied this situation, applying the methods of spectralsequences to obtain combinatorial conditions which determine certain families ofinvariants.

These have become known as Vassiliev invariants.They are stable

20JOAN S. BIRMANvalues of elements in the group ¯H3d−1(Γd ∩Σ) ∼= eH0(Γd −Γd ∩Σ), as d →∞,evaluated on a component K of M−Σ. As such, they are elements of eH0(M−Σ; Q).A Vassiliev invariant vi(K) of order i takes into account information about thesingular knot types near K in M−Σ, M1 −Σ1, M2 −Σ2, .

. .

, Mi −Σi, where i isa positive integer. The sum of two ith order invariants is another invariant of orderi.

In fact, these invariants lie in a finite-dimensional vector space Vi of dimensiondi. Clearly, these invariants have much to do with crossing changes.The manuscript [V] ends with a combinatorial recipe for the calculation of vi(K).5.

Axioms and initial data for Vassiliev invariantsIn 1990, when preprints of [V] were first circulated, topologists were in possessionof an overflowing cornucopia of knot and link invariants. In addition to the Jonespolynomial and its generalizations, we mention the knot group invariants of [Wa],the energy invariants of [FH], and the algebraic geometry invariants of [CCG], allof which seemed to come from mutually unrelated directions!

In addition, therewere generalizations of the Jones invariants to knotted graphs [Y1] and, finally, thenumerical knot invariants of [V].The first hint that the Vassiliev and Jones invariants might be related was thatboth extended to invariants of singular knots.A second hint was a tiny bit ofevidence: the first nontrivial Vassiliev invariant is v2, which was identified in [V]as coinciding with the second coefficient in the Conway-Alexander polynomial. Infact, that coefficient has another interpretation, as the second derivative of the Jonespolynomial evaluated at 1.

Studying [V] and having in mind a possible relationshipwith Jones invariants, the author and Lin were led to reformulate the results in [V]in terms of a set of “axioms and initial data”. We describe them in this section.Let v : M −Σ →Q denote a function from the space M −Σ of all knots to therational numbers.

This function determines a Vassiliev invariant if it satisfies thefollowing conditions:Axioms. To state the first axiom, we note that the singular knots in [V] are subjectto an equivalence relation which is known as rigid vertex isotopy—a neighborhoodof each double point of the singular knot spans an open disc R2 ⊂R3—and this discmust be preserved under admissible isotopies.

For example, if one begins with thestandard diagram of the trefoil and of its mirror image (both shown in Figure 4) andthen replaces any one crossing in each diagram by a double point, one will obtaintwo singular knots which are equivalent under isotopy but not under rigid vertexisotopy. This restriction is quite natural if one thinks of singular knots as objectswhich keep track of crossing changes in a passage from one knot to a neighbor inM −Σ.

Knowing that there is such a disc, or plane, it then makes sense to speakof the two resolutions (see Figure 5) at a singular point p of the singular knot Kj:We denote these two resolutions Kj−1p+and Kj−1p−respectively. The first axiom is atype of crossing-change formula:(10a)v(Kjp) = v(Kj−1p+ ) −v(Kj−1p−).This axiom in itself places no constraints on v as a knot invariant.

Rather, it servesto define v on singular knots, assuming that it is known on knots.The second axiom is what makes v computable:(10b)∃i ∈Z+ such that v(Kj) = 0 if j > i.

NEW POINTS OF VIEW IN KNOT THEORY21Figure 9. Actuality table for i = 2.The smallest such i is the order of v. To stress it, we call our invariant vi from nowon.Initial data.

In addition to (10a) and (10b) we need initial data. The first pieceof initial data relates to the normalization mentioned earlier:(10c)vi(O) = 0 .To describe the second piece of initial data, we need a definition.

A singular pointon a knot diagram will be called nugatory if its positive and negative resolutionsdefine the same knot type, in the obvious manner indicated in Figure 10. It is clearthat if we are to obtain a true knot invariant, its value on vi(Kj−1p+ ) and vi(Kj−1p−)must agree when p is a nugatory crossing, so by (10a) the initial data must satisfy(10d)vi(Kjp) = 0 if p is a nugatory crossing.The final piece of initial data is in the form of a table, but before we can describeit we need to discuss a point that we avoided earlier: The space Mj −Σj has anatural decomposition into components, such that two singular knots cannot definethe same singular knot type if they belong to distinct components.

