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Three Dimensional Chern-Simons Theory as a Theory of Knots

arXiv:hep-th/9111063v1 28 Nov 1991Madras IMSc-91/33IISc-CTS 7/91November 26, 2024Three Dimensional Chern-Simons Theory as a Theory of Knotsand LinksR.K. Kaul∗Centre for Theoretical Studies, Indian Institute of ScienceBangalore 560 012, IndiaandT.R.Govindarajan†The Institute of Mathematical SciencesC.I.T.

Campus, Taramani, Madras 600 113, IndiaAbstractThree dimensional SU(2) Chern-Simons theory has been studied as a topological fieldtheory to provide a field theoretic description of knots and links in three dimensions.A systematic method has been developed to obtain the link-invariants within this fieldtheoretic framework.The monodromy properties of the correlators of the associatedWess-Zumino SU(2)k conformal field theory on a two-dimensional sphere prove to beuseful tools. The method is simple enough to yield a whole variety of new knot invariantsof which the Jones polynomials are the simplest example.e-mail∗: kaul@vigyan.ernet.in†: trg@imsc.ernet.in

1. IntroductionTopological quantum field theories have attracted a good deal of interest in recentyears.

This started particularly with the field theoretic interpretations of two importantdevelopements in mathematics: Donaldson’s theory for the integer invariants of smooth4-manifolds in terms of the moduli spaces of SU(2)-instantons1 and Jones work on knotsin three dimensions2. These were developed by Witten some years ago3,4.

Cohomologicalfield theories involving monopoles of the three dimensions have also been developed.5These applications of topological quantum field theories reflect the enormous interestat present in building a link between quantum physics on one hand and geometry andtopology of low dimensional manifolds on the other. It appears that the properties oflow dimensional manifolds can be very sucessfully unravelled by relating them to infinitedimensional manifolds of the fields.

This is done through the functional integral formu-lation of the such quantum field theories. In fact an axiomatic formulation of topologicalquantum field theories has already been developed by Atiyah6.

This relates the functionalintegrals of quantum field theory with of the notion of modular functors.Witten in his pioneering work4 has demonstrated that Jones polynomials of the knottheory are the expectation values of the Wilson loop-operators in a three dimensionalSU(2) Chern-Simons theory where the fundamental representation of the gauge groupSU(2) lives on the knots. The two variable generalization7 of the Jones knot invariantsare obtained as the expectation values of the Wilson loop operators with N-dimensionalrepresentation living on the knots in an SU(N) Chern-Simons theory.

The most importantimport of the Witten’s formulation of the knot theory is that it provides an intrinsicallythree dimensional description of knots and links. This field theoretic formulation is alsopowerful enough to study knots and links in any arbitrary three-manifold.1

The knot invariants have also been studied from the point of view of exactly solv-able models in statistical mechanics8,9,10. The intimate connection that knot invariantshave with the exactly solvable lattice models has been exploited by Akutsu, Deguchi andWadati to obtain a general method for constructing invariant polynomials for knots andlinks.

The Yang-Baxter relation is an important tool in this development. These au-thors in particular have derived explicitly new knot invariants from a three-state exactlysolvable model.

These new invariants (both one variable and their two variable gener-alizations) are indeed more powerful than the Jones polynomials (and their two-variablegeneralizations).The knot invariants have also been studied from the quantum group point of view11,12.All these have an intimate connection with two dimensional conformal field theory13.In this paper, following Witten, we shall study the knot theory in terms of a topologicalquantum field theory. The link invariants are given by the expectation value of Wilsonlink-operators in a Chern-Simons theory4.

For definiteness we shall restrict ourselves tothe SU(2) gauge group. Generalizations to other groups are straight forward and will bediscussed elsewhere.

By placing other than the doublet representation of SU(2) on allthe component knots of a link we obtain link invariants other than those of Jones. Inparticular, a triplet representation on all the component knots leads to the one-variableinvariants of Akutsu, Deguchi and Wadati obtained from the three state exactly solvablemodel10.

These invariants obey generalized Alexander-Conway skein relations containingn + 2 elements where n is the number of the boxes in the Young tableaux correspondingto the representation that lives on the knots. For n = 1, these relations are the standardAlexander-Conway skein relations that relate Jones one-variable polynomials.

These wereobtained within the field theoretic framework by Witten in ref.[4]. This construction of2

Witten can be extended to exhibit the generalized Alexander-Conway skein relations too.After presenting a brief discussion of the non-Abelian Chern-Simons theory in Section 2,we shall present a short derivation of the generalised skein relations in Section 3.These havealso been obtained in ref.[14]. While these relations for n = 1 case (Jones polynomials) canbe recursively solved to obtain Jones polynomials for any knot or link, those for n ̸= 1 donot contain enough information to allow us to obtain the new invariants for any arbitraryknot or link.

Therefore, there is need to develop methods to obtain these invariants. Thisis what we attempt to do in this paper.

In Section 4, we shall present two new types ofrecursion relations for links for arbitrary n. In Section 5, invariants for links obtained asthe closures of braids with two strands, both parallely oriented and anti-parallely orientedshall be obtained. In Section 6, we shall present some useful theorems for the functionalintegrals over three-balls containing Wilson lines which meet the boundary at four points,two incoming and two outgoing.

Next in Section 7, we shall develop the method furtherand obtain building blocks for the calculation of the link invariants. These are used tocalculate the invariants for the knots upto seven crossing points as illustrations of themethod in Section 8.

Then we conclude with some remarks about generalizations of ourresults in Section 9. Appendix A contains some useful formulae for the SU(2) quantumRacah coefficients and the duality matrix which relates the four point correllators ofthe SU(2)k Wess-Zumino conformal field theory.

The field theoretic method developedhere can be used to rederive many nice properties of the Jones polynomials (n = 1).As an illustration of these, we shall present a proof of the generalization of numerator-denominator theorem of Conway for Jones polynomials in Appendix B. In Appendix C welist some functional integrals over three manifolds containing Wilson lines ending on one ortwo boundaries, each an S2 with four punctures, obtained by the method developed earlier.These functional integrals are used to obtain the knot invariants presented explicitly inSection 8.3

2. Wilson Link-Operators in SU(2) Chern-Simons Theory in 3DThe metric independent action of Chern-Simons theory in a three-manifold M3 isgiven bykS = k4πZM3tr(AdA + 23A3)(2.1)where A is a matrix valued connection one-form of the gauge group G. In most of whatfollows we shall take the three manifold to be a three-sphere, S3.

The topological operatorsof this topological field theory are given in terms of the Wilson loop (knot) operators:WR[C] = trRPexpICA(2.2)for an oriented knot C. These operators depend both on the isotopy type of the knotas well as the representation R living on the knot through the matrix valued one-formAµ = AaµT aR where T aR are the representation matrices corresponding to the representationR of the gauge group.For a link L = (C1, C2, ..., Cs) made up of component knots C1, C2, ...Cs carryingrepresentations R1, R2, ..., Rs respectively, the Wilson link operator is defined asWR1R2...Rs [L] =sYi=1WRi [Ci](2.3)Unless indicated explicitly otherwise, in the following we shall place the same representa-tion R on all the component knots of a link and hence write the link operator WRR....R[L]simply as WR[L]. The functional average of this link operator are the topological invari-ants of this theory.These we define as follows :VR[L] = ⟨WR[L]⟩≡RM3[dA]sQi=1 WR[Ci]eikSRS3[dA]eikS.

(2.4)Clearly these invariants depend only on the isotopy type of the oriented link L and therepresentation R placed on the component knots of the link. This is so because both the4

integrand in the functional integral as well as the measure15 are independent of the metricof three-manifold on which the theory is defined.These functional integrals can be evaluated by exploiting the connection of Chern-Simons theory on a three dimensional manifold with boundary to corresponding Wess-Zumino conformal field theory on the boundary4. Consider a three manifold M3 witha set of two-dimensional boundaries Σ(1), Σ(2)....Σ(r).

Each of these boundaries Σ(i) mayhave a certain number of Wilson lines carrying representations R(i)j (j = 1, 2...) ending orbeginning (at the punctures P (i)j ) with representation R(i)j (j = 1, 2..) on them. With sucha manifold we associate a state in the tensor product of Hilbert spaces ⊗r H(i) associatedwith these boundaries, each with a certain number of punctures.

This state then willrepresent the functional integral of the Chern-Simons theory over such a manifold. Thedimensionality of each of these Hilbert spaces H(i) is given by the number of conformalblocks of the corresponding Wess-Zumino conformal field theory on the respective twodimensional boundaries with punctures P (i)j , j = 1,2 ..., carrying the primary fields inrepresentations R(i)j , j = 1,2 ..., at these punctures.Exploiting this connection with the Wess-Zumino conformal field theories, Witten hasproved the following two very simple theorems for the link invariants (2.4).Theorem 1: For the union of two distant (disjoint) oriented links (i.e., with no mutualentanglements)L1, L2 carrying representations R1 and R2VR1R2[L] = VR1[L1]VR2[L2](2.5)where L = L1F L2.Theorem 2: Given two oriented links L1 and L2 (Fig.1a) their connected sum L1#L2,is obtained as shown in Fig.lb.The strands that are joined have to have the same5

representation living on them and also the orientations on them should match. Thenthe invariant for the connected sum is related to the invariant for the individual links asVR[L1#L2] = VR[L1]VR[L2]VR[∪](2.6)where VR[∪] is the invariant of an unknot ∪(i.e., ⃝).Besides these two theorems, within this field theoretic framework Witten has alsoproved the Alexander-Conway skein relation for the case where the fundamental repre-sentation lives on all the components of the link.

