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1. Lecture 1: Knot Polynomials

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1. Lecture 1: Knot Polynomials NEW ZEALAND JOURNAL OF MATHEMATICS Volume 21 (1992), 1-16 KNOTS, BRAIDS, STATISTICAL MECHANICS AND VON NEUMANN ALGEBRAS V a u g h a n J o n e s (Received June 1991) Abstract. In the first lecture I will define several invariants of knots including the classical Alexander polynomial. I will show how they are related to statistical me­ chanics and quantum field theory. In the second lecture I will trace the origin of the new knot invariants through braids and von Neumann algebras and show how totally new methods in knot theory seem to be necessary for understanding and manipulating these invariants. 1. Lecture 1: Knot Polynomials 1.1 Definitions A knot will be a smooth curve in the three sphere S 3 (or R 3) which is the image of an embedding of S'1. A link will be a disjoint union of knots. Links may or may not be oriented, they will be considered up to diffeomorphisms of S3 which are generally required to preserve the orientation of S3 (this is the same as isotopy in S3). Links may always be projected onto some plane so that the only singularities are double points. If one records which part of the knot is closest to the plane of projection one obtains “link diagrams”. Here are some examples. The double points of the projection are called crossings for obvious reasons. the unknot v V / the figure eight knot the right trefoil Conway’s knot QO the left trefoil Knots have been tabulated up to 13 crossings [Thl]. Forgetting obvious symme­ tries and composite knots (e.g. Q>! is a composite knot made up of two trefoils), there are about 12^ thousand in the tables. The number of different knots grows exponentially with crossing number. 9 VAUGHAN JONES The relation of equality of links is translated into the equivalence relation on link diagrams generated by (planar diffeomorphisms and) the Reidemeister moves of types I, II and III shown below Type I These moves are applied locally, affecting only that part of the link seen in the drawings above. An invariant is a quantity associated to a knot or link which depends only on the knot in three dimensions. If this invariant is calculated from diagrams, this just means that the answer is invariant under the Reidemeister moves. A very elementary but important example of an invariant is the linking number between two oriented curves. We will call ^ ( a positive crossing and a negative one. Given two nonintersecting smooth curvesC\ and C 2, lk(C \,C 2 ) is defined to be the signed sum of the crossings between C\ and C2 in some generic projection. It is simple to verify invariance under the type II and III moves, the only ones that can involve more than one component. In mathematics, knots and links have been most important in the study of three- manifolds via a process called surgery. Given a link L in a 3-manifold M, one removes a tubular neighbourhood ofL (a solid torus per component ofL ) then one glues it back with some new identification on the boundary. A good exercise is to see that S l x S 2 may be obtained by surgery on the unknot in S 3 by a diffeomorphism of the boundary which reverses the meridian and longitude of the torus. A basic theorem of Lickorish and Wallace shows that any closed oriented 3-manifold may be obtained from S 3 by surgery on a link (see [Ro]). 1.2 Five polynomial invariants in order of appearance 1.2.1 The Alexander polynomial Given an oriented link L in S3, one may realise elements of (S'1 — L) as smooth oriented curves. Taking the sum of the linking numbers with the components ofL gives a homomorphism from 7Ti (S3 —L) —► 7Z . This in turn defines an infinite KNOTS, BRAIDS. VON NEUMANN ALGEBRAS 3 cyclic covering space S 3~L whose first homology becomes aZZ[t,t x] module, t acting by the generator of the deck transformations. Using van Kampen’s theorem one sees that H i(S 3-L ) has a square presentation matrix as a 2 Z [t,t~ l] module. The principal ideal generated by this polynomial is then an invariant of the link L. It is called the Alexander polynomial A i{t) of L. As defined above A i s ambiguous up to multiplication by ± a power of t. It can be shown (see [Co]) that Al(£) may be normalized in such a way that it has the following (skein) properties: 1. A K{t) = 1 if K is the unknot. 