To make thisassertion precise, let Kj be a singular knot of order j, i.e., the image of S1 under aj-embedding φ ∈Mj −Σj. Then φ−1(Kj) is a circle with 2j distinguished points,arranged in pairs, where two distinguished points are paired if they are mappedto the same double point on Kj.

The [j]-configuration which Kj respects is thecyclically ordered collection of point pairs. We will use a picture to define it, i.e., acircle with arcs joining the paired points, as in the top row of Figure 9, where weshow the two possible [2]-configurations, together with a choice of a singular knotwhich respect each.

The initial data must take account of the following (see [ST 1]for a proof):Lemma 1. Two singular knots Kj1, Kj2 become equivalent after an appropriate se-ries of crossing changes if and only if they respect the same [j]-configuration.

22JOAN S. BIRMANFigure 10. Nugatory crossing.The table which we now construct to complete the initial data is called an ac-tuality table.

Figure 9 is an example, when i = 2. It gives the values of vi(Kj) fora representative collection of singular knots of order j ≤i.

The table contains achoice of a singular knot Kj which respects each [j]-configuration, for j = 1, 2, . .

., i.The choice is arbitrary (however, the work in completing the table will be increasedif poor choices are made). Next to each Kj in the table is the configuration it re-spects, and below it is the value of vi(Kj).

These values are, of course, far fromarbitrary, and the heart of the work in [V] is the discovery of a finite set of ruleswhich suffice to determine them. The rules turn out to be in the form of a systemof linear equations.

The unknowns are the values of the functionals on the finite setof singular knots in the actuality table. The linear equations which hold betweenthese unknowns are consequences of the local equations (which may be thoughtof as crossing-change formulas) illustrated in Figure 11.

These equations are notdifficult to understand: use (10a) to resolve each double point into a sum of twocrossing points. Then each local picture in Figure 11 will be replaced by a linearcombination of four pictures.

The equations in Figure 11 will then be seen to reduceto a sequence of applications of Reidemeister’s third move. See §3 of [BL] for adescription of the method that allows one to write down the full set of equations.See §2.4 of [BN] for a proof that solutions to the equations, in the special case whenj = i, may be constructed out of information about the irreducible representationsof simple Lie algebras.

It is not known whether the methods of [BN] yield all so-lutions in the case j = i. The extension of the solutions for the case j = i to thecases 2 ≤j ≤i −1 must be handled by the less routine methods described in [BL],at this time.An example should suffice to illustrate that (10a)–(10d) and the actuality tableallow one to compute vi(K) on all knots.

For our example we compute v2(K) whenK is the trefoil knot. The first picture in Figure 12 shows our representative of thetrefoil, with a crossing which is marked.

Changing it, we will obtain the unknot O.The crossing we selected is positive and so (10a) yieldsv2(K) = v2(O) + v2(N 1)where N 1 is the indicated singular knot. It does not have the same singular knottype as the singular knot K1 in the table, so (using the lemma) we introduceanother crossing change to modify it to the singular knot in the table which respectsthe unique [1]-configuration.

In so doing, we obtain a singular knot N 2 with twosingular points. It does not have the same singular knot type as the representativein the table which respects the same [2]-configuration, but by (10b) that does notmatter.

Thus, using (10b), the calculation comes to an end in finitely many steps.6. Singular braidsIn §3 we showed that any generalized Jones invariant may be obtained from a

NEW POINTS OF VIEW IN KNOT THEORY23Figure 11. Crossing change formula for Vassiliev invariants of relatedsingular knots.Figure 12.

Computing v2(K) for the trefoil knot.Markov trace on an appropriate family of finite-dimensional matrix representationsof the braid groups. Up to now, braids have not entered the picture as regardsVassiliev invariants, but that is easy to rectify.

