Louis Kauffman- From Knots to Quantum Groups (and Back)

8/3/2019 Louis Kauffman- From Knots to Quantum Groups (and Back)

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& jubm^ted manuscript has been authoredv a contractor of the U. S. Governmentnder contract No. W-31-709-ENG-38-cco rdingly, the U. S- Government retains anexclusive, royalty-free license to publish

reproduce the published form g! thisntribution, or alia** others to do so. for S. Government purposes.

ANL-HEP-CP-90-36

April, 1990

ANL-HEP-CP90-36

DE90 013688

From Knots to Quantum Groups (and Back)

Louis Kauffman

Dept. of Mathematics, Statistics and Computer Science

The University of Illinois at Chicago

Chicago, IL 60680

and

Argonne National Laboratory

High Energy Physics DivisionArgonne, IL 60439

DISCLAIMER

This report was prepared as an account of work sponsored by an agency of the United StatesGovernm ent. Neith er the United States Government nor any agencythereof, nor any of theiremployees, makes any warranty, express or implied, or assumes any legal liability or responsi-bility for the accuracy^ completeness, or usefulness of any information, apparatus, product, orprocess disclosed, or represents thai its use would not infringe privately owned rights. Refer-ence herein to any specific commercial product, process, or service by trade name, trademark,manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recom-mendation, or favoring by the United States Government or any agencythereof. The viewsand opinions of authors expressed herein do not necessarily state or reflect those of theUnited Stales Government or any agencythereof.

M A S T E RDISTRIBUTION OF THIS D OCU MEN T IS UNLIM

Talk given at the Informal Spring Workshop on Quantum Groups, Argonne National Laboratory,April 26 -M ay 11 ,19 90 . Work supported by the U.S. Department of Energy, Division of HighEnergy Physics, Contract W-31-1Q9-ENG-38 and NSF grant DMS-8822602.


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From Knots to Quantum Groups (and Back)by Louis H. Kauffman

I . In t roduc t ionThis paper traces how the Jones polynomial leads naturally to the

notion of quantum group, and how quantum groups give rise toinvariants of links via solutions to the Yanc-Baxter equation.Section 5, is an original treatment of the construction of theuniversa l R-matrix. All the other material has , or will appearelsewhere in similar form.

II. Knot TheoryLet's begin by recalling the Reidemeister moves:

X.


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These crossings are taken to indicate the projection of arcs

embedded in a three-space, and projected to the plane. The brokenarc pair at a crossing indicates the arc that passes underneath theother arc in space . Any link (A link is a collection of circlesimbedded in a three-sphere or Euclidean three space.) has a point ofprojection to the surface of a two dimens ional sphere or to a plane,so that the projection (with under and over-crossing indications)becomes a diagram for that link.

Two links are said to be am bien t isot op ic if there is acontinuous time-param eter family of embeddings starting with onelink and ending with the other one. The theory of knots and links isthe theory of link embsddings under the equivalence relation ofambient isotopy. (A kn ot is a link with one component, t h a t is. aknot is an embedding of a single circle into three-space.)

It is assumed that all the embeddings are represented (up toambient isotopy) by an embedding that is a differentiate curve(s) inthe three-space. Links that do not admit such a representation arecalled wild , and must be treated separately.

The Reidemeister moves generate the theory of knots and linksin three-dimensional space in the sense of the following theorem:

Theorem (Reidemeister [18]). Let K and K1 be two links embeddedin three-dim ensional space (either the three-dimensiona l sphere ,or the Euclidean space R 3). Then K and K1 are ambient isotopic ifand only if diagrams for K and K' are related by a finite sequence ofthe moves 1,11, II I.

