Link Invariants of Finite Type and

arXiv:hep-th/9207041v1 13 Jul 1992Link Invariants of Finite Type andPerturbation TheoryJohn C. BaezDepartment of MathematicsWellesley CollegeWellesley, Massachusetts 02181(on leave from the University of California at Riverside)July 13, 1992AbstractThe Vassiliev-Gusarov link invariants of finite type are known to be closelyrelated to perturbation theory for Chern-Simons theory.In order to clarifythe perturbative nature of such link invariants, we introduce an algebra V∞containing elements gi satisfying the usual braid group relations and elementsai satisfying gi −g−1i= ǫai, where ǫ is a formal variable that may be regardedas measuring the failure of g2i to equal 1. Topologically, the elements ai sig-nify crossings.

We show that a large class of link invariants of finite type arein one-to-one correspondence with homogeneous Markov traces on V∞.Wesketch a possible application of link invariants of finite type to a manifestlydiffeomorphism-invariant perturbation theory for quantum gravity in the looprepresentation.1IntroductionThe manner in which the braid group Bn takes the place of the symmetric group Snin the representation theory of quantum groups is by now well known. Recall thatthe braid group Bn has a presentation with generators si, 1 ≤i < n, and relationssisj=sjsi|i −j| > 1,sisi+1si=si+1sisi+1.The symmetric group Sn is the quotient of Bn by the further relations s2i = 1.

If Gis a semisimple Lie group, then the corresponding quantized enveloping algebra Uqgis a deformation of the universal enveloping algebra Ug as Hopf algebras. Given anyrepresentation of the group G, there is a natural action of the group algebra CSn asintertwining operators on the representation E⊗n.

Similarly, given any representationE of Uqg there is a natural action of CBn as intertwining operators on E⊗n.Naively, one might be led to hope that CBn is a kind of deformation of CSn.While this is not true according to the standard definition - after all, CBn is infinite-dimensional while CSn is finite-dimensional - we show here that some sense can be1

made of this idea. Roughly speaking, one can form an algebra Vn over C[ǫ], where ǫis a formal variable, by adjoining to CBn elements ai, 1 ≤i < n, such thatsi −s−1i= ǫai.The parameter ǫ should be thought of as measuring the failure for s2i to equal 1.Setting ǫ equal to any nonzero constant gives the algebra CBn, while setting ǫ = 0gives an algebra containing CSn.

More generally, for any d ≥0, the quotient algebraVn/⟨ǫd+1⟩should be regarded as a dth-order perturbative approximation to CBn.These quotients are closely related to the theory of link invariants of finite type, asdeveloped by Vassiliev, Gusarov, Birman, Lin, Bar-Natan, and others [3, 5, 10, 13, 19].The basic idea here is that one can canonically extend an invariant L of oriented linksto an invariant of generalized links admitting nice self-intersections (transverse doublepoints) by means of the ruleL(L+) −L(L−) = ǫL(L×),Here L+ is as in Figure 1, L−is as in Figure 2, and L× is as in Figure 3, with thestrands oriented so as to be pointing downwards.❙❙❙❭❭❭✓✓✓✓✓✓Figure 1.✓✓✓✓✓✓❙❙❙❙❙❙Figure 2.✓✓✓✓✓✓❙❙❙❙❙❙Figure 3.Those invariants vanishing on all generalized links with more than d self-intersectionsare said to be of degree d. The space of link invariants of degree d can be regarded asa dth-order approximation to the dual of the space with basis given by isotopy classesof links. If one takes one of the C(q)-valued link invariants derived from quantum2