We shall now present a derivation ofthe generalized Alexander-Conway skein relations for arbitrary representations living onthe links.3. Recursion Relations Among the Link InvariantsBefore deriving recursion relations among the link invariants, let us introduce a fewuseful definitions.Following, Lickorish and Millet16, we shall call a compact 3-dimensional submanifoldin S3 with the boundary carrying a finite set of points marked by arrows as “in” or “out”,a room.

An inhabitant of the room is a properly embedded smooth, compact oriented1-manifold which meets the boundary of the room at the given set of points with its ori-entation matching with the “in” and “out” designations. Examples of rooms are shownin Fig.2.

Here 2(a) is a three-ball with no markings on its boundary; 2(b) is a three-ballwith two marked points on its boundary, one “in” and one “out”; 2(c) is a three-ball withfour marked points on the boundary, two “in” and two “out”. Fig.2(d) shows a room withtwo inlets and two outlets with an example of an inhabitant of the room drawn explicitly.6

Now let us consider a link Lm(A) as shown in Fig.3(a) made up of a room✒✑✓✏A✻✻✻✻withits parallely oriented lower two strands containing a certain number of half-twists m andthen joined to the upper two strands of the room as shown. The half-twists are taken tobe positive or negative if these are right-handed or left-handed respectivily as shown inFig.4.

Thus m can be 0, ±1, ±2, ±3... in the link Lm(A) of Fig.3(a). On each componentof this link, we place the spin n/2 representation Rn of SU(2) given by Young tableauxwith n horizontal boxes.Following Witten4, we cut out a ball B1 containing the room✒✑✓✏A✻✻✻✻as well as allthe twists in the lower two strands of this room as shown in Fig.5(b).

The boundary S2of this ball is punctured at two “in” and two “out” points. The normalized functionalintegral over this ball will be represented by ψm(A).

The rest of S3 containing the re-maining part of the link Lm(A) is also a three-ball (B2) with an oppositely oriented S2with four punctures (two “in” and two “out”) as its boundary as shown in Fig.5(b). Thenormalized functional integral over this ball is represented by ψ0.

These two normalizedfunctional integrals, ψm(A) and ψ0 are vectors in two mutually dual Hilbert spaces, Hand H, associated with the two oppositely oriented S2’s forming the boundaries of thetwo balls B1 and B2 respectively. Each of this S2 has four punctures with representationsRn attached to them.

The dimensionality of each of these two vector spaces is given bythe number of conformal blocks of the correlator for four primary fields, each in represen-tation Rn, of the corresponding Wess-Zumino SU(2)k conformal field theory on S2. Thefusion rules of this conformal field theory are given by: Rn ⊗Rn = ⊕nj=0R2j, for k ≥2n.

(For k < 2n the representations with 2j > k on the right hand side are not integrableand hence are not to be included in the fusion rules). We shall restrict our discussion tothe case k ≥2n.

However, the method developed here can also be used for k < 2n.Thenumber of conformal blocks for the above four-point correlator is the number of singletsin the decomposition of Rn ⊗Rn⊗Rn ⊗Rn, which by the fusion rules is simply n + 1.7

Hence the dimensionality of each of the vector spaces H and H is (n + 1) and normalizedfunctional integrals | ψm(A)⟩and | ψ0⟩are (n + 1) dimensional vectors in these spaces,respectively.Now glueing back the balls B1 and B2 gives us the original link Lm(A).Thus theinvariant Vn[Lm(A)] for this link can be represented through the natural contraction ofthe vectors | ψm(A)⟩and | ψ0⟩in the mutually dual Hilbert spaces,H and H, respectively:Vn[Lm(A)] = ⟨ψ0 | ψm(A)⟩(3.1)Here the subscript n indicates that the representation Rn with its Young tableaux con-taining n boxes is living on all the component knots.Now the vector | ψm(A)⟩represents normalized functional integral over the ball B1with m half-twists in the inner two strands as shown in the Fig.3(b).The operation ofintroducing a half-twist in these two strands can be represented by an (n + 1) × (n + 1)matrix, ˜B operating on these vectors | ψm(A)⟩in the (n + 1) dimensional Hilbert spaceH. Thus we may write( ˜B)j | ψl(A)⟩=| ψl+j(A)⟩,j = ±1, ±2....The characteristic equation of the (n + 1) × (n + 1) half-twist monodromy matrix ˜B canbe written asn+ℓ+1Xj=ℓ˜αj−ℓ( ˜B)j = 0,ℓ= 0, ±1, ±2, ...(3.2)where the coefficients ˜α0, ˜α1, .....˜αn+1 in terms of the eigen-values ˜λi, i = 0, 1, ....n, of thematrix ˜B are˜α0=(−1)n+1nYi=0˜λi,˜α1=(−1)nnX0i1̸=i2̸=....̸=in˜λi1˜λi2....˜λin,8

...˜αn−1=(−1)2nX0i1̸=i2˜λi1˜λi2,˜αn=nX0˜λi,˜αn+1=1(3.3)The eigen-values of the matrix ˜B are given by the monodromy properties of the four-point correlator for the primary fields, all in representation Rn, of SU(2)k Wess-Zuminoconformal field theory on S2 (ref. 17):˜λj = (−)n−jexp[iπ(2hn −h2j)],j = 0, 1, · · ·n(3.4)where hj = j2( j2 + 1)/(k + 2) is the conformal weight of the primary field of spin j/2 rep-resentation Rj of the SU(2)k Wess-Zumino conformal field theory.Applying Eqn.

(3.2) on the vector | ψo(A)⟩with no twists, yields an equation amongstthese vectors with successively increasing number of half-twists as :n+ℓ+1Xj=ℓ˜αj−ℓ| ψj(A)⟩= 0,ℓ= 0, ±1, ±2, · · ·(3.5)which in turn, using (3.1), yields a relation amongst link invariants Lm(A) asℓ+n+1Xj=ℓ˜αj−ℓVn[Lj(A)] = 0,ℓ= 0, ±1, ±2, · · ·(3.6)The link invariants may be given with respect to some reference framing. Generallythese are given in standard framing wherein each component knot of the link has its self-linking number as zero.

Notice that the half-twist matrix ˜B above does not preserve theframing. Thus the various link invariants in Eqn.

(3.6) are not in the same framing. Inorder to have all the links in standard framing, the coefficients, ˜αm, m = 0, 1 · · ·n + 1 in(3.6) need to be multiplied by exp(−2iπmhn), respectively to cancel the change in framing9

due to twisting.We implement this change in Eqns. (3.2)-(3.6) and for convenience divideall of them by an over all factor exp(−2iπ(n + 1)hn).This makes the effective half-twistmonodromy matrix for parallely oriented strands to be B = exp(2iπhn) ˜B.

Its eigenvaluesare given by :λj = (−)n−jq(n(n+2)−j(j+1))2j = 0, 1, · · ·n(3.7)with q = exp[2iπ/(k + 2)], instead of those in Eqn.(3.4). Thus we rewrite the final gener-alized Alexander-Conway skein recursion relation in the form of the following theorem:Theorem 3: The link invariants for links Lm(A) of Fig.3 are related as:ℓ+n+1Xj=ℓαj−ℓVn[Lj(A)] = 0,ℓ= 0, ±1, ±2 · · ·(3.8)where the coefficients αi are given in terms of the eigenvalues of the effective half-twistmatrix B above (Eqn.3.7) as :α0=(−1)n+1nYi=0λi,α1=(−1)nnX0i1̸=i2̸=....̸=inλi1λi2....λin,...αn−1=(−1)2nX0i1̸=i2λi1λi2,αn=nX0λi,αn+1=1(3.9)This recursion relation has also been obtained in ref.14 within the field theoretic formu-lation.

An earliar derivation of this recursion relation within the frame-work of exactlysolvable models is presented in ref.10.10

Now let us list some special cases of this relation : (i)n=1: Here the two eigenvalues of the effective monodromy matrix B are λo = −q3/2 and λ1 = q1/2, so thatαo = −q2, α1 = q3/2 −q1/2 andα2 = 1. These lead to the three-element recursionrelation for the Jones polynomials as−qV1[Lℓ(A)]+(q1/2 −q−1/2)V1[Lℓ+1(A)]+q−1V1[Lℓ+2(A)] = 0,ℓ= 0, ±1, ±2 · · · (3.10)(ii)n=2: Here λo = q4, λ1 = −q3,λ2 = q, and therefore, αo = q8, α1 = −q7 + q5 −q4, α2 = −q4 + q3 −q and α3 = 1.

This leads to the four-element recursion relation:q4V2[Lℓ(A)] −(q3 −q + 1)V2[Lℓ+1(A)]−(q−3 −q−1 + 1)V2[Lℓ+2(A)]+q−4V2[Lℓ+3(A)] = 0,ℓ= 0, ±1, ±2 · · ·(3.11)(iii)n=3:Here λo = −q15/2,λ1 = q13/2, λ2 = −q9/2, λ3 = q3/2, and hence, αo =q20, α1 = −q37/2 −q27/2 + q31/2 + q25/2, α2 = −q14 + q12 −q11 −q9 + q8 −q6, α3 =q15/2 −q13/2 + q9/2 −q3/2, α4 = 1. The recursion relation here has five elements:q10V3[Lℓ] −(q17/2 −q11/2 + q7/2 −q5/2)V3[Lℓ+1] −(q4 −q2 + q + q−1 −q−2 + q−4)V3[Lℓ+2]−(q−17/2 −q−11/2 + q−7/2 −q−5/2)V3[Lℓ+3] + q−10V3[Lℓ+4] = 0,ℓ= 0, ±1, ±2 · · ·(3.12)Now we shall present two more recursion relations which can be obtained on the samelines as above.4.