Re mark. In Reidemeister's day the notion of ambient isotopy wasalso com binatorial. Let R denote Reidemeister. For R, ambientisotopy was generated by a single move type called anelem enta ry co m bin ato ria l iso to py (or elementary isotopy forshort). The knots and links for R are piec ew ise linea r - meaningthat they consist of interconnections of straight line segmentsembedded in Euclidean space . Vertices are regarded as the endpointsof these segments, and any straight segment can be regarded as theconnection of two segments . by adding a vertex at an interior point.The elementary combinatorial isotopy has two directions:ex pa ns ion , and c on tr ac tio n . In an expansion, one takes twovertices on the link, and a new vertex in the complement such that


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the (two dimensional) triangle spanned by these vertices intersectsthe link only at one of its three edge s. Expansion cons ists inreplacing this edge in the link by the two remaining edges in thetriangle. Contraction is the opposite of expansion - three points onthe link span a triangle intersecting the link only along two edges;

these edges are replaced by the third edge of the triangle.

The Reidemeister moves come about via examination of the forms ofplanar projections of the elementary isotopies. For example , thediagram below shows how a type I Reidemeister move is the shadowof an elementary isotopy.

Reidemeister's approach to his theorem is a good way to get ageom etric feel for the situation . For a modern treatment of the

Theorem, using the continuous (or differentiate) notion of isotopy,see [5].

While Reidemeister's Theorem is an excellent starting pointfor a combinatorial theory of knots and links, it does not make lifeeasy. The easiest way to illustrate this is to exhibit a demon (Thisdemon - shown to the author by Ken Millett- improves over previousculprits, and is the smallest possible for projections on a sphere.)such as the one shown below:


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This demon D is unknotted, but does not admit any simplifyingReidemeister m oves, nor does it admit any type three moves. (AReidemeister move is said to be simplifying if it reduces thenumber of crossings in the diagram.) In order to unknot D it isnecessary to first make the diagram more complex before it canbecome simpler. Exam ples of this sort show that the equivalencerelation generated by the Reidemeister moves is subtle, and that thematter of constructing invariants is non-trivial.

There are many accounts of the classical construction of knot andlink invariants ( [1], [8],[4J,[11], [21]). In the next section I shall godirectly to a model for the Jones polynomial and discuss its physicalinterpretations. For these purposes it does make sense to make oneremark about the process of abstraction leading to mathematicalkno ts. If we were to make a knot or link from rope or othermaterial, then the amount of twisting on the rope would make adifference in the behav iour of the resulting knotted form . Suchtwisting has been abstracted when we go to the diagram or to themathematical curve embedded in space. We can recover some of thisstructure by considering framed links. A framed link is a linksuch that each component has a continuous normal vector field. Thisis equivalent to thinking about embeddings of bands rather thancircles. Thus the figure below indicates a framed trefo il, withstandard framing inherited from its planar embedding.


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If we keep track of the framing then one no longer has invariance

under the type I move:

For this reason it is useful to have the concept of regular isotopy.Two links are said to be regularly isotopic if one can be obtainedfrom the other by a sequence of type II and type III moves only.Regular isotopy is the equivalence relation generated by the type IIand type ill moves. Note:

Opposite curls cancel. This regular isotopy is the knot theoreticversion of the Whitney trick [24]. Actua lly, regular isotopy is a bitsubtler than simple framing. The bands shown below


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are isotopic, but the corresponding string diagrams are not regularlyisotopic ( They have different Whitney degree [24].)-

A useful invariant of regular isotopy is thewrithe, w(K).writhe is the sum of the crossing signs

The

in a given diagram. Thus

- + 3

It is easy to see that the writhe is a regular isotopy invariant fordiagram s. It is very useful for norm alizing other invariants ofregular isotopy. It turns out that most of the invariants we shall

discuss need such a normalization.

III . Link Invariants as Vacuum-Vacuum AmplitudesFirst, a quick description of the bracket model [12] of the Jones

polynomial [9]: We give a method of associating a. well-definedpolynomial in three variables, (A,B,d), to an unoriented link K.This polynomial is defined recursively by the formulas:


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.

< o > = d .

The first formula asserts that the polynomial for a given diagram isobtained as an additive combination of the polynomials for thediagrams obtained by splicing away the given crossing in twopossible ways. Thus the small diagrams indicate larger diagramsthat differ only as indicated. The second formula says that the valueof a loop (simple closed curve in the plane) is d, and that if a loopoccurs (isolated) inside a larger diagram, then the value of thepolynomial acquires a factor of d from this loop. In particular, thevalue of a disjoint union of N simple closed curves is d N .