To do so, we need to extend theusual notions of braids and closed braids to singular braids and closed singularbraids.A representative Kj of a singular knot or link Kj will be said to be a closedsingular braid if there is an axis A in R3 (think of it as the z-axis) such that ifKj is parametrized by cylindrical coordinates (z, θ) relative to A, the polar anglefunction restricted to Kj is monotonic increasing. This implies that Kj meets each

24JOAN S. BIRMANhalf-plane θ = θ0 in exactly n points, for some n. In [Al1] Alexander proved thewell-known fact that every knot or link K may be so represented, and we begin ourwork by extending his theorem to singular knots and links.Lemma 2. Let Kj be an arbitrary representative of a singular knot or link Kj.Choose any copy A of R1 in R3 −Kj.Then Kj may be deformed to a closedsingular n-braid, for some n, with A as axis.Proof.

Regard A as the z-axis in R3. After an isotopy of Kj in R3 −A we mayassume that Kj is defined by a diagram in the (r, θ)-plane.

By a further isotopy wemay also arrange that each singular point pk has a neighborhood N(pk) ∈Kj suchthat the polar angle function restricted to Sjk=1 N(pk) is monotonic increasing.The proof then proceeds exactly as in [Al1], vis: ModifyKj −Sjk=1 N(pk)toa piecewise linear family of arcs A, subdividing the collection if necessary so thateach α ∈A contains at most one undercrossing or overcrossing of the knot diagram.After a small isotopy we may assume that the polar angle function is nonconstanton each α ∈A. Call an arc α ∈A bad or good accordingly as the polar anglefunction is increasing or decreasing on α.

If there are no bad arcs, we will have aclosed braid, so we may assume there is at least one, say β. Modify Kj by replacingβ by two good edges β1 ∪β2 as in Figure 13.

The only possible obstruction is if theinterior of the triangle which is bounded by β ∪β1 ∪β2 is pierced by the rest of Kj,but that may always be avoided by choosing the new vertex β1 ∩β2 so that it liesvery far above (resp. below) the rest of Kj if the arc β contains an under- (resp.over-) crossing of the diagram.

Induction on the number of bad edges completesthe proof.□In view of Lemma 2, every singular knot in the actuality table of §5 may bechosen to be a closed singular braid. To continue, we split these closed singularbraids open along a plane θ = θ0 to “open” singular braids, which we now define.

In§3 we described a geometric braid as a pattern of n interwoven strings in R2×I ⊂R3which join n points, labeled 1, 2, . .

., n in R2 × {0} to the corresponding points inR2 × {1}, intersecting each intermediate plane R2 × {t} in exactly n points. Toextend to singular braids, it is only necessary to weaken the last condition to allowfinitely many values of t at which the braid meets R2 ×{t} in n−1 points instead ofn points.

Two singular braids are equivalent if they are isotopic through a sequenceof singular braids, the isotopy fixing the initial and end points of each singular braidstrand. Singular braids compose like ordinary braids: concatenate two patterns,erase the middle plane, and rescale.Figure 13.

Replacing a bad arc by two good arcs.Choose any representative of an element of SBn.After an isotopy we mayassume that distinct double points occur at distinct t-levels. From this it follows

NEW POINTS OF VIEW IN KNOT THEORY25that SBn is generated by the elementary braids σ1, . .

. , σn−1 and the elementarysingular braids τ1, .

. .

, τn−1 of Figure 8. We distinguish between the σi’s and theτi’s by calling them crossing points and double points respectively in the singularbraid diagram.

Both determine double points in the projection.The manuscript [Bae] lists defining relations in SBn as:(11a)[σi, σj] = [σi, τj] = [τi, τj] = 0if |i −j| ≥2,(11b)[σi, τi] = 0,(11c)σiσjσi = σjσiσjif |i −j| = 1,(11d)σiσjτi = τjσiσjif |i −j| = 1,where in all cases 1 ≤i, j ≤n −1. The same set of relations also occurs in [KV] asgeneralized Reidemeister moves.The validity of these relations is easily establishedvia pictures; for example, see Figure 14 for special cases of (11a)–(11d).

To thebest of our knowledge, however, there is not even a sketch of a proof that theysuffice in the literature, so we sketch one now, as it will be important for us thatno additional relations are needed.Lemma 3. The monoid SBn is generated by {σi, τi; 1 ≤i ≤n −1}.