New Recursion RelationsLet us consider a link ˆL2m( ˆA) as shown in Fig.5a, which has even number of half-twists2m, in the oppositely oriented inner two strands.This link can be obtained by glueingtwo balls B1 and B2 as shown in Fig.5b, along their oppositely oriented boundaries.The normalized functional integrals over these two balls are represented by two (n+1)11

dimensional vectors | χ2m( ˆA)⟩and | ¯χo⟩in the two (n + 1) dimensional mutually dualHilbert spaces associated with the two four-punctured S2’s forming the boundaries ofthese two balls respectively. The invariant for this link is given by the natural contractionof these two vectors :Vn[ˆL2m( ˆA)] = ⟨χo | χ2m( ˆA)⟩(4.1)Let ˆB be the (n + 1) × (n + 1) matrix introducing half-twists in oppositely oriented innertwo strands in the ball B1.

Then( ˆB2)j | χ2m( ˆA)⟩=| χ2m+2j( ˆA)⟩,j = ±1, ±2, · · ·(4.2)The eigen-values of this matrix ˆB introducing half-twists in oppositely oriented strandson a four-punctured S2, are given byˆλℓ= (−)ℓqℓ(ℓ+1)/2ℓ= 0, 1, · · ·n(4.3)The characteristic equation for ˆB2 yields:ℓ+n+1Xj=ℓˆαj−ℓˆB2j = 0(4.4)and thereforen+ℓ+1Xj=ℓˆαj−ℓ| χ2j( ˆA)⟩= 0(4.5)where ˆαi, i = 0, 1, · · ·n+1 are related to the eigen values of the ˆB2 matrix in the usual waygiven below in Eqn. (4.7).Thus this yields us a recursion relation for the link invariants:Theorem 4: The invariants for the links in Fig.5a are related asn+ℓ+1Xj=ℓˆαj−ℓVn[ˆL2j( ˆA)] = 0,ℓ= 0, ±1, ±2 · · ·(4.6)where the coefficients ˆαi are given byˆα0=(−1)n+1nYi=0ˆλ2i ,12

ˆα1=(−1)nnX0i1̸=i2̸=....̸=inˆλ2i1ˆλ2i2....ˆλ2in,...ˆαn−1=(−1)2nX0i1̸=i2ˆλ2i1ˆλ2i2,ˆαn=nX0ˆλ2i ,ˆαn+1=1(4.7)Here ˆλi are the eigenvalues of ˆB matrix given in Eq. (4.3).Now let us present a few special cases of this theorem: (i)n=1: Here the two eigen-values of the monodromy matrix ˆB are ˆλ0 = 1,ˆλ1 = −q.

Thus ˆα0 = q2, ˆα1 = −(1 +q2), ˆα2 = 1. The recursion relations for ℓ= 0, ±1, ±2 · · ·, then read :qV1[ˆ L2ℓ( ˆA)] −(q + q−1)V1[ˆL2ℓ+2( ˆA)] + q−1V1[ˆL2ℓ+4( ˆA)] = 0(4.8)(ii)n=2: Here the eigenvalues of ˆB are ˆλ0 = 1, ˆλ1 = −q,ˆλ2 = q3 so that ˆα0 =−q8,ˆα1 = q2 + q6 + q8, ˆα2 = −(1 + q2 + q6) and ˆα3 = 1.

The recursion relations forℓ= 0, ±1 ± 2....,then read :−q4V2[ˆL2ℓ( ˆA)] + (q−2 + q2 + q4)V2[ˆL2ℓ+L( ˆA)]−(q−4 + q−2 + q2)V2[ˆL2ℓ+4( ˆA)]+q−4V2[ˆL2ℓ+6( ˆA)] = 0(4.9)Similar recursion relations can be obtained for links shown in Fig.6 which have oddnumber of half-twists in the oppositely oriented middle two strands with the room✒✑✓✏ˆA′✻❄✻❄as indicated in the Fig.6. This room is to be contrasted with the room✒✑✓✏ˆA✻❄✻❄in Fig.5a.An analysis as above then leads to the theorem:13

Theorem 5: For links ˆL2m+1( ˆA′) as shown in Fig.6, the link invariants are related asn+ℓ+1Xj=ℓˆαj−ℓVn[ˆL2j+1( ˆA′)] = 0(4.10)where ˆαi,i = 0, 1, · · ·n + 1 are given by equations (4.7) above.The recursion relation given in Theorems 3 - 5 do relate various link invariants. How-ever, it is only for n = 1 (Jones Polynomials) that the Alexander-Conway skein relation(3.10) along with the factorization property of disjoint links given by Theorem 1 is com-plete.

That is, this can recursively be solved for the link invariant of an arbitrary link.For n = 2 and higher, this is not so. Additional information is required to obtain thelink invariants.

Alternatively, methods need to be developed to obtain the link invariantsdirectly. In the following sections , we shall present one such method within the fieldtheoretic frame work.5.

Invariants for Links Obtained as Closures of Two Strand BraidsConsider the link Lm(A1, A2) obtained by glueing two balls along their oppositelyoriented boundaries S2’s as shown in Fig.7. The ball B1 contains the room✒✑✓✏A1✻✻✻✻withthe lower two strands oriented in the same direction and containing m half-twists.

Theball B2 contains the room✒✑✓✏A2✻✻✻✻. The two boundaries are two S2 ’s with four punctureseach, two “in′′ and two “out′′.

The normalized functional integrals over these two balls arerepresented by vectors | ψm(A1)⟩and | ψ0(A2)⟩in two mutually dual (n+1) dimensionalvector spaces H and H respectively. Let | φℓ⟩, ℓ= 0, 1, · · ·, n be a complete orthonormalset of eigen-vectors in H of the braid matrix B which introduces half-twists in the innertwo parallel strands in B1 in Fig.

7a :B | φℓ⟩= λℓ| φℓ⟩,14

λℓ= (−)n−ℓqn(n+2)−ℓ(ℓ+1)2,ℓ= 0, 1, 2 · · ·, n(5.1)The corresponding vectors in dual Hilbert space are denoted by | φℓ⟩and the pairingof these is given by ⟨φℓ| φj⟩= δℓj . The vectors | ψ0(A1)⟩and | ψ0(A2)⟩can then beexpanded in terms of these bases :| ψ0(A1)⟩=nXℓ=0µℓ(A1) | φℓ⟩⟨ψ0(A2) |=nXℓ=0µℓ(A2)⟨φℓ|(5.2)The vector with m half-twists (m = 0, ±1, ±2, · · ·) is| ψm(A1)⟩=nXℓ=0µℓ(A1)(λℓ)m | φℓ⟩(5.3)The invariant for the link Lm(A1, A2) of Fig.7, then can be written as the contraction ofthe vector | ψm(A1)⟩and | ψ0(A2)⟩as :Vn[Lm(A1, A2)] =nXℓ=0µℓ(A1)µℓ(A2)(λℓ)m(5.4)Now if we can obtain a systematic method of finding the coefficientsµℓ(A1) and µℓ(A2), we can write down these link invariants.

To do so let us take thetwo rooms to be✒✑✓✏A1✻✻✻✻=✒✑✓✏A2✻✻✻✻=✒✑✓✏✻✻.Then the link Lm(A1, A2) is simply the linkLm obtained as the closure of m times twisted braid of two parallely oriented strands asshown in Fig.8.Writing simply µℓfor µℓ(✒✑✓✏✻✻) , the invariant for this link isVn[Lm] =nXℓ=0µℓµℓ(λℓ)m(5.5)Now we need to determine µℓ. Notice L0 is simply two unknots unlinked ∪F ∪; L±1 areboth one unknot ∪; L±2 are right/left handed Hopf links (H, H∗) and L±3 are right/lefthanded trefoil knots (T, T ∗).

Thus we may writeVn[L0] =(Vn[∪])2=nXℓ=0µℓµℓVn[L±1] =Vn[∪]=nXℓ=0µℓµℓ(−)n−ℓq± n(n+2)2∓ℓ(ℓ+1)2(5.6)15

where we have used Theorem 1 to write Vn[∪⊔∪] = (Vn[∪])2. From the second relationabove, it is clear that Vn[∪] is invariant under q →q−1 .This is so because unknot has nochirality.These two equations can be solved recursively for various values of n. For n =0, we have µ0 = 1.

Then using this, for n = 1, we obtain from these relations µ1 =q[3].For n = 2, next we obtain µ2 =q[5]. Thus in general µℓ=q[2ℓ+ 1], where [m] is theq-number defined as[m] = qm/2 −q−m/2q1/2 −q−1/2(5.7)Using these in the first equation in (5.6) and the identity Pnℓ=0[2ℓ+ 1] = [n + 1]2 we havethe knot invariant for the unknot asVn[∪] = [n + 1](5.8)This is not surprising, because the link invariant for two cabled knots such as the twounlinked unknots obey the fusion rules.

That is, as can be checked readily, the expression(5.8) for unknot obey the fusion rule Vn1[∪]Vn2[∪] = Pmin(n1,n2)j=0V|n1−n2|+2j[∪]. We cannow put all this together in the form of a theorem :Theorem 6: For links Lm obtained as the closure of a braid of two parallely orientedstrands with m half-twists (Fig.8) the link invariant is given byVn[Lm] =nXℓ=0(−)m(n−ℓ)qm2 (n(n+2)−ℓ(ℓ+1))[2ℓ+ 1],m = 0, ±1, ±2 · · ·(5.9)A similar discussion can be carried through for the links of the type shown in Fig.9and Fig.10.