group representations by the procedure of Turaev [18], sets q = exp(ǫ) to obtain aformal power series in ǫ, and takes the coefficient of ǫd, one obtains a link invariant ofdegree d. More generally, we show that there is a one-to-one correspondence betweena large class of link invariants of degree d and Markov traces τ: V∞→C[ǫ] that are“homogeneous of degree d” in a certain sense. Such Markov traces may be thoughtof as defined on V∞/⟨ǫd+1⟩.The connection between link invariants of finite type and physical perturbationtheory is presently clearest in the context of Chern-Simons theory with semisimplegauge group G. Here the action is given byS = k4πZS3 tr(A ∧dA + 23A ∧A ∧A),where A is a G-connection and the level k ≥0 is an integer.From the work ofWitten [20] and others it is clear that, for example, the Jones polynomial VL(q) maybe obtained from the vacuum expectation values of Wilson loops in Chern-Simonstheory with G = SU(2) and q = exp(2πi/(k + 2)).

The coefficient of the ǫd termof the Jones polynomial, an invariant of finite degree, should thus be calculable bya dth-order perturbation expansion in Chern-Simons theory. This has been pursuedby Cotta-Ramusino et al, Smolin, and others [8, 15].

In particular, the relation toknot invariants of finite type has been studied by Bar-Natan [3], who dealt with anarbitrary classical Lie group, and by abstraction obtained a general combinatorialscheme for constructing knot invariants of finite type.Link invariants of finite type may also be expected to play a role in a novelperturbation theory for 4-dimensional quantum gravity. In the loop representationof quantum gravity developed by Rovelli and Smolin [14], states are described bylinear combinations of isotopy classes of framed unoriented links (or tangles), possiblyadmitting self-intersections.We briefly comment on the relation between Chern-Simons perturbation theory, link invariants of finite type, and perturbative quantumgravity in the final section of this paper.The author thanks Dror Bar-Natan, J´ozef Przytycki, and Stephen Sawin for usefuldiscussions concerning link invariants of finite type, and thanks Micheal Weiss for helpwith drawing the figures.3

2The Vassiliev AlgebraLet the generalized braid monoid, GBn, denote the monoid with generators gi, g−1i , ai,1 ≤i < n, and relations[gi, gj]=[ai, aj] = [ai, gj] = 0|i −j| > 1,gig−1i=g−1i gi = 1,aigi=giai,gigi+1gi=gi+1gigi+1,gi+1aig−1i+1=g−1i ai+1gi,g−1i+1aigi+1=giai+1g−1i .In pictures of generalized braids, gi represents a right-handed crossing and g−1irep-resents a left-handed crossing of the ith and (i + 1)st strands, as in Figures 1 and 2.The element ai represents an intersection of the ith and (i+1) strands, as in Figure 3.The relations above express topological facts about generalized braids, which admitintersections as well as crossings; the reader is strongly encouraged to draw theserelations. The generalized braid monoid appears in the work of Kaufmann [11], aswell as in the work of Br¨ugmann, Gambini and Pullin [7, 9] on quantum gravity.Let CGBn denote the monoid algebra of the generalized braid monoid.

We definethe Vassiliev algebra, Vn, to be the quotient of CGBn ⊗C[ǫ] by the ideal generatedby the elementsgi −g−1i= ǫai.It is clear that there is a homomorphismv: CBn →Vngiven by si 7→gi. The basic properties of this homomorphism are as follows.Lemma 1.

Let C(ǫ) denote the algebra of Laurent polynomials in ǫ. Thenv ⊗1: CBn ⊗C(ǫ) →Vn ⊗C[ǫ] C(ǫ)is an isomorphism.Proof - The inverse is given by gi 7→si, ai 7→ǫ−1(si −s−1i ).⊓⊔Corollary 1. Let j: Vn →Vn/⟨ǫ −x⟩denote the quotient map.

The mapj ◦v: CBn →Vn/⟨ǫ −x⟩is an isomorphism if x ∈C is nonzero, while if x = 0 it factors through CSn.4

Proof - The composite j ◦v is an isomorphism for x ̸= 0 by Lemma 1, while ifx = 0, s2i = 1 in Vn/⟨ǫ⟩, so j ◦v factors through CSn.⊓⊔Corollary 2. The homomorphism v: CBn →Vn is one-to-one.Proof - This is immediate from Lemma 1.⊓⊔We conclude this section with a word on the universal role the Vassiliev algebraplays in the context of braided tensor categories.