Definingrelations are (11a)–(11d).Proof. We have already indicated a proof that the σi’s and the τi’s generate SBn,so the only question is whether every relation is a consequence of (11a)–(11d).We regard our braids as being defined by diagrams.

Let z, z′ be singular braidswhich represent the same element of SBn, and let {zs; s ∈I} denote the family ofsingular braids which join them. The fact that the intermediate braid diagrams zshave no triple points implies that there is a well-defined order of the singularitiesFigure 14.

Relations in SBn.

26JOAN S. BIRMANalong each braid strand which is preserved during the isotopy, and this allows us toset up a 1-1 correspondence between double points in z and z′. We now examinethe other changes which occur during the isotopy.

Divide the s-interval [0,1] intosmall subintervals, during which exactly one of the following changes occurs in thesequence of braid diagrams:(i) Two double points in the braid projection interchange their t-levels.Seerelations (11a) and (11b) and Figure 14. (ii) A triple point in the projection is created momentarily as a “free” strandcrosses a double point or a crossing point in the projection.

See (11d) and Figure14. (iii) New crossing points in the knot diagram are created or destroyed.SeeReidemeister’s second move in Figure 2.All possible cases of (i) are described by relations (11a) and (11b).

Noting thatthe σi’s are invertible and that the mirror image of σi (resp. σ−1i, τi) is σ−1i(resp.σi, τi), it is a simple exercise to see that consequences of (11c) and (11d) cover allpossible cases of (ii).

As for (iii), if we restrict to small s-intervals about the instantof creation or destruction, these will occur in pairs and be described by the trivialrelation σiσ−1i= σ−1iσi = 1. Outside of these special s-intervals the singular braiddiagram will be modified by isotopy, which contributes no new relations.

Thusrelations (11a)–(11d) are defining relations for SBn.□Now something really interesting happens. Let eσi denote the image of the el-ementary braid σi under the natural map from the braid group Bn to its groupalgebra CBn.Theorem 2 (cf.

[Bae, Lemma 1]). The map η : SBn →CBn which is defined byη(σi) = eσi, η(τi) = eσi −eσ−1iis a monoid homomorphism.Proof.

Check to see that relations (11a)–(11d) are consequences of the braid rela-tions in CBn.□Corollary 1. Every finite-dimensional matrix representation ρn : Bn →GLn(E)extends to a representation ˜ρn of SBn, defined by setting ˜ρn(τi) = ρn(σi)−ρn(σ−1i).Proof.

Clear.□Thus, in particular, all R-matrix representations of Bn extend to representationsof the singular braid monoid SBn.Remark 1. Recall that in Figure 11 we gave picture examples of some of the relationswhich need to be satisfied by a functional on the knot space in order for the indicesin an actuality table to determine a knot type invariant.

Unlike the relations whichwe depicted in Figure 14, the ones in Figure 11 are, initially, somewhat mysterious.However, if one passes via Theorem 2 to the group algebra CBn of the braid group,replacing each τi by σi −σ−1i, it will be seen that these relations actually hold inthe algebra, not simply in the space of VBL-functionals. This fact is additionalevidence of the naturality of the map η of Theorem 2 and indeed of the Vassilievconstruction.

We conjecture that the kernel of η is trivial, i.e., that a nontrivialsingular braid in the monoid SBn never maps to zero in the group algebra CBn.Remark 2. Various investigators, for example, Kauffman in [Kau1], have considereda somewhat different monoid which we shall call the tangle monoid, obtained byadding the “elementary tangle” εi of Figure 8 to the braid group.The tangle

NEW POINTS OF VIEW IN KNOT THEORY27monoid, however, does not map homomorphically to CBn, and one must pass toquotients of CBn, for example, the so-called Birman-Wenzl algebra [BW, We1],to give it algebraic meaning. In that sense the tangle monoid appears to be lessfundamental than the singular braid monoid.7.