The link ˆL2m( ˆA1, ˆA2) is obtained by glueing the two connected balls B1 andB2 as shown in Fig.9. Here we have two rooms✒✑✓✏ˆA1✻❄✻❄and✒✑✓✏ˆA2✻❄✻❄with 2m half-twists inthe oppositely oriented lower two strands of the first room.

The functional integrals overthese two balls are again given by vectors | χ2m( ˆA1)⟩and | χ0(A2)⟩in mutually dual n+116

dimensional Hilbert spaces associated with oppositely oriented four-punctured S2. Theinvariant of this link is given by natural contraction of these two vectors.

Here the braidmatrix ˆB introduces half-twists now in oppositely oriented middle two strands of ball B1, | χ2m( ˆA1)⟩= ˆB2m | χ0( ˆA1)⟩. Let | ˆφℓ⟩be a complete set of eigenfunctions, in vectorspace H, of this braid matrix ˆB,ˆB | ˆφℓ⟩= ˆλℓ| ˆφℓ⟩,ˆλℓ= (−)ℓqℓ(ℓ+1)2,ℓ= 0, 1, · · ·n(5.10)The corresponding basis in the dual Hilbert space are denoted by | ˆφℓ⟩, with their naturalcontraction as ⟨ˆφℓ| ˆφj⟩= δℓj.

Expand | χ0( ˆA1)⟩and ⟨χ0( ˆA2) | in these bases :| χ0( ˆA1)⟩=nXℓ=0ˆµℓ( ˆA1) | ˆφℓ⟩⟨χ0( ˆA2) |=nXℓ=0ˆµℓ( ˆA2) < ˆφℓ|(5.11)Then the vector with 2m half-twists is| χ2m( ˆA1)⟩=Xˆµℓ( ˆA1)(ˆλℓ)2m | ˆφℓ⟩(5.12)so that the link invariant for the links of Fig.9 are given byVn[ˆL2m( ˆA1, ˆA2)] =nXℓ=0ˆµℓ( ˆA1)ˆµℓ( ˆA2)(ˆλℓ)2m(5.13)In particular, if we take the two rooms to be✒✑✓✏ˆA1✻❄✻❄=✒✑✓✏ˆA2✻❄✻❄=✒✑✓✏❄✻, then the linkˆL2m( ˆA1, ˆA2) simply becomes the link ˆL2m obtained as the closure of two oppositely ori-ented strands with 2m half-twists as shown in Fig.10. The invariants for these links canbe written asVn[ ˆL2m] =nXℓ=0ˆµℓˆµℓ( ˆλℓ)2m(5.14)where ˆµℓhere refer to the room✒✑✓✏❄✻.

Now, since ˆL0 = ∪F ∪and ˆL±2 are right/lefthanded Hopf links (H, H∗), using Vn[∪] = [n + 1] and Vn[H] = 1 + q + q2 + · · · qn(n+2) and17

Vn[H∗] = 1 + q−1 + q−2 + · · · + q−n(n+2) as obtained using Theorem 6, we can solve forthese ˆµℓ’s successively for n = 0, 1, 2, · · ·.This yieldsˆµℓ= µℓ=q[2ℓ+ 1].Thus we collect these results into a theorem.Theorem 7: For links ˆL2m obtained as the closure of a braid made up of two oppositelyoriented strands containing 2m half- twists (Fig.10) the invariants are :Vn[ ˆL2m] =nXℓ=0[2ℓ+ 1]qmℓ(ℓ+1),m = 0, ±1, ±2, · · ·(5.15)After presenting these two simple theorems, we wish to develope this method furtherto obtain link invariants for more complicated links. This we do in the next section.6.

Some Useful Theorems for Link InvariantsConsider the two rooms QVm and QH2p+1 with four markings as indicated in Fig 11a.The first one has m half-twists vertically in the parallely oriented strands and the latterhas 2p + 1 half-twists horizontally in the oppositely oriented strands.The functionalintegral over the ball containing these rooms (Fig 11b) may be represented by vectors| ψ(QVm)⟩and | ψ(QH2p+1)⟩respectively. The vector | ψ(QVm)⟩is obtained by applying thebraid matrix B (with eigenvalues λℓ= (−)n−ℓqn(n+2)2−ℓ(ℓ+1)2and eigenfunctions denotedby | φℓ⟩, ℓ= 0, 1, ..n) of the parallely oriented central two strands m times on the vector| ψ(QVo )⟩.

This vector from the discussion of the previous section can be represented interms of the normalised eigenfunctions of B as| ψ(QV0 )⟩=nXℓ=0q[2ℓ+ 1] | φℓ⟩(6.1)18

On the other hand, the vector | ψ(QH2p+1)⟩can be thought of as obtained by applying thebraiding matrix ˆB (with eigenvalues ˆλℓ= (−)ℓqℓ(ℓ+1)2and eigenfunctions denoted by | ˆφℓ⟩,ℓ= 0, 1, 2, ...n) of anti-parallely oriented side two strands in Fig11b on the vector| ψ(QH0 )⟩=nXℓ=0q[2ℓ+ 1] | ˆφℓ⟩(6.2)which has been expanded in terms of the eigenfunctions | ˆφℓ⟩following the discussion ofthe previous section.The basis referring to the first two strands on the left (or equivalentlythe last two on the right) in Fig.11b, | ˆφℓ⟩and those refering to the inner two strands | φℓ⟩are connected by the matrix ajℓ= ⟨φℓ| ˆφj⟩where ⟨φℓ| refers to the basis with respectto the central two strands in the dual Hilbert space obtained by changing the orientationof the boundary S2 of the ball. This discussion immediately allows us to write down thefollowing theorem.Theorem 8: The functional integrals | ψ(QVm)⟩and | ψ(QH2p+1)⟩for the balls as shown inFig 11b can be written in terms of the basis | φℓ⟩refering to the parallely oriented middletwo strands as :| ψ(QVm)⟩=nXℓ=0µℓ(QVm) | φℓ⟩| ψ(QH2p+1)⟩=Xµℓ(QH2p+1) | φℓ⟩withµℓ(QVm) = (−1)m(n−ℓ)qm2 (n(n+2)−ℓ(ℓ+1))q[2ℓ+ 1]µℓ(QH2p+1) =nXj=0(−1)jq(2p+1) j(j+1)2q[2j + 1] ajℓ(6.3)Here µℓ(QH2p+1) is obtained by first expanding in terms of the basis |ˆφℓ⟩referring to the firsttwo strands and then changing the basis to |φℓ⟩which refers to the middle two strands,|ˆφj⟩= Pnℓ=0 ajℓ|φℓ⟩.19

Similar discussion can be gone through with regard to the rooms ˆQVm, ˆQH2p and ˆQH′pas shown in Fig 12a and the corresponding functional integrals for these balls (redrawnin Fig 12b),| χ( ˆQVm)⟩, | χ( ˆQH2p⟩and | χ( ˆQH′p )⟩respectively. The middle two strands here(Fig 12b) in all cases are oppositely oriented.

The basis | ˆφℓ⟩referring to these are eigenfunctions of braid matrix ˆB with eigenvalues (−)ℓqℓ(ℓ+1)/2, ℓ= 0, 1, ...n. For the vector| χ( ˆQH2p)⟩, the braid matrix with respect to the first two strands which are also oppositelyoriented has the same eigenvalues.On the other hand the braid matrix with respectto the first two strands in | χ( ˆQH′p )⟩which are parallely oriented, has the eigenvalues(−)n−ℓqn(n+2)2−ℓ(ℓ+1)2, ℓ= 0, 1, ...n.This discussion, therefore, immediately leads to thetheorem:Theorem.9:The functional integrals | χ( ˆQVm)⟩,| χ( ˆQH2p)⟩and | χ( ˆQH′p )⟩for the balls asshown in Fig 12b can be written in terms of the basis | ˆφℓ⟩referring to the anti-parallelyoriented middle two strands as| χ( ˆQVm)⟩=nXℓ=0ˆµℓ( ˆQVm) | ˆφℓ⟩| χ( ˆQH2p)⟩=Xˆµℓ( ˆQH2p) | ˆφℓ⟩| χ( ˆQH′p )⟩=Xµℓ( ˆQH′p ) | ˆφℓ⟩whereˆµℓ( ˆQVm) = (−)mℓqmℓ(ℓ+1)2q[2ℓ+ 1]ˆµℓ( ˆQV2p) =nXj=0qpj(j+1)q[2j + 1] ajℓˆµℓ( ˆQH′p ) =nXj=0(−)p(n−j)qp(n(n+2)2−j(j+1)2)q[2j + 1] ajℓ(6.4)Here again the matrix ajℓrelates basis with respect to the first two strands (or equiva-lently last two strands) with the basis with respect to the middle two strands in Fig 12b.20

The matrix ajℓhas very interesting properties.To see this, let us glue two ballscontaining the rooms ˆQH2m and ˆQH2p respectively to obtain the link ˆL0( ˆQH2m, ˆQH2p) as shownin Fig 13a. This link is the same as link ˆL2m+2p obtained by the closure of the braid oftwo anti-parallel strands with 2m+ 2p half-twists discussed in the previous section.

Thenthe invariant for this link Vn[ ˆLo( ˆQH2m, ˆQH2p)] can be written asnXℓ=0ˆµℓ( ˆQH2m) ˆµℓ( ˆQH2p) =nXℓ=0q(m+p)ℓ(ℓ+1)q[2ℓ+ 1]In this we substitute ˆµℓ( ˆQH2m) and ˆµℓ( ˆQH2p) from Eqn 6.4. Since this equation is validfor arbitrary m and p, we can equate the coefficients of various powers of qm, qp.