2. If L +, L - and L q admit diagrams identical except near one crossing where they are as below, then then AL+(t) - AL_(t) = \^ /t- It is easy to see that the skein relation provides an algorithm for the calculation of Az,(t). Examples of Alexander polynomials are: • The trefoils have A^(£) =t~ l — 1 + 1. • The figure 8 has A Lit) = 3 — t — t -1 . • Conway’s knot has A^(i) = 1. In general a split link (one with two or more non empty components separated by a 2-sphere) has Alexander polynomial zero. The Alexander polynomial first appeared in 1928 in [A] and has been a powerful tool in knot theory. It may be calculated in polynomial time with respect to the number of crossings. Since Ai(f) is defined in terms of S 3-L, it may be calculated from any construc­ tion of this manifold. One way to do this is to glue together infinitely many copies of S 3 - L. indexed by 7 Z . each one joined to the next by identifying along an oriented surface in S 3 whose oriented boundary is L. By considering surfaces of the type shown below, it is possible to gain complete control over (t).A For instance using a companion matrix one may easily show that, if p{t) is any given polynomial with • P(t)=p(t~1), • P{ 1) = 1 then there is a knot K with A x{t) = p(t). This is due to Seifert in [Se]. 4 VAUGHAN JONES 1.2.2 The polynomial Vi,{t) It was shown in [J 1] that there is another polynomialV l (£) of oriented links defined by the skein relation \ v L+(t) - tVL_(t) = fr f- ± ) VLo(t) (and Kinknot = 1)> where L +, L _ and L q are as for the Alexander polynomial. Examples are: • The Right trefoil has Vl (t) —t + t3 — t4. • The figure 8 has Vi,{t) = t ~2 — t~ 1 + 1 — t + t2. • Conway’s knot has Vl(£) = —t~ 4 + 2t~3 — 2t~2 + 2t_1 + t2 — 2t3+ 2t4 — 2t5 + t6 Mirror image symmetry is reflected in the polynomial by t —> \ so that the left trefoil has Vl(£) =t ^ 1 -f t~ 3 — t~ 4. Note that the Alexander polynomial of a knot is insensitive to this operation. The ^-polynomial of a distant union of two links is the product of theirV- polynomials multiplied by the factor — ^s/t + Thus split links have non-zero F-polynomial and in fact no link can have F-polynomial equal to zero. It is not known at this stage whether there is a non-trivial knot whoseV- polynomial is that of the unknot, nor is there any idea of how to characterize polynomials of the formV (t). It is unlikely that V can be calculated in polynomial time. In marked contrast to A(£), V{t) is essentially insensitive to the orientation of a link (it changes by a power of t under orientation changes). There is no known interpretation of Vl(£) in terms of the algebraic topology of S^-L. 1.2.3 The skein (or HOMFLY or FLYPTHOM) polynomialP l (1,tti) The similarity between the skein relations of A and V prompted many people ([F+], [PT]) to prove that the coefficients in the skein relation can be chosen arbitrarily. A standard choice is to define Pi,(l,m ) by l~1PL+(l,m ) + lPL_(l,m ) + m PLo(l,m ) = 0 and .Punknot = 1- Examples are • Pright trefoil = I 2 / + I Ul . • P figure 8 = {~l~2 ~ 1 ~ I2) + m2. • PConway1~2 += 7 (2+ 6l2 + 214) + (-31~2 - 11 - ll/2 -l4)m2 3 + ( r 2 + 6 + 612 + l4)m4 + (— 1 —l2)m6. The polynomial P (l,m ) is known to contain more information than just the combination ofV and A (see [LM]). KNOTS, BRAIDS, VON NEUMANN ALGEBRAS 5 1.2.4 The absolute polynomial of Ql (x ) [BLM], [Ho] Considering unoriented links led to the discovery of a polynomial invariant Q l {x ) obeying the following generalized skein relation. If L +, L_, L q and Loo are unoriented links admitting diagrams identical except near one crossing where they are as below, then Q l + + Ql _ = x (Ql 0 + Ql J Examples are: • Qtrefoil = ~ 3 +2x + 2x2. • Qfigure 8 = - 3 - 2X + 4x2 + 2x3. • QConway = 17 - 24x - 52x2 + 54x3 + 76a:4 - 28x5 - 48x6 - 4x7 + 8 x 8 + 2x9. This polynomial is understood no better than V ^ t). 1.2.5 The Kauffman polynomial F i,(a , x ) ([K al]) Regular isotopy is defined to be the relation on link diagrams generated by Rei­ demeister moves of type II and III. It is very little different from isotopy. Kauffman first defines an invariant F(a, x) of regular isotopy of unoriented link diagrams by F l + + Fl _ = x (Fl 0 + Fl ^ ) as in section 1.2.4 (.Funknot = 1) and Fl ±i = cl± xFl if L ±l differs from L by a type I Reidemeister move.
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