Together, the two formulas completely determine , and iswell-defined just so long as A,E and d commute with one another.Thus this polynomial takes values in the ring Z[A,B,d] ofpolynomials in three variables with integer coefficients.

As it stands, is not an invariant of any of the Reidemeistermoves. However, the following formulais an easy consequence of 1.and 2. above.

Proposition.

As a result we see that is invariant under the type II move ifwe select B=A" 1 and d = -A 2 - A" 2 . Furthermore, it nowfollows directly that is invariant under the move III:


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Thus, with B=A *1 and d = -A2 - A~2, we have that is aninvariant of regular isotopy. To obtain an invariant of ambientisotopy for oriented links, we form

fK(A) = ( - A 3 ) - W ( K ) / < O >where K is oriented , w(K) is the writhe of K as defined in theprevious section, and is the bracket evaluated on the unorientedlink underlying K. The reason for this factor of -A3 :s that

One then has the

Theore m [12 ]. For any oriented link K, VK(t) = fK (t '1 /4) whereV denotes the original one-variable Jones polynom ial.

Thus, the bracket, suitably normaiizeci, gives a direct model for theJones polynomial.


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The Vacuum-Vacuum Ampl i tudeIn the rest of this section I shall stick to the bracket, and show

how it can be seen as a "vacuum -vacuum am plitude" in acom binatorial version of topologica l quantum field theory [25].

More generally, we can consider an amplitude associated to a givendiagram by regarding the plane as 1+1 spacetime. By convention, lettime run vertically up the page, and space proceed from left to right(This is the convention of the reader of English.). Position the linkdiagram so that it is transversal to the space levels except atcritical points corresponding to maxima, minima and crossings.

n

X XEach minimum can be regarded as a creation of two particles fromthe vacuum, each maximum an annihilation, and each crossing is aninteraction (thought of as involving braiding in the extra spatialdimension orthogona l to the page). To each of these events weassociate a matrix whose indices go over (say) the spins of theparticles , and whose values are the amplitudes for each of theseprocesses.

\J

The amplitudes used here are a generalization of amplitudes inquantum mechanics, suitably generalized for the purposes oftopology (!n the process we take leave of the usual interpretationsof observation in quantum m echanics. In the topology the amplitude


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10

itself is a real property of the system. There is no "collapse of thewave function".) Therefore the amplitudes will take values in acommutative ring (e.g. in Z[A,A"1]), and the spins will run over anarbitrary finite index set (e.g. {-1,+1})- Amplitudes are calculatedaccording to the principles of quantum mechanics [7j:

1. If an event occurs in a way that can be decomposed into aset of individual steps (e.g. creations, annihilations,interactions), then the amplitude of the given event is theproduct of the amplitudes of the individual steps.2. If an event may occur in several disjoint alternative ways,then the amplitude of this event is the sum of the amplitudesof the ways.

Given a diagram K , and a set of matrices as above, we cancalculate the amplitude for particles to be created from the vacuum,interact in the pattern of the link diagram , and return to thevacuum. This amplitude decomposes as a sum of the amplitudes forconfigurations of the diagram. Eachconfiguration a is an assignment of spins to the nodes of thediagram. (The nodes are the input and output nodes of the smalldiagrams corresponding to the matrices.) Given a configuration,

each matrix has a well-defined value , and the amplitude of thisconfiguration is the product of these values. Thus the vacuum-vacuum amplitude , T(K), for a given diagram K is the sum (over theconfigurations) of the product of the matrix values for eachconfigurat ion.

Symbolically, this works out in accord with the usual Einsteinconvention for repeated indices: Write down a product of all thematrices for the given diagram , in indices ,with one index for eachnode. The am plitude is then the value of this express ion interpretedas a sum over all cases of repetitions of an index in lower and upperpositions.


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11

Having defined the vacuum-vacuum amplitude T(K), we must seewhen it will be an invariant of regular isotopy, and when it will

model the bracket. In order for T(K) to be an invariant of regularisotopy, we need the following restrictions on the matrices [15]:

i e

(In 3. there is a corresponding left-hand twist, and in 4. there isalso the same equation for all crossings reversed.) Equation 4. is


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12

called the Yang - Baxter Equation [3] (here given without rapidityparameter).