Thisimmediately yields :nXℓ=0aiℓajℓ= δij(6.5)On the other hand, if we compose two balls containing the rooms ˆQH2m and ˆQV2p asshown in Fig 13b, we have the links ˆLo( ˆQH2m, ˆQV2p).This link is the same as the linkˆLo( ˆQH2p, ˆQV2m) where m and p have been interchanged. For the invariant for this link, wecan writenXℓ=0ˆµℓ( ˆQH2m)q[2ℓ+ 1] qpℓ(ℓ+1) =nXℓ=0ˆµℓ( ˆQH2p)q[2ℓ+ 1] qmℓ(ℓ+1)where we have substituted for ˆµℓ( ˆQV2p) and ˆµℓ( ˆQV2m) on the two sides of this equation from(6.4).

Now if we substitute for ˆµℓ( ˆQH2m) and ˆµℓ( ˆQH2p) also and since this relation is validfor arbitrary m and p, we haveaiℓ= aℓi(6.6)Thus this matrix is symmetric and orthogonal. Further in the link in Fig.13b if we takep = 0, ˆLo( ˆQHm, ˆQVo ) is simply an unknot for any value of m. This impliesnXℓ=0ˆµℓ( ˆQH2m)q[2ℓ+ 1] = [n + 1]Substitute for ˆµℓ( ˆQH2m) and equate coefficients of various powers of qm.

This leads tonXℓ=0q[2ℓ+ 1] aℓj = [n + 1]δjo(6.7)21

Next we consider the link Lo(QH2p+1, QH2m+1) obtained by composing two balls contain-ing the rooms QH2p+1 and QH2m+1 respectively as shown in Fig 14a. This link is the sameas the link ˆL2m+2p+2 obtained as the closure of the braid with two oppositely orientedstrands with 2m + 2p + 2 half-twists.

Hence its link invaraint can be written asnXℓ=0µℓ(QH2p+1)µℓ(QH2m+1) =nXℓ=0q(p+m+1)ℓ(ℓ+1)[2ℓ+ 1]Here we substitute from Eqn(6.3) for µ(QH2p+1) and µ(QH2m+1) and then equate coefficientsof qpi(i+1), qmj(j+1) and q(p+m)ℓ(ℓ+1) for various values of i, j, ℓ. Thus we obtain again theorthogonality condition (6.5) for the matrix ajℓ.

Similarly, the links Lo(QH±1, QVm) obtainedby glueing the two balls containing the rooms QH±1 and QVm respectively as shown in Fig14b, are the same as the links Lm±1 obtained by the closure of braids with two parallelyoriented strands with m ± 1 half-twists for which link invariants are given by Theorem 6.ThusnXℓ=0µℓ(QH±1)(−)m(n−ℓ)qm2 (n(n+2)−ℓ(ℓ+1))[2ℓ+ 1] =nXℓ=0[2ℓ+ 1](−)(m±1)(n−ℓ)q(m±1)2(n(n+2)−ℓ(ℓ+1))Here substitute from Eqn(6.3) for µℓ(QH±1). This yieldsnXj=0q[2j + 1](−)jq± j(j+1)2ajℓ=q[2ℓ+ 1](−)n−ℓq± 12 (n(n+2)−ℓ(ℓ+1))(6.8)Same equation would emerge if we had considered instead the link Lo(QV2p+1, QV±1) ≡ˆL2p+2, ˆL2p obtained by composing two balls containing the rooms QH2p+1 and QV±1 respec-tively.The matrix ajℓsatisfying conditions (6.5) - (6.8) is given in terms of quantum Racahcoefficients for SU(2)11−13ajℓ= (−)ℓ+j−nq[2j + 1][2ℓ+ 1]n/2n/2jn/2n/2ℓ(6.9)where the quantum Racah coefficients are as given in Appendix A.

We have also presentedsome useful formulae for the Racah coefficients and the matrix ajℓin this Appendix.22

The fact that the matrix ajℓrelating the two bases is the quantum Racah coefficient isnot surprising. The two bases referring to the side two strands and the middle two strandsattached to the four-punctured S2’s forming the boundaries of various balls consideredabove are related by duality of the conformal blocks for four-point correlations of thecorresponding SU(2)k Wess-Zumino conformal field theory on S2.

The duality matrix forthese is indeed given by the quantum Racah coefficients13.It is clear that the Theorems 8 and 9 with the duality matrix ajℓgiven by Eqn. (6.9) canbe used to construct a variety of link invariants.

We can also use these theorems for n = 1to rederive in the present framework many of the results obtained for Jones polynomial.As an example we shall present a new proof of the generalization of the numerator-denominator theorem of Conway for the Jones polynomials derived by Lickorish andMillet in ref.16 within the present framework in Appendix B.7. Building Blocks for Link InvariantsAlthough the results in Theorems 8 and 9 do allow us to write down the link invari-ants for a class of links, these are not enough to study the invariants for arbitrary links.Now we shall attempt to develope several other building blocks which will be useful inobtaining link invariants for an arbitrary link.

For this, consider the three - manifold S3from which several three-balls (say their number is r) have been removed. This yields athree-manifold with r boundaries, each an S2.

Wilson lines are placed inside this mani-fold such that each boundary S2 is punctured in four places, two ingoing and outgoing.The normalized functional integral over such a manifold defines an operator in the tensorproduct of Hilbert spaces, H(1) ⊗H(2) ⊗· · · ⊗H(r) where Hi is the (n + 1) dimensionalHilbert space associated with the i th boundary. This operator can be expanded in termsof a convenient set of basis vectors corresponding to each of these Hilbert spaces.23

For example, we have already argued that for a ball with boundary as a four-puncturedS2, and Wilson lines as shown in Fig.15a, the normalised functional integral is(Fig.15a)=nXℓ=0q[2ℓ+ 1] | ˆφ(1)ℓ⟩(7.1)Here the functional integral is multiplied by normalization N−1/2 with N as the func-tional integral over boundaryless empty S3. The basis | ˆφ(1)ℓ⟩of the (n+1) dimensionalHilbert space H(1) associated with the boundary in terms of which above expansion hasbeen made refers to the middle two (oppositely oriented) strands.

The half-twist matrixoperating on these strands is diagonal in this basis with eigenvalues given by (5.10). Thesignature ǫ = + refers to the orientation of the boundary.

For opposite orientation, ǫ = −,we expand in terms of the corresponding basis ⟨ˆφ(1)ℓ| of the associated dual vector spaceH(1). Thus the normalised functional integral over the manifold in Fig.15b is(Fig.15b)=nXℓ=0q[2ℓ+ 1]⟨ˆφ(1)ℓ|(7.2)In contrast, the normalized functional integral over the ball with the structure shownin Fig.16a is:ˆν1 =nXj,ℓ=0q[2ℓ+ 1][2j + 1] aℓj | ˆφ(1)j ⟩= [n + 1] | ˆφ(1)0 ⟩(7.3)Next, let us consider a three-manifold with two boundaries, each a four-punctured S2,with four Wilson lines connecting them as indicated in Fig.16b.

The normalized functionalintegral over this manifold can be expanded in terms of the complete set of vectors | ˆφ(1)ℓ⟩and | ˆφ(2)ℓ⟩(referring to the middle two strands in both cases) in the Hilbert spaces H(1)24

and H(2) associated with the two boundaries :ˆν2 =nXi,j=0ˆAij | ˆφ(1)i ⟩| ˆφ(2)j ⟩(7.4)Similarly for the manifold with three boundaries as shown in Fig.16c, each a four-punctured S2, we expand in terms of the bases | ˆφ(1)ℓ⟩, | ˆφ(2)ℓ⟩, | ˆφ(3)ℓ⟩of the three Hilbertspaces H(1),H(2),H(3) associated with these boundaries and referring to the inner twostrands in each case :ˆν3 =nXijℓˆAijℓ| ˆφ(1)i ⟩| ˆφ(2)j ⟩| ˆφ(3)ℓ⟩(7.5)Here ˆν3 represents the functional integral normalized by multiplying with N1/2 where Nis the functional integral over the empty S3.Now in Eqn. (7.4), the matrix-element ˆAij has to be δij.This is obvious, becauseglueing the manifold shown in Fig.16b onto the manifolds in Figs.16a-c along an S2 doesnot change any of these manifolds.

That is, this functional integral (7.4) is an identityoperator. On the other hand comparing the manifold in (7.5) with those in (7.2) - (7.4)allows us to conclude that ˆAijℓ= Pm aimajmaℓm/q[2m + 1] where aim is the dualitymatrix.

Now glueing two manifolds of the type in Fig.16c(Eqn.7.5) with three boundariesalong one each boundary with opposite orientations, would yield a manifold with fourboundaries. Repeating this composition several times then yields the following theorem :Theorem 10: The normalized functional integral for a manifold with r boundaries, eachan S2, with Wilson lines as indicated in Fig.17 expanded in terms of the bases | ˆφ(j)ℓ⟩of the Hilbert spaces H(j), j = 1, 2 · · ·r associated with the boundaries referring to the25

middle two strands in each case,isˆνr =nXt=0Qrj=1 aℓjtq[2t + 1]r−2 | ˆφ(j)ℓj ⟩(7.6)Here the functional integral is normalised by multiplying it by a factor (N1/2)r−2, whereN is the functional integral over boundaryless empty S3.This functional integral may be acted upon by matrix ˆB(j), j = 1, 2 · · ·r which intro-duces half-twists in the central two strands of jth boundary :ˆB(j)| ˆφℓ(j)⟩= (−)ℓqℓ(ℓ+1)2|ˆφ(j)ℓ⟩(7.7)This matrix when operated mj times then will introduce mj half-twists in the central twoanti-parallel strands of the jth boundary leading to factor (−)mjℓjqmjℓj(ℓj+1)/2 inside thesummation on the right hand side of Eqn.7.6. On the other hand if we operate by thebraiding matrix ˆB′(j) which introduce half-twists in the first two (or similarly in the lasttwo) strands (again oppositely oriented) of the jth boundary, then( ˆB′(j))mj|ˆφ(j)ℓ⟩=nXs=0(−)smjqmjs(s+1)/2aℓsasr|ˆφ(j)r ⟩(7.8)This way a whole variety of half-twists can be introduced in the diagram in Fig.17 cor-responding to Eqn.