The proof of Theorem 1We are now ready to prove Theorem 1. We refer the reader to the introductionfor its statement.Theorem 1 was first proved by the author and Lin in [BL] for the special cases ofthe HOMFLY and Kauffman polynomials and then in full generality for all quantumgroup invariants in [L].

The proof we give here is new. It is modeled on the prooffor the special cases in [BL].

We like it because it is simple and because it givesus an opportunity to show that the braid groups, which are central to the studyof the Jones invariants, are equally useful in Vassiliev’s setting. The tools in ourproof are the R-matrix representations of §3, the axioms and initial data of §5, andthe singular braid monoid of §6.

Theorem 1 also is implicit in [Bae], which waswritten simultaneously with and independently of this manuscript. The techniquesused there are very similar to ours, but the goal is different.Proof of Theorem 1.

By hypothesis, we are given a quantum group invariant Jq :M −Σ →E, where E denotes a ring of Laurent polynomials over the integersin powers of q (or in certain cases powers of roots of q).Using Lemma 2, wefind a closed braid representative Kβ of K, β ∈Bn.We then pass to the R-matrix representation ρn,R : Bn →GLmn(E) associated to Jq. By Corollary 1that representation extends to a representation ˜ρn,R of SBn.

By equation (9) theLaurent polynomial Jq(K) is the trace of ρn,R(β) · µ, where µ is the enhancementof ρn,R.As was discussed in §3, our representation ρn,R is determined by the choice of amatrix R which acts on the vector space V ⊗2. Lin notes in Lemma 1.3 of [L] thaton setting q = 1 the matrix R = (Rj1j2i1i2 ) goes over to idV ⊗idV .

From this it followsthat ρn,R(σi) (which acts on V ⊗n) has order 2 at q = 1. Hence, we conclude thatρn,R goes over to a representation of the symmetric group if we set q = 1, with σigoing to the transposition (i, i + 1).

In particular, this means that at q = 1 theimages under ρn,R of σi and σ−1iwill be identical, which, in turn, means that theimages of σi and σ−1iunder ρn,R coincide at q = 1.Armed with this knowledge, we change variables, as in the statement of Theorem1, replacing q by ex. Expanding the powers of ex in its Taylor series, the image ofan arbitrary element β under ρn,R will be a matrix power series(12)ρn,R(β) = M0(β) + M1(β) + M2(β) + · · ·where each Mi(β) ∈GLmn(Q).Lemma 5 (cf.

[Bae, Corollary 1]). In the extended representation, M0(τi) = 0.Proof.

Since M0(σi) = M0(σ−1i), the assertion follows.□Now let us turn our attention to the power series expansion of Jq, i.e., to Jx(K) =P∞i=0 ui(K)xi. The coefficients ui(K) in this series, as K ranges over all knot types,

28JOAN S. BIRMANdetermine a functional ui : M −Σ →Q. We wish to prove that ui is a Vassilievinvariant of order i.By §5 it suffices to prove that, if we use (10a) to extend the definition of ui tosingular knots, then (10b)–(10d) will be satisfied and a consistent actuality tableexists.

The first thing to notice is that, since we began with a knot-type invariantJq, the functional ui is also a knot-type invariant. From this it follows that itsextension to singular knots is also well defined, so using our knowledge of ui onknots we can fill in the actuality table.The second observation is that (10c) is satisfied, because every generalized Jonesinvariant satisfies Jq(O) ≡1, and from this it follows that Px(O) ≡1.

As for (10d),it is also satisfied, for if not ui could not be a knot-type invariant. Thus the onlyproblem which remains is to prove that ui satisfies axiom (10b).

However, noticethat by Lemma 5 we have˜ρn,R(τi) = M1(τi)x + M2(τi)x2 + M3(τi)x3 + · · · ,and from this it follows that if Kj is a singular knot which has j singular points,then any singular closed braid Kjγ, γ ∈SBr, which represents Kj will also havej singularities. The singular braid word γ will then contain j elementary singularbraids.

From this it follows that˜ρn,R(γ) = Mj(γ)xj + Mj+1(γ)xj+1 + · · · .The coefficient of xi in this power series is ui(Kj). But then, ui(Kj) = 0 if i < j,i.e., axiom (10b) is also satisfied and the proof of Theorem 1 is complete.□8.