Remark. It is interesting to speculate about the physical meaningof these res trictions . The twist cond ition 3. is the most

my sterious since it relates R and R-1

via creations andannihilations. A simpler physical situation may lend some insighthere. In the simplified scenario, we assume that 1. holds, and thatparallel identity lines are interchangeable with pairs of creationsand annihilations:

f\Then one has the sequence of identifications

\o~\c~jO

STATISTICS

This has been interpreted as a depiction of the equivalence of spinand statistics (see e.g. [22] and references therein) where spin isregarded as catalogued by the twist of framing (become curl ofdiagram) and statistics corresponds to the braiding of the two lines.This shows that part of our diagrammatics correspond to ordinaryphysical interpretations, and that where the topology begins the

equivalence of spin and statistics leaves off. In this sense,the topology is an index of the non-standard statistics.

Modelling the BracketIn order to model the bracket with a vacuum-vacuum amplitude

we need to find creation and annihilation matrices that are inverse


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1 3

to one another , and that give a loop value of -A2 - A"2.an answer to that puzzle:

Here is

ONote that the matrix M has square the identity, and that the loopvalue is therefore the sum of the squares of the entries of M.(See [13],[14],[15] for motivations for this cons truction.)

With a given choice for the creations and annihilations, there isone choice for the R matrix to give the bracket:

With this choice, T(K) will satisfy the defining equations of thebrack et, and therefo re =T(K) (since we have correctlyadjusted the loop value) .

Remark. In fact it is interesting to note that if the creation andannihilation are inverse matrices , and R is defined as above, then2. follows easily, while 3. goes as below


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14

and 4. is proved by first checking

and then performing the following variation on our bracketderivation of the invariance under the III move

position

The upshot of this discussion is that by simply adjusting thec r e a t i o n a n d a n n i h i l a t i o n m a t r i c e s c o r r e c t l y, w eautom atically produce a model of the bracket and asolution to the Yang-Baxter equation. This is the simplestinstance of a solution to the Yang-Baxter equation appearingnaturally from the knot theory. This is the well-known [19] R-matrix corresponding to the SL(2) quantum group. In fact, thestructure that we have created so far will now enable us to see onemotivation for the construction of the quantum group.


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IV. The SL(2) Quantum GroupNote that we can write

e =- A ' O

and that , as A approaches 1 ,the matrix approaches

The matrix e is significant in linear algebra because it expressesthe determinant of a 2x2 ma trix. That is, let P be a 2x2 matrixwith comm uting entries. Then

P sP T"= DeKf)6.)

Now SL(2) (over a commutative ring) is the set of matrices ofdeterminant one, and can therefore be characterized as the set ofmatrices leaving the epsilon invariant:

SL(2) = {PHere T denotes matrix transpose.


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1 6

At A=1 the bracket dos not discriminate between under and Gver-crossings, and the identity

=1

corresponds directly to the Fierz identity

= &ac

&bd - 5ad

5bc

Thus at A=1 (and also at A=-1) the diagrams become interpretedas tensor diagrams for SL(2) invariant expressions .

It is then natural to ask whether there is ageneralization of this symmetry for the topology and linkdiagrams. Specifically , we ask whether has a symmetrygroup analogous to SL(2). Some experimentation shows that the

way to ask this question is to consider

with associative, possibly non-commutative entries, and ask for theinvariances:

*L1*

It is then an exercise in elementary algebra to see that theseconditions are equivalent to the* equa tions: (q


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17

ca=qac db=qbdba=qab dc=qcdbc=cb

ad-da = (q" 1 -q)bcad-q-1bc = 1

These are the defining relations for the algebra U*=SL(2)q([6],[17]), sometimes called the SL(2) quantum group. It is not agroup , but rather a Hopf algebra . The co-algebra structure is givenby the map

A: U * >U* 0 U*

where

ThusA(a) = a a +A(b) = a b +A(c) = c a + d bA(d) = c b +

In this case the Hopf algebra has an antipode and this is directlyrelated to the fact that the matrix P has an inverse y(P):

a.