(7.6).And then composing the resultant manifold with functionalintegrals of the type in Eqns. (7.1-3) would lead to various link invaraints.In Eqn.

(7.6), we have expanded the functional integral with respect to the basis re-ferring to the middle two strands on each of the boundary. The expansion with respectto the basis referring to the first two strands (or equivalently the last two strands) isobtained by recognising thatPnℓ=0 aℓt | ˆφ(j)ℓ⟩,t = 0, 1, · · ·n form such a basis for eachof the boundary.

Furthermore, the normalized (by factor (N1/2)r−2) functional integralwith orientation on the Wilson lines different from what are given in Fig.17 can also be26

obtained in the same manner. For example, the normalized functional intergal for themanifold with r-boundaries, each an S2, with Wilson lines as indicated in Fig.18 is givenby:νr =nXℓ=0| φ(1)ℓ⟩| φ(2)ℓ⟩· · · | φ(r)ℓ⟩q[2ℓ+ 1]r−2(7.9)where we have now expanded this functional integral with respect to the basis referringto the first two parallely oriented strands (or equivalently the last two strands) on eachof the boundaries.Again, here this functional integral is multiplied by normalization(N1/2)r−2.

Now the half-twists in the first two strands of the jth boundary which havethe same orientation are introduced through the matrix B(j), j = 1, 2, · · ·r withB(j) | φ(j)ℓ⟩= (−)n−ℓq12(n(n+2)−ℓ(ℓ+1)) | φ(j)ℓ⟩(7.10)so that mj half-twists would mean a factor (−)mj(n−ℓ)qmj2 (n(n+2)−ℓ(ℓ+1)) inside the summa-tion in the right-hand side of Eqn.(7.9). The mj half-twists in the last two strands (againparallely oriented) also introduce the same factor because | φ(j)ℓ⟩are also eigen-vectorswith the same eigenvalues of the braid matrix for these two strands.

On the other handfor mj half-twists in the middle two strands (which are oppositely oriented) of the jthboundary are introduced through the braid matrix B′(j) :(B′(j))mj | φ(j)ℓ⟩=nXs=0(−)mjsqmjs(s+1)2aℓsasr | φ(j)r ⟩(7.11)Using the method described here, the functional integral over various three-manifoldscontaining Wilson lines can be constructed. As another example, the normalized func-tional integral over the manifold as indicated in Fig.19 containing 2r + 2 boundaries,r = 0, 1, 2, · · · can be expanded as :(Fig.19) = 1[n + 1]!rnXiℓ=0| φ(1)i1 ⟩| φ(2)i1 ⟩• | φ(3)i2 ⟩| φ(4)i2 ⟩•· · · | φ(2r+1)ir+1⟩| φ(2r+2)ir+1⟩(7.12)27

Here | φ(j)ℓ⟩are the basis of the Hilbert space H(j) associated with the jth boundary andreferring to the first two strands or equivalently the last two strands (oriented in the samedirection) attached to that boundary.It should be noted that in construction of all these functional integral that half-twistsare to be introduced in the strands attached to a four-punctured S2 though braid matriceswhich have eigenvalues (−)n−ℓq12(n(n+2)−ℓ(ℓ+1)), ℓ= 0, 1, · · ·n if the strands are oriented inthe same direction and (−)ℓq12ℓ(ℓ+1), ℓ= 0, 1, · · ·n if the strands are oriented in oppositedirections. A number of useful building blocks can be constructed in this manner whichthen readily yield the link invariants by appropriate composing of the three-manifolds.

InAppendix C, we have listed some of these building blocks for reference. They have beenused to calculate the knot invariants discussed in the next section.The method described above can further be generalized to obtain building blocks rep-resenting normalized functional integrals over three manifolds with S2 boundaries whichare punctured by Wilson lines at 6,8,10 ... points.

The dimensionality of the Hilbertspaces associated with such boundaries is again given by the number of conformal blocksof 6-point, 8-point, 10-point, ....correlators of the associated SU(2)k Wess - Zuminomodel on these boundaries. The duality matrix relating different bases of each of theseHilbert spaces associated with S2 boundaries punctured at 2m+2 points with m = 0, 1, · · ·are given in terms of quantum 6mj symbols.8.

Explicit Calculation of Knot InvariantsHere we shall present the knot invariants for some knots calculated from the buildingblocks listed in Appendix C as illustrartions. It is possible to obtain the invariants forall the knots and links listed in, for example, Rolfsen’s book or the book by Burde and28

Zieschang18. However below we shall list the answers only for knots upto seven crossingnumber.

For a knot these invariants are unchanged if its orientation is reversed.For themirror reflected knot the invariants are given by the conjugate expressions obtained byreplacing q by q−1.The invariants Vn[L] for knots with seven crossing number listed in Fig.20 are givenby01 : Vn=[n + 1]31 : Vn=nXℓ=0[2ℓ+ 1](−)(n−ℓ)q−32(n(n+2)−ℓ(ℓ+1))41 : Vn=nXℓ,j=0q[2j + 1][2ℓ+ 1]ajℓqℓ(ℓ+1)−j(j+1)51 : Vn=nXℓ=0[2ℓ+ 1](−)n−ℓq−52 (n(n+2)−ℓ(ℓ+1))52 : Vn=nXj,ℓ=0q[2j + 1][2ℓ+ 1]ajℓ(−)jqn(n+2)−ℓ(ℓ+1)+ 32 j(j+1))61 : Vn=nXj,ℓ=0q[2j + 1][2ℓ+ 1]aℓjqℓ(ℓ+1)−2j(j+1)62 : Vn=nXi,j,ℓ=0q[2i + 1][2j + 1]aℓiaℓj(−)n−ℓ−jq−32 (n(n+2)−j(j+1))−ℓ(ℓ+1)2+i(i+1)63 : Vn=nXj,ℓ,r,sq[2j + 1][2s + 1]ajℓaℓrars(−)ℓ+rq−j(j+1)+s(s+1)+ ℓ(ℓ+1)2−r(r+1)271 : Vn=nXℓ=0[2ℓ+ 1](−)n−ℓq−72(n(n+2)−ℓ(ℓ+1))72 : Vn=nXj,ℓ=0q[2j + 1][2ℓ+ 1]ajℓ(−)jq−n(n+2)+ℓ(ℓ+1)−52j(j+1)73 : Vn=nXj,ℓ=0q[2j + 1][2ℓ+ 1]ajℓ(−)jq32j(j+1)+2n(n+2)−2ℓ(ℓ+1)74 : Vn=nXj,ℓ,rq[2j + 1][2r + 1]ajℓarℓ(−)n−j−ℓ−rqn(n+2)2−ℓ(ℓ+1)2+ 32j(j+1)+ 32r(r+1)29

75 : Vn=nXijℓq[2i + 1][2j + 1]aℓiaℓj(−)n−jq−52n(n+2)+ 32j(j+1)+i(i+1)−ℓ(ℓ+1)76 : Vn=nXjℓrsq[2j + 1][2s + 1]ajℓaℓrars(−)ℓq−n(n+2)+j(j+1)−ℓ(ℓ+1)2+r(r+1)−s(s+1)77 : Vn=nXjℓrspq[2j + 1][2p + 1]ajℓaℓsasrarp(−)n−r−s−ℓq−n(n+2)2+j(j+1)−ℓ(ℓ+1)2−r(r+1)2+ s(s+1)2Here ajℓis the duality matrix given in Appendix A.Using the building blocks for normalized functional integrals over three manifolds with4 and higher punctures on their S2 boundaries, the above calculations can be extended toobtain the invaraints for the whole tables of knots and links given in ref.17 in a straightforward manner.Using the explicit representation for the duality matrix ajl for n = 1 as given inAppendix A, the above expressions can easily be seen to yield the Jones one-variablepolynomials2 for these knots.On the other hand,with the help of explicit representationfor ajlfor n = 2 as given in Appendix A,we obtain the polynomials19 calculated explicitlyby Akutsu,Deguchi and Wadati from the three-state exactly solvable model10.9.Concluding RemarksFollowing Witten4, here we have studied the SU(2) Chern-Simons theory in threedimensions as a theory of knots and links. A systematic method has been developed toobtain link invaraints.

The relation of a Chern-Simons theory on three-manifold withboundary to the Wess-Zumino conformal field theory on the boundary has been exploitedin doing so. Expectation value of Wilson link operators with the same spin n/2 represen-tation of SU(2) living on each of the component knots of the link yields a whole varietyof link invaraints.

The Jones one variable polynomial corresponds to n = 1. For higher n,30

these are the new link invaraints discussed by Wadati, Deguchi and Akutsu from the pointof view of N = n + 1 state exactly solvable statistical models10. As illustration of ourmethod we have also computed explicitly these invariants for knots upto seven crossingpoints.

The method can be generalized to links where different representations of SU(2)are placed on the component knots. Such links may be called multicoloured.

Expecta-tion values of Wilson link operators associated with such multicoloured links would thenprovide new link invariants. The knowledge of the expectation value of Wilson operatorsfor any link with arbitrary representations of the gauge group living on the componentknots would then provide a complete solution of the non-abelian Chern-Simons theory inthree dimensions.