Open problemsOur story is almost ended, so it is appropriate to recapitulate and ask whatwe have gained by our interpretation of the quantum group invariants as Vassilievinvariants? The goal has been to bring topology into the picture, and indeed wehave done so because we have shown that, when the Jones invariant of a knotK is expanded in a power series in x, the coefficient of xi gives information aboutcertain stable homology groups of the discriminant Σ, in a neighborhood of K. Theinformation concerns its structure at “depth” i; that is, of course, only a beginning.The discriminant Σ is a complicated subset of an infinite-dimensional space, M,and there seems to be no way to begin to visualize Σ.

Indeed, the work has justbegun.The study of Vassiliev invariants is fairly new. We now review some of the thingsthat have been learned about them during the past few years.

A natural place tostart is with the special cases when i is small, and this was already done in [V]. Theinvariant v1(K) is zero for all knots K. The first nontrivial Vassiliev invariant hasorder 2, and there is a one-dimensional vector space of such invariants.

However,v2(K) was well known to topologists in other guises before either Jones or Vassilievcame on the scene, vis:(i) It is the second coefficient in Conway’s version of the Alexander polynomial[C]. (ii) Its mod 2 reduction is the Arf invariant of a knot, which has to do withcobordism.It now has other interpretations too:

NEW POINTS OF VIEW IN KNOT THEORY29(iii) It is the “total twist” of a knot, as defined in [LM]. It may be computedfrom the following recursive formula:v2(Kp+) −v2(Kp−) = Lk(Kp0),where Lk denotes the linking number of the two-component link Kp0.

(iv) It is the second derivative of the Jones polynomial Jq(K), evaluated at q = 1.Unfortunately, all of these facts do not allow us to understand the significanceof v2 for the topology of Σ. There is much work to be done.

As for v3(K), italso belongs to a one-dimensional vector space of invariants, but to the best ofour knowledge no one has yet identified it as a classical invariant—indeed, nothingmuch seems to be known about it.With regard to the higher-order invariants, recall that Vassiliev invariants oforder i belong to a linear vector space Vi. This space is the space of ith orderinvariants modulo those of order i −1.

The first characteristic of this vector spaceto question is its dimension mi, for i > 2. This seems to be a deep and difficultcombinatorial problem for arbitrary i, and at this writing the best we can do is tocompute, the issue being the construction of the actuality table for an invariant oforder i.

This problem divides naturally into two parts: the first is to determine theVassiliev invariants of the singular knots Ki in the top row. By axiom (10b) theyonly depend upon the [i]-configuration which they respect, so their determination iseasier than the corresponding problem for the remaining rows.

The former problemis solved by setting up and solving the system of linear equations (3.11) of [BL].The dimension of the space of solutions is the sought-for integer mi. One year agoBar Natan wrote a computer program which listed the distinct [i]-configurationsand calculated the dimensions (top row only) for i ≤7.

Very recently Stanforddeveloped a different computer program which checked that all of Bar Natan’s“top row” solutions, for i ≤7, extend to the remaining rows of the actuality table,i.e., to solutions to the somewhat more complicated set of equations (3.17) of [BL],so Bar Natan’s numbers are actually the dimensions mi that we seek. The resultsof the two calculations are:i1234567mi01134914The data we just gave leads us to an important question.

We proved, in Theorem1, that{Quantum group invariants} ⊆{Vassiliev invariants}.The question is: Is the inclusion proper?The data is relevant because it wasthought that the question might be answered by showing that for some fixed i thequantum group invariants spanned a vector space of dimension di < mi; however,a dimension count to i = 6 shows that there are enough linearly independentinvariants coming from quantum groups to span the vector spaces Vi. Bar Natan’scalculations showed that d7 is at least 12, whereas m7 = 14; however, the data onthe quantum group invariants is imprecise because the invariants which come fromthe nonexceptional Lie algebras begin to make their presence felt as i increases.The only one of those which has been investigated, to date, is G2.

30JOAN S. BIRMANFor i = 8 the computations themselves create difficulties. Bar Natan’s calcula-tion is close to the edge of what one can do, because to determine m8 he foundhe had to solve a linear system consisting of 334,908 equations in 41,874 unknowns(the number of distinct [8]-configurations with no separating arcs).