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Recall that the antipode is a mapfollowing diagram commutes

U* such that the

where e and r) are the co-unit and unit respectively and m denotesthe multiplication in the algebra . Here

Thus the condition that y be an antipode is just that

And this is the same as saying that P and are inverse matrices.

We could now and discuss a number of things about therelationship of this quantum group to solutions to the Yang-Baxterequation, and to its dual form as a deformation of the Lie Algebrafor SL(2). (See [6] , [19] ) But here there is not space for this. Thepurpose of this section has been to show how the quantum grouparises naturally from a com bination of the topology and a desire tcextend the algebraic symmetry Inherent in a significant special caseof the vacuum-vacuum expectation model.


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1 9

V. And BackIn order to indicate how the trail looks going back from quantum

groups to link invariants I shall make a leap to the formalism behindthe so-called quantum double construction of Drinfeld [6]. We shallthen see how a Hopf algebra structure can give rise through itsmatrix representations, to invariants of links.

We begin with an algebra with generators eQ,e 1,...,e n and

e,e 1 , . . . ,e n and the following relations describing multiplication inthe algebra: >

CA e* e* = z

Diagrammatically , I shall write O&&& >

where it is understood that the boxes denote the product expansioncoefficients, and hence the boxes commute with the e-nodes () and with each other. We further assume the following relationshipbetween multiplying upper and Sower e's:

- e


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2 0

Then we have

Theorem. With the above assumptions, the element

satisfies the algebraic form of the Yang-Baxter equation:

i s U * """' a3 *13 \ i*-

Proof. (See [15]).

In a matrix represen tation the e-nodes sprout indices , and analgebraic solution to the Yang-Baxter Equation becomes an knottheorist's matrix solution via an added permutation. Thus if

denotes the algebraic solution, then


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2 1

denotes the corresponding knot theoretic R-matrix in somerepresentation (the indices of this representation correspond to thenew lines).

As we know from the previous section , the knot theory demandsa relationship between the creation and annihilation matrices andthe R-matrix. This "twist relation" is given diagrammatically asf o l l o w s :

Thus

We conclude that there should be an automorphismx > y(x)

of the abstract algebra that corresponds to the map in therepresentat ion

and we need that v(es) 0 e s is the inverse of es 0 e s

summ ation on repeated lower and upper indices). It is then not hard


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to see that if ws were to make the algebra into a Hopf algebra suchthat the co-multiplication for the lower index e's is themultiplication for the upper index e's and vice versa (this is thedouble construction) , then this inverse requirement isequivalent to y being an antipode. (See the appendix.) The twistconditions of the knot theory are intimately tied with the Hopfalgebra structure for the quantum group. This completes the journeyback, albeit in an abstract mode.

A p p e n d i xHere we verify that y , as defined in section 5 , is an antipode in

the Hopv algebra. Recall that we demanded that y(e s) es be

inverse to es e s . Multiplying this out , we find

[ y(ej) e1 ][ ej d ] = y(6j)eJ ei

= y(ej)ei

'"K ' x~"~~ "** ^

= m(.uijy(ei) ei) ek

= m ((y 1)(jj'Jej eJ)) ek

e k

We are summ ing on repeated indices, and in the last step we usedthe fact that in the Drinfeld double construction, the diagonal in thealgebra of the e's with lower subscripts is given by themultiplication coefficients in the algebra of the upper e's. Note alsothat m denotes multiplication, and the the formula for y to be anantipode is

m((y 1)(A(ek))) = e0 if k equals 0m((y 1)(A(ek))) = 0 if k is not 0.

Here we take eo to be 1 , the identity element in the Hopf algebra.Thus we have shown that if y is an antipode for the Drinfeld doubleconstruction, then y(es) e s and e s e s are inverses. This , inturn, shows that the diagramm atic twist condition for the


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vacuum-vacuum expectation corresponds, through representation ofthe Drinfeld construction, to the existence of an antipode in theHopf algebra. In this sense , the abstract quantum group defined bythe formalism of the Drinfeld double construction gives a universallink invariant.