A detailed discussion of this will be presented elsewhere.Generalization of our discussion to an arbitrary compact gauge group, say SU(N), israther straight forward. We shall take up this elsewhere.AcknowledgementsWe thank R.Jagannathan and K.Srinivasa Rao for discussions on quantum Racah coeffi-cients.Appendix AHere we shall list some useful properties of the quantum Racah coefficients and theduality matrix ajℓ.31

The quantum Racah coefficients are given by 11−13j1j2j12j3j4j23=∆(j1, j2, j12)∆(j3, j4, j12)∆(j1, j4, j23)∆(j3, j2, j23)Xm≥0(−)m[m + 1]!n[m −j1 −j2 −j12]! [m −j3 −j4 −j12]!

[m −j1 −j4 −j23]! [m −j3 −j2 −j23]!

[j1 + j2 + j3 + j4 −m]! [j1 + j3 + j12 + j23 −m]!

[j2 + j4 + j12 + j23 −m]!o−1(A.1)and∆(a, b, c) =vuut[−a + b + c]! [a −b + c]!

[a + b −c]! [a + b + c + 1]!

(A.2)Here [a]! = [a][a −1][a −2]...[2][1].

The SU(2) spins are related as −→j 1 + −→j 2 + −→j 3 =−→j 4, −→j 1 + −→j 2 = −→j 12, −→j 2 + −→j 3 = −→j 23The duality matrix ajℓwhere three spins, each n/2 is combined into spin n/2 is givenbyajℓ= (−)ℓ+j−nq[2j + 1][2ℓ+ 1]n/2n/2jn/2n/2ℓ(A.3)The q-Racah coefficients satisfy the following properties11 :j1j2jj3j4ℓ=j1j4ℓj3j2j(A.4)j1j20j3j4ℓ= (−1)ℓ+j2+j3δj1j2δj3j4q[2j2 + 1][2j3 + 1](A.5)Xj[2j + 1][2ℓ+ 1]j1j2jj3j4ℓj1j2jj3j4ℓ′= δℓℓ′(A.6)Xx(−)j+ℓ+x[2x + 1]q−Cxj1j2xj3j4jj1j2xj4j3ℓ=j1j3jj2j4ℓqCj/2+Cℓ/2q−Cj1/2−Cj2/2−Cj3/2−Cj4/2(A.7)32

where Cj = j(j + 1) is the Casimir invariant in the spin j representation R2j. Theserelations respectively imply the following properties for the duality matrix ajℓdefined in(A.3)ajℓ= aℓj(A.8)ajo =q[2j + 1][n + 1](A.9)nXℓ=0aℓiaℓj = δij(A.10)Xx(−)n−xq± 12(n(n+2)−x(x+1)) ajxaℓx = (−)j+ℓq±(j(j+1)+ℓ(ℓ+1))/2 ajℓ(A.11)Further notice that (A.9) and (A.10) imply:nXℓ=0q[2ℓ+ 1] aℓj = δjo[n + 1](A.12)and (A.9) and (A.11) imply :nXj=0q[2j + 1] (−)j q± j(j+1)2ajl = (−)n−lq[2ℓ+ 1] q± 12 (n(n+2)−ℓ(ℓ+1))(A.13)Also (A.10) and (A.11) imply:Xr,s(−)r+sq±( r(r+1)2+ s(s+1)2) aℓrarsasj = δℓj(−)n−ℓq±( n(n+1)2+ ℓ(ℓ+1)2)(A.14)Another useful relation isXr,sq[2r + 1][2s + 1] ars qr(r+1)+s(s+1) =X[2ℓ+ 1] (−)ℓq32 ℓ(ℓ+1)(A.15)It is instructive to write down this duality matrix ajℓexplicitly for various low valuesof n. We need this to compare the results obtained here with those of Jones2 and Akutsu,Deguchi and Wadati10 for the link invariants for n = 1 and n = 2 respectively.Theduality matrix for these two values of n reads :33

(i) n = 1ajℓ= 1[2]1q[3]q[3]−1(A.16)34

(ii) n = 2ajℓ= 1[3]1q[3]q[5]q[3][3]([5]−1)[4][2]−[2]√[5][3][4]q[5]−[2]√[5][3][4][2][4](A.17)Appendix BHere we shall illustrate by an example how the framework developed in this paper canbe used to rederive many of the results already known for Jones polynomials.The generalization of numerator-denominator theorem of Conway for Jones polyno-mials has been proved by Lickorish and Millet16. We shall present a new proof for thistheorem here .Theorem : For the links depicted in Fig.21,this theorem states that∪(∪2 −1)ˆLo( ˆA, ˆB)=∪nˆLo( ˆA, ˆQHo )ˆLo( ˆQHo , ˆB) + ˆLo( ˆA, ˆQVo )ˆLo( ˆQVo , ˆB)o−nˆLo( ˆA, ˆQHo )ˆLo( ˆQVo , ˆB) + ˆLo( ˆA, ˆQVo )ˆLo( ˆQHo , ˆB)o(B.1)where the symbol for each of the link diagrams itself represents the Jones link invariantV1[L].The proof of this theorem is rather straight forward in our framework.

We write thelink invariants above as :ˆLo( ˆA, ˆB) = ˆµo( ˆA) ˆµo( ˆB) + ˆµ1( ˆA) ˆµ1( ˆB)(B.2)35

ˆLo( ˆQVo , ˆB) = ˆµo( ˆB) +q[3] ˆµ1( ˆB)(B.3)ˆLo( ˆQHo , ˆB) = ˆµo( ˆB) [2](B.4)ˆLo( ˆA, ˆQVo ) = ˆµo( ˆA) +q[3] ˆµ1( ˆA)(B.5)ˆLo( ˆA, ˆQHo ) = ˆµo( ˆA) [2](B.6)where we have used from Theorem 9.ˆµℓ( ˆQVo ) =q[2ℓ+ 1],ˆµℓ( ˆQHo ) =nXj=0q[2j + 1] ajℓ= δℓo[n + 1]Now if we solve for ˆµℓ( ˆA) and ˆµℓ( ˆB),ℓ= 0, 1 from Eqns(B.3-B.6), we haveˆµo( ˆX) = ∪ˆL0( ˆQH0 , ˆB)ˆµ1( ˆX) = ∪ˆL0( ˆX, ˆQV0 ) −ˆL0( ˆX, ˆQH0 )∪q(∪2 −1)(B.7)where ˆX = ˆA, ˆB and the link invariant for the unknot (n = 1) ∪= [2] and [3] = ∪2 −1.Substituting these values (B.7) into (B.2) then, immediately yields the theorem above.Appendix CIn the following, we list various normalized functional integrals expanded in termsof the bases vectors referring to the middle two of the four strands attached to each ofthe boundaries ,S2. The diagram for the manifold itself will represent the normalisedfunctional integral.In the following numbers on the left hand side refer to the diagrams inFig.22,respectively:(1)=nXℓ=0q[2ℓ+ 1] | ˆφ(1)ℓ⟩(2)=[n + 1] | ˆφ(1)o ⟩36

(3)=nXℓ=0q[2ℓ+ 1]qmℓ(ℓ+1) | ˆφ(1)ℓ⟩,(4)=nXℓ=0q[2ℓ+ 1](−)q2m+12ℓ(ℓ+1) | ˆφ(1)ℓ⟩(5)=nXj,ℓ=0q[2j + 1]ajℓqmj(j+1) | ˆφ(1)ℓ⟩,(6)=nXjℓq[2j + 1]ajℓ(−)jq2m+12j(j+1) | ˆφ(1)ℓ⟩(7)=Xjqmj(j+1) | ˆφ(1)j ⟩| ˆφ(2)j ⟩(8)=nXj=0(−)jq2m+12j(j+1) | ˆφ(1)j ⟩| ˆφ(2)j ⟩(9)=Xj,ℓvuut[2ℓ+ 1][2j + 1]aℓjqm(n(n+2)−ℓ(ℓ+1)) | ˆφ(1)j ⟩| ˆφ(2)j ⟩(10)=Xj,ℓvuut[2ℓ+ 1][2j + 1]aℓjqmℓ(ℓ+1) | ˆφ(1)j ⟩| ˆφ(2)j ⟩(11)=Xj,ℓ=0vuut[2ℓ+ 1][2j + 1]aℓj(−)n−ℓq2m+12(n(n+2)−ℓ(ℓ+1)) | ˆφ(1)j ⟩| ˆφ(2)j ⟩(12)=Xvuut[2ℓ+ 1][2j + 1]aℓj(−)ℓq2m+12ℓ(ℓ+1)) | ˆφ(1)j ⟩| ˆφ(2)j ⟩(13)=Xjℓrajraℓrq(m+p)r(r+1) | ˆφ(1)j ⟩| ˆφ(2)ℓ⟩(14)=Xjℓrajraℓrq(m+p+1)r(r+1) | ˆφ(1)j ⟩| ˆφ(2)ℓ⟩(15)=Xajraℓr(−)rq( 2m+2p+12)r(r+1) | ˆφ(1)j ⟩| ˆφ(2)ℓ⟩(16)=Xijℓraijajraℓrq[2ℓ+ 1]q[2r + 1]qmℓ(ℓ+1) | ˆφ(1)i ⟩| ˆφ(2)j ⟩(17)=Xijℓrairajraℓrq[2ℓ+ 1]q[2r + 1]qm(n(n+2)−ℓ(ℓ+1)) | ˆφ(1)i ⟩| ˆφ(2)j ⟩(18)=Xijℓrairajraℓrq[2ℓ+ 1]q[2r + 1](−)ℓq( 2m+12)ℓ(ℓ+1) | ˆφ(1)i ⟩| ˆφ(2)j ⟩37