It is clear that, given any object E ina C[ǫ]-linear strict braided monoidal category, if R ≡R−1 mod ǫ, where R: E ⊗E →E ⊗E is the braiding, then there is a canonical action of Vn as endomorphisms ofE⊗n. In particular, this applies to the category of quantum group representations,where we write q = exp(ǫ).3Link Invariants of Finite TypeBy a link invariant we will always mean an ambient isotopy invariant of oriented links.It is easy to see that any C-valued link invariant L uniquely extends to a C(ǫ)-valuedinvariant of generalized links admitting transverse double points, which we also callL, by means of the ruleL(L+) −L(L−) = ǫL(L×),where L+, L−, and L× denote link diagrams with a right-handed crossing, a left-handed crossing, and an intersection, respectively, at a given point, the rest of thediagrams being the same.

We define a C-valued link invariant to be of degree d if itvanishes on all generalized links with d+1 or more self-intersections. A link invariantof degree d for some d is said to be of finite type.For all n there are algebra inclusions Vn ֒→Vn+1 and CBn ֒→CBn+1.

Let V∞and CB∞denote the inductive limits of the algebras Vn and CBn, respectively, andletv: CB∞→V∞denote the inductive limit of the maps v: CBn →Vn. We define a Markov trace onV∞to be a C[ǫ]-linear map τ: V∞→E, where E is some C[ǫ]-module, satisfyingτ(xy) = τ(yx)for all x, y ∈V∞, and for some fixed z ∈Cτ(g±1n x) = zτ(x)for all x ∈Vn ⊂V∞.

A similar definition is standard for Markov traces tr: CB∞→C.We say that a Markov trace τ: V∞→C[ǫ] is homogeneous of degree d if for everyx ∈CB∞, τ(v(x)) is homogeneous of degree d as a polynomial in ǫ.5

For any braid x ∈Bn, let ˆx denote its closure. Of course, given x ∈B∞, ˆx dependson a choice of n such that x ∈Bn.

Given a link L, let L ∪◦denote the distant unionof L with the unknot.Theorem 1. There is a one-to-one correspondence between C-valued link invariantsL of degree d such that for some z ̸= 0 and all links L,L(L ∪◦) = z−1L(L),(1)and Markov traces τ: V∞→C[ǫ] that are homogeneous of degree d. The invariant Ldetermines the trace τ, and conversely, by the property thatτ(v(x)) = ǫdzn−1L(ˆx)(2)for x ∈Bn.Proof - Let E be a vector space, and suppose L is a E-valued link invariant satisfy-ing equation (1).

Then by Markov’s theorem [4] there is a Markov trace tr: CB∞→E,given bytr(x) = zn−1L(ˆx)(3)for all x ∈Bn. Note in particular that tr is well-defined because if y ∈CBn+1 is theimage of x ∈CBn under the inclusion CBn ֒→CBn+1, thentr(y) = znL(ˆy) = znL(ˆx ∪◦) = zn−1L(ˆx) = tr(x).Moreover tr is a trace because cxy = cyx for all x, y ∈Bn, and tr has the Markovproperty:tr(s±1n x) = znL((s±1n x)b) = znL(ˆx) = z tr(x).In fact, Markov’s theorem implies that equation (3) gives a one-to-one correspon-dence between E-valued link invariants satisfying equation (1) and Markov tracestr: CB∞→E.Now let L be a C-valued link invariant of degree d satisfying equation (1).