His approxi-mate calculation shows that the solution space is of dimension 27. However, even ifhis answer is correct, we would still need to compute the rest of the actuality tablebefore we could be sure that m8 = 27 rather than m8 ≤27.Vassiliev invariants actually form an algebra, not simply a sequence of vectorspaces, because the product of a Vassiliev invariant of order p and one of q is aVassiliev invariant of order p + q.

This was proved by Lin (unpublished), usingstraightforward methods, and more indirectly by Bar Natan [BN]. Thus the dimen-sion ˆmi of new invariants is in general smaller than mi, because the data in our tableincludes invariants which are products of ones of smaller order.

It is a simple matterto correct the given data to find ˆmi. For example, an invariant of order 4 could bethe product of two invariants of order 2; so when we correct for the fact that thereis a one-dimensional space of invariants of order 2, we see that ˆm4 = 3 −1 = 2.For i = 1, 2, 3, 4, 5, 6, 7 we find that ˆmi = 0, 1, 1, 2, 3, 5, 8, i.e., the beginning of theFibbonaci sequence!

This caused some excitement until Bar Natan’s computationof m8 showed that ˆm8 was at most 12, not 13. The asymptotic behavior of mi asi →∞is a very interesting problem indeed.We can approach the question of whether the inclusion is proper from a differentpoint of view.

One of the earliest problems in knot theory concerned the funda-mental symmetries which are always present in the definition of knot type. Wedefined knot type to be the topological equivalence class of the pair (S3, K) underhomeomorphisms which preserve the given orientation on both S3 and K. A knottype is called amphicheiral if it is equivalent to the knot type obtained by reversingthe orientation of S3 (but not K) and invertible if it is equivalent to the knot typeobtained by reversing the orientation on K (but not S3).

As noted earlier, MaxDehn proved in 1913 that nonamphicheiral knots exist [De], but remarkably, it tookover forty years before it became known that noninvertible knots exist [Tr]. Therelevance of this matter to our question is: While the quantum group invariantsdetect nonamphicheirality of knots, they do not detect noninvertibility of knots.So, if we could prove that Vassiliev invariants distinguished a single noninvertibleknot from its inverse, the answer to our question, Is the inclusion proper?

wouldbe yes. We know we cannot answer the question this way for i ≤7, and as notedabove i = 8 presents serious computational difficulties.

On the other hand, thetheoretical problem seems to be unexpectedly subtle. Thus, at this moment, thematter of whether Vassiliev invariants ever detect noninvertibility remains open.Setting aside empirical evidence and unsolved problems, we can ask some easyquestions which will allow us to sharpen the question of whether Jones or Vassilievinvariants determine knot type.

As was noted in §2, there are three very intuitiveinvariants which have, to date, proved to be elusive: the crossing number c(K),the unknotting number u(K), and the braid index s(K). Clearly these determinefunctionals on the space M −Σ, and so they determine elements in the groupeH0(M−Σ).

Vassiliev invariants lie in a sequence of approximations to eH0(M−Σ).So a reasonable question to ask is, Are c(K), u(K), and s(K) Vassiliev invariants?Theorem 5.1 of [BL] shows that u(K) is not, and the proof given there is easilymodified to show that c(K) and s(K) are not either. So, at the very least, we havelearned that there are integer-valued functionals on Vassiliev’s space of all knots

NEW POINTS OF VIEW IN KNOT THEORY31which are not Vassiliev invariants. This leaves open the question of whether thereare sequences of Vassiliev invariants which converge to these invariants.Another question which has been asked is, How powerful are the Vassiliev in-variants, if we restrict our attention to invariants of bounded order?

The answer tothat question is, not very good, based on examples which were discovered, simul-taneously and independently, by Lin [Li2] and by Stanford [St2]. We now describeStanford’s construction, which is particularly interesting from our point of viewbecause it is based on the closed-braid approach to knots and links.

See Remark(ii) for a description of Lin’s construction.To state his theorem, we return to braids. Let Pn be the pure braid group, i.e.,the kernel of the natural homomorphism from Bn to the symmetric group Sn.