(19)=Xijℓrairajraℓrq[2ℓ+ 1]q[2r + 1](−)n−ℓq( 2m+12)(n(n+2)−ℓ(ℓ+1)) | ˆφ(1)i ⟩| ˆφ(2)j ⟩(20)=Xijℓrsairajraℓrasrq[2ℓ+ 1][2s + 1][2r + 1]qmℓ(ℓ+1)qps(s+1) | ˆφ(1)i ⟩| ˆφ(2)j ⟩(21)=Xijℓrsairajraℓrasr[2r + 1]q[2ℓ+ 1][2s + 1](−)ℓ+sq( 2m+12)(n(n+2)−ℓ(ℓ+1))• q( 2p+12)(n(n+2)−s(s+1)) | ˆφ(1)i ⟩| ˆφ(2)j ⟩(22)=Xijℓrsairajraℓrasr[2r + 1]q[2ℓ+ 1][2s + 1]qm(n(n+2)−ℓ(ℓ+1))• qp(n(n+2)−s(s+1)) | ˆφ(1)i ⟩| ˆφ(2)j ⟩(23)=Xijℓrsairajraℓrasrq[2ℓ+ 1][2s + 1][2r + 1](−)ℓ+sq( 2m+12)ℓ(ℓ+1)• q( 2p+12)s(s+1) | ˆφ(1)i ⟩| ˆφ(2)j ⟩(24)=Xjℓrarjarℓqm(n(n+2)−r(r+1)) | ˆφ(1)j ⟩| ˆφ(2)ℓ⟩(C.1)Now let us present some functional integrals expanded in terms of the bases vectorsreferring to the first two strands on the left (and equivalently the last two on the right)attached to the boundaries, each of which is an S2 with four punctures as shown inFig.23.For the diagrams listed in this figure,the normalized functional integrals are asfollows :(1)=nXℓ=0q[2ℓ+ 1] | φ(1)ℓ⟩,(2)=nXℓ=0q[2ℓ+ 1](−)m(n−ℓ)qm2 (n(n+2)−ℓ(ℓ+1)) | φ(1)ℓ⟩(3)=nXj,ℓq[2j + 1](−)jajℓq( 2m+12)j(j+1) | φ(1)ℓ⟩,(4)=nXjℓq[2j + 1]ajℓqmj(j+1) | φ(1)ℓ⟩38

(5)=nXℓ=0(−)(m+p)(n−ℓ)q( m+p2)(n(n+2)−ℓ(ℓ+1)) | φ(1)ℓ⟩| φ(2)ℓ⟩(6)=Xjℓvuut[2j + 1][2ℓ+ 1](−)jajℓq( 2m+12)j(j+1) | φ(1)ℓ⟩| φ(2)ℓ⟩(7)=Xjℓvuut[2j + 1][2ℓ+ 1]qmj(j+1)ajℓ| φ(1)ℓ⟩| φ(2)ℓ⟩(8)=Xjℓvuut[2j + 1][2ℓ+ 1]qm(n(n+2)−j(j+1))ajℓ| φ(1)ℓ⟩| φ(2)ℓ⟩(9)=Xjℓvuut[2j + 1][2ℓ+ 1](−)n−jq( 2m+12)(n(n+2)−j(j+1))ajℓ| φ(1)ℓ⟩| φ(2)ℓ⟩(10)=Xijℓqmℓ(ℓ+1)aℓiaℓj | φ(1)i ⟩| φ(2)j ⟩(11)=Xijℓ(−)ℓq( 2m+12)ℓ(ℓ+1)aℓiaℓj | φ(1)i ⟩| φ(2)j ⟩(12)=Xijℓrairajraℓrq[2ℓ+ 1]q[2r + 1]qmℓ(ℓ+1) | φ(1)i ⟩| φ(2)j ⟩(13)=Xijℓrairajraℓrq[2ℓ+ 1]q[2r + 1](−)ℓq( 2m+12)ℓ(ℓ+1) | φ(1)i ⟩| φ(2)j ⟩(14)=Xijℓrairajraℓrq[2ℓ+ 1]q[2r + 1]qm(n(n+2)−ℓ(ℓ+1)) | φ(1)i ⟩| φ(2)j ⟩(15)=Xijℓrairajraℓrq[2ℓ+ 1]q[2r + 1](−)n−ℓq( 2m+12)(n(n+2)−ℓ(ℓ+1)) | φ(1)i ⟩| φ(2)j ⟩(C.2)References[1] S.K.Donaldson, J.Diff.Geom. 18 (1983) 269 ; Polynomial invariants for smooth four-manifolds, Oxford preprint.

[2] V.F.R. Jones, Bull.

AMS 12 (1985) 103 ; Ann. of Math.

128 (1987) 335.39

[3] E.Witten, Commun. Math.

Phys. 117 (1988) 353.

[4] E.Witten, Commun. Math.

Phys. 121 (1989) 351.

[5] D.Birmingham, M.Rakowski and G.Thomson, Nucl. Phys.

B315 (1989) 577 ;L.Baulieu and B.Grossman,Phys. Letts.

B214 (1988) 223 ;R.K.Kaul andR.Rajaraman, Phys. Letts.B (in press).

[6] M.F.Atiyah in ”Oxford Seminar on Jones-Witten Theory”, 1988 ; M.F.Atiyah, Publ.Math. IHES 68 (1988) 175.

[7] P.Freyd, D.Yetter, J.Hoste, W.B.R.Lickorish, K.Millet and A.Ocneanu, Bull. AMS12 (1985) 239 ; J.H.Przytycki and K.P.Traczyk, Kobe J.Math.

4 (1987) 115. [8] L.Kaufmann, Topology 26 (1987) 395.

[9] V.G.Turaev, Inv. Math.

92 (1988) 527. [10] M.Wadati, T.Deguchi and Y.Akutsu, Phys.

Rep. 180 (1989) 247 and referencestherein. [11] A.N.Kirillov and N.Yu.Reshetikhin, Representation algebra Uq(SL(2)), q-orthogonalpolynomials and invariants of links, LOMI preprint E-9-88; See also ‘New Develop-ments in the Theory of Knots’, World Scientific Ed.T.Kohno (1989).

[12] L.Alvarez Gaume, G.Gomez and G.Sierra, Phys. Letts.

B220 (1989) 42. [13] For a review see, L.Alvarez Gaume and G.Sierra, CERN preprint TH.5540/89.

[14] K.Yamagishi, M-L Ge and Y-S Wu, Letts. Math.

Phys. 19 (1990) 15.

[15] R.Kaul and R.Rajaraman, Phys. Letts.

B249 (1990) 433. [16] W.B.R.Lickorish and K.C.Millet, Topology 26 (1987) 107.40

[17] G.Moore and N.Seiberg, Lectures on RCFT in ”Superstring ’89”, ed. M.Green et.al.,World Scientific, Singapore (1990).

[18] D.Rolfsen, ”Knots and Links”, Publish or Perish Inc., Berkely (1976) ; G.Burde andH.Zieschang, ”Knots”, de Gruyter Studies in Mathematics 5, Walter de Gruyter,Berlin, New York (1985). [19] Our normalization differs from those in Refs.2 and 10, where the invaraint for anunknot is taken to be one.

Thus to compare our results for n = 1 and n = 2 withthose of these references, we need to multiply their expressions by Vn[∪] = [n + 1].41

Figure CaptionsFig.[1]. Connected sum of two links L1, L2.Fig.[2].

Examples of rooms.Fig.[3]. Composition of two balls B1, B2 leads to link Lm(A) in S3.Fig.[4].

Half twists in two strands.Fig.[5]. Composing of balls B1 and B2 gives link ˆL2m( ˆA) in S3.Fig.[6].

Link ˆL2m+1( ˆA′) with room✒✑✓✏ˆA′✻❄✻❄.Fig.[7]. Composition of balls B1 and B2 yields link L(A1, A2) in S3.Fig.[8].

(a)Parallely oriented two-strand braid with m half-twists and(b) its closure Lm.Fig.[9]. Composition of balls B1 and B2 yields link ˆL2m( ˆA1, ˆA2) in S3.Fig.[10].

(a)Oppositely oriented two-strand braid with 2m half-twists and(b)its closure ˆL2m.42

Fig.[11]. Rooms QVm and QH2p+1.Fig.[12].

Rooms ˆQVm, ˆQH2p and ˆQ′HpFig.[13]. The links (a)ˆLo( ˆQH2m, ˆQH2p) ≡ˆL2m+2p and(b)ˆLo( ˆQH2m, ˆQV2p) ≡ˆLo( ˆQV2m, ˆQH2p)Fig.[14].

The links (a)Lo(QH2p+1, QH2m+1) = ˆL2p+2m+2 and(b)Lo(QH1 , QVm) = Lm+1Fig.[15]. Vectors corresponding to Eqns.7.1 and 7.2 respectively.Fig.[16].

Diagrammatic representations of functional integrals ˆν1, ˆν2, ˆν3in Eqns.7.3-5 respectively.Fig.[17]. Functional integral over a manifold with r boundaries ,ˆνr of Eqn.7.6.Fig.[18].

Functional integral over a manifold with r boundaries,νr of Eqn.7.9.Fig.[19]. Functional integral over a manifold with 2r + 2 boundaries (Eqn.7.12).Fig.[20].

Knot projections upto seven crossing points.Fig.[21]. Link diagrams for numerator-denominator theorem (Eqn.B.1).43

Fig.[22]. Building blocks corresponding to the Eqns.

(C.1) respectively.Fig. [23].Building blocks corresponding to the Eqns.

(C.2) respectively.44

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