References

1. Alexander, J.W. . Topological invariants of knots and links. Trans.Amer. Math. Soc. 20 (1928), 275-306.

2. Akutsu ,Y., Deguchi, T., Wadati ,M.. The Yang-Baxter Relation: ANew Tool for Knot Theory. Braid Group, Knot Theory and

Statistical Mechanics, ed. Yang and Ge. World Scientific (1989)

3. Baxter, R.J.. Exactly Solved Models in StatisticalMechanics. Academic Press (1982).

4. Birman.J.S.. Braids, Links and Mapping Class Groups. Annalsof Math. Study 82, Princeton Univ. Press (1976).

5. Burde.G. and Zieschang, H.. Knots. deGruyter (1985).

6. Drinfeld, V.G.. Quantum groups. Proc. Intl. Congress Math. ,Berkeley, Calif., USA, 798-820 (1986).

7. Feynman, R.P.. Lectures on Physics. Addison Wesley (1964).

7.1 Finkelstein ,D. and Rubenstein.J. J.Math.Phys. Vol.9, 1762 (1968).

8. Fox, R.H.. A quick trip through knot theory. Topology of 3-Manifolds. M.K.Fort ed. Prentice Hall (1962),120-167.

8.1 Hennings, M. Invariants of links and three manifolds obtainedfrom Hopf algebras (preprint 1989).

9. Jones, V.F.R.. A polynomial invariant for links via von Neumannalgebras. Bull. A.M.S. 12, (1985), 103-112.

10. Jones, V.F.R.. On knot invariants related to some statisticalmechanics models. Pacific J. Math. 137, No.2, (1989), 311-334.


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11. Kauffman, L.H.. On Knots, Annals of Mathematics Study 115.Princeton Univ. Press (1987).

12. Kauffman, L.H.. State models and the Jones polynomial. Topology26, (1987) , 395-407.

13. Kauffman, L.H.. Statistical mechanics and the Jones polynomial.AMS Contemp. Math. 78 (1988), 263-297.

14. Kauffman, L.H.. Spin networks and knot polynomials. Int. J. Mod.Phys. A. ,Vol.5, N o.1 , Jan . 1990. 93-115.

15. Kauffman, L.H.. Knots, abstract tensors and the Yang-Baxterequation, (to appear).

16. Lawrence, R.. A universal link invariant using quantum groups,(preprint 1988).

16.1. Lee, H.C.. G-deformation of sl(2,C) x Z N and link invariants,(preprint 1989).

17. Manin, Y.I .. Quantum groups and non-commutative geometry. CRM -Universite de Montereal (1988).

18. Reidemeister, K.. Knotentheorie. Chelsea Pub. Co, New York(1948), Copyright 1932, Julius Springer, Berlin.

19. Reshetikhin.N.Y.. Quan tized universal enveloping algebras , theYang-Baxter equation and invariants of links, I and II. LOMI reprintsE-4-87 and E-17-87. Steklov Institute , Leningrad, USSR.

20. Reshetikhin , N.Y. and Turaev, V.G.. Invariants of 3-manifolds vialirk polynomials and quantum groups, (preprint 1989).

21. Rolfsen.D.. Knots and Links. Publish or Perish Press (1976).

22. Sorkin, R.. A general relation between kink-exchange and kink-rotation. Comm.Math.Phys. Vol.115, 421-434 (1988).

23. Turaev, V.G.. The Yang-Baxter equations and invariants of links.Invent. Math. 92 (1988), 527-553.

24. Whitney, H.. On regular closed curves in the plane. Comp. Math. 4

(1937), 276-284.


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25. Witten, E.. Quantum field theory and the Jones polynomial.Commun.Math.Phys. 121 (1989) , 351-399.

Department of Mathematics, Statistics and Computer ScienceThe University of Illinois at Chicago

Chicago, Illinois 60680

Louis Kauffman- From Knots to Quantum Groups (and Back)

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Transcript of Louis Kauffman- From Knots to Quantum Groups (and Back)