Thegroups of the lower central series {P kn; k = 1, 2, . .

. } of Pn are defined inductivelyby P 1n = Pn, P kn = [Pn, P k−1n].

Notice that if β ∈Bn, with the closed braid bβ aknot, then cαβ will also be a knot for every α ∈Pn.Theorem 3 [St2]. Let K be any knot type, and let Kβ, where β ∈Bn, be any closedbraid representative of K. Choose any α ∈P kn.

Then the Vassiliev invariants oforder ≤k of the knots Kβ and Kαβ coincide.Remarks. (i) Using the results in [BM1], Stanford has constructed sequences α1, α2,.

. .

of 3-braids such that the knot types obtained via Theorem 3 are all distinct andprime. Intuition suggests that distinct αj’s will always give distinct knot types, butat this writing that has not been proved.

(ii) One may choose Kβ in Theorem 3 to represent the unknot and obtain in-finitely many distinct knots all of whose Vassiliev invariants of order ≤k, for anyk, are zero. Lin’s construction gives other such examples.

In particular, he provesthat, if K is any knot and if K(m) is its mth iterated (untwisted) Whitehead dou-ble, then all Vassiliev invariants of order ≤(m + 1) of K(m) are zero. This allowshim to construct infinitely many composite knots which have the same propertiesas Stanford’s knots.

It is not clear whether his construction is a special case ofStanford’s construction. (iii) The special case of the unknot and the one-variable Jones polynomial isinteresting.

Theorem 3 says that if we expand the Jones polynomials of the knotsin the collection in power series and truncate those series by cutting offall termsof x-degree bigger than k, the result will be zero. This is far from saying thatthe polynomials themselves are trivial.

Indeed, the question of whether there is anontrivial knot which has the same one-variable Jones polynomial as the unknotremains an open problem and an intriguing one. Note that our lack of knowledgeabout this problem is in striking contrast to the control mathematicians now haveover the Alexander polynomial: understanding its topological meaning, we alsoknow precisely how to construct knots with Alexander polynomial 1.

(iv) For a brief period (before a mistake was discovered in the proof) it seemedas if there might be an affirmative answer to the following question: Given anytwo distinct knots K and K∗, does there exist a sequence of Vassiliev invariants{wi | i = 1, 2, 3, . .

.} and an integer N such that wi(K) ̸= wi(K∗) ∀i ≥N?

Noticethat even if the answer was yes, it would not solve the knot problem, because ifwe had explicit examples K and K∗and if we knew the sequence, we still wouldnot know how large N had to be. So after letting the computer run all weekendwithout a definitive answer, we would not know whether the knots were really the

32JOAN S. BIRMANsame or whether we simply had given up too soon. However, this is probably thevery best that one could hope for from the algebraic invariants.

(v) By contrast to all of this, the work of Haken [Ha] and the work of Hemion [He]show that there is an algorithm which distinguishes knots. In recent years effortshave been directed at making that algorithm workable (for example, see [JT] for adiscussion of recent results), but much work remains to be done before it could beconsidered “practical”, even for the simplest examples.

(vi) The work in [BM2] and related papers referenced therein is aimed at adifferent algorithm which is based upon the theory of braids. At present it hasresulted in a very rapid hand calculation for definitively distinguishing knots andlinks which are defined by closed braids with at most 3 strands, with partial resultsfor the general case.

(vii) Among the special cases for which an effective classification scheme existswe mention (in addition to the work just cited on links defined by closed 3-braids)the cases of links with 2-bridges and of algebraic (in the sense of algebraic geometry)links. For more information on these and other classical topics, see [Ro1] and [BZ].

(viii) An extensive guide to the “pre-Jones” literature may be found in [BZ].AcknowledgmentsMany people helped us to understand this subject. We single out, especially: V.I.

Arnold (who first stimulated our interest in Vassiliev invariants), Xiao-Song Lin,and Dror Bar Natan. Discussions with Vaughan Jones, Vladimir Turaev, MichaelPolyak, Alex Sossinsky, and Ted Stanford have also been very helpful to us.

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