Thenthere is a unique Markov trace tr: CB∞→C given by equation (3). We claim thatthere exists a Markov trace τ0: V∞→C(ǫ) such thatǫdtr = τ0v.To see this, note that by Lemma 1 there is a Markov trace ˜τ0: V∞⊗C[ǫ] C(ǫ) →C(ǫ)such thatǫd(tr ⊗1) = ˜τ0(v ⊗1)as maps from CB∞⊗C(ǫ) to C(ǫ).

Let τ0 denote the restriction of ˜τ0 to V∞, whichmay be regarded as a subalgebra of V∞⊗C[ǫ] C(ǫ). It follows that ǫdtr = τ0v, asdesired.6

Note that τ0(x) = cǫd−ℓfor some c ∈C if x ∈V∞is a product of ℓelements of theform ai and arbitrarily many of the form gi. Moreover, if ℓ> d then τ0(x) vanishes,since L is of degree d. It follows thatτ0(x) =dXi=0ciǫi,with ci ∈C, if x is a product of elements of the form ai and gi.

It follows that τ0factors through a map τ: V∞→C[ǫ]. It is easy to check that τ0, hence τ, is a Markovtrace that is homogeneous of degree k. Moreover, equation (2) holds by construction.Conversely, suppose that τ: V∞→C[ǫ] is a Markov trace homogeneous of degreed.

We may define a Markov trace tr: CB∞→C byτv = ǫdtr.Associated to this trace there is a link invariant L satisfying equation 1, given byequation (3). We claim that L is an invariant of degree d. For this, we need ananalog of Alexander’s theorem for generalized links.

Given an element x ∈GBn, wemay form a generalized link ˆx, the closure of x, in a manner analogous to the usualclosure of a braid.Lemma 2. For every generalized link L, for some n there is an element x ∈GBnsuch that ˆx is ambient isotopic to L.Proof - We omit the proof, as it is similar to the usual proof of Alexander’s theorem[4], but would be quite long with all the details included.⊓⊔Now let L be a generalized link with ℓself-intersections, and let x0 ∈GBn be suchthat ˆx0 is ambient isotopic to L. Let x1 be the image of x0 in V∞.

Thenx1 = y1ai1y2ai2 · · · aiℓyℓ+1where the elements yi are products of elements gj, g−1j . Define x ∈CB∞byx = y1(gi1 −g−1i1 )y2 · · ·(giℓ−g−1iℓ)yℓ+1.Note thatǫdtr(x) = τ(v(x)) = ǫℓτ(x1).Since tr(x) ∈C and τ(x1) ∈C[ǫ], so if ℓ> d we must have tr(x) = 0.

By construction,L(L) = z1−ntr(x),so it follows that L(L) = 0.Now let us show that the invariant L uniquely determines the trace τ, and viceversa, by equation (2). Since every link is the closure of some braid, L is determined7

by τ. Conversely, L determines τ on the image of v, since v is one-to-one by Corollary2. It then follows by the C[ǫ]-linearity of τ that τ is determined on all of V∞, sinceǫai = si −s−1i .⊓⊔The reader may be puzzled by the fact that every link invariant of degree d isobviously of degree d + 1, while a normalized Markov trace τ: V∞→C[ǫ] that ishomogeneous of degree d is definitely not homogeneous of degree d + 1.

The point isthat ǫτ will be a normalized Markov trace homogeneous of degree d + 1.Note that to reconstruct the link invariant coming from a normalized Markovtrace τ: V∞→C[ǫ] that is homogeneous of degree d, it suffices to know the compositeof τ with the quotient map C[ǫ] 7→C[ǫ]/⟨ǫd+1⟩, which may be regarded as a Markovtraceτ: V∞/⟨ǫd+1⟩→C[ǫ]/⟨ǫd+1⟩.As an illustration of the theorem, let tr0: CB∞→C(q) be one of the tracesobtained from quantum group representations by Turaev’s procedure. Writing q =exp(ǫ), we regard tr0 as having values in C[[ǫ]].

For some invertible z ∈C[[ǫ]],tr0(s±1n x) = z tr0(x)for all x ∈CBn. Since z /∈C, the theorem above is not directly applicable.

However,we may define a new Markov trace tr: CB∞→[[ǫ]] withtr(s±1n x) = tr(x)for all x ∈CBn by settingtr(x) = zm−ntr0(x),for x ∈Bn, where m denotes the number of components of ˆx. There is a uniqueMarkov trace τ: V∞→C[[ǫ]] such thatτv = tr.Moreover, we may writeτ =∞Xd=0τdwhere τd: V∞→C[ǫ] is a Markov trace homogeneous of degree d, with τd(g±1n x) =τd(x) for all x ∈Vn.

By the theorem, each such trace gives a link invariant Ld ofdegree d, such thatLd(ˆx) = ǫ−dτd(v(x))for x ∈Bn.It is also important that the space of Markov traces τ: V∞→C[ǫ] that are homoge-neous of degree d is finite-dimensional. This may be seen by graph-theoretic reasoningalong the lines of Birman and Lin [5], Bar-Natan [3], and Stanford [17].

Suppose, forexample, that we fix the value of z. Then if d = 0, τ is determined by its value on 1.If d = 1, τ is determined by its value on 1, a1, and a1s1.

If d = 2, τ is determined byits value on 1, a1, a1s1, a21, a21s1, a1a2, a1a2s1s2, a1a3, a1a3s1, and a1a3s1s3.8

4Connections to PhysicsThe Vassiliev algebra formalism makes clear that link invariants of finite type arisefrom a sort of “topological perturbation theory.” We have pointedly used the symbolǫ in our paper, instead of the suggestive letter h (common in the quantum groupliterature), because the physical interpretation of this sort of perturbation theory isan interesting issue.In SU(n) Chern-Simons theory one makes the identificationǫ =2πik + n.The limit ǫ →0 thus corresponds to k →∞, which may regarded either as a weakcoupling limit or as a classical limit. In the weak coupling approach [20] we may writeA = A0 + k1/2B, where A0 is a flat connection, and obtainS = kI + 18πZǫabctr(BaDbBc + 23k−12Ba[Bb, Bc]),where I is the Chern-Simons invariant of A0 and D denotes the covariant derivativewith respect to A0.

Here we see that the k →∞limit is closely related to a de-formation of the Lie algebra su(n) to an abelian Lie algebra by scaling the bracket.Alternatively, since k appears where one would expect a factor of ¯h−1 in the partitionfunctionZ =ZDA exp ik4πZtr(A ∧dA + 23A ∧A ∧A!,one may also regard k →∞as a classical limit. This is consistent with the weakcoupling interpretation, of course, since the classical solutions of Chern-Simons theoryare flat connections.In quantum gravity the two obvious limits to consider are the classical limit ¯h →0and the G →0 limit, where G is Newton’s gravitational constant.

The latter has beenconsidered in the loop representation of Euclidean quantum gravity by Smolin [16],with the aim of developing a new perturbation theory for quantum gravity whichis manifestly diffeomorphism-invariant at every order, as opposed to perturbationabout a flat background spacetime. In this approach an SU(2) connection is a keydynamical variable [1], and Smolin shows that the role of G is to scale the Lie bracketin su(2).

The analogy with the k →∞limit of Chern-Simons theory is no accident,since states of quantum gravity may be obtained from Wilson loops in SU(2) Chern-Simons theory [12].In this connection, the extension of the Jones polynomial togeneralized links by Br¨ugmann, Gambini and Pullin [7] is quite intriguing.It is thus natural to hope that link invariants of finite type will play an importantrole in the G →0 limit of quantum gravity, or similar perturbation expansions inmore general tangle field theories [2]. Roughly, we may expect that the true physicalHilbert space H is an inverse limit of spaces Hd, where two states in H are identifiedin Hd if they cannot be distinguished by link (or tangle) invariants of degree d.9

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