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Home , Jones polynomial

The Jones polynomial in terms of representations Liang Tee (supervisor: Peter McNamara)

Overview The braid group and Markov moves Representations of the braid group (cont.)

3 A is a simple closed curve in R ; we draw a knot diagram by projecting a A braid is a set of n strands in 2 × {0, 1}, each with distinct We consider the case where V is a 2-dimensional vector space and R is of 2 R knot onto R . A is a disjoint union of , each considered a component endpoints: one in {1, 2, ..., n}×{0}×{0, 1} and one in {1, 2, ..., n}× the form   of the link. Two knots are isotopic if one can be continously deformed into each {1} × {0, 1}. Any braid can be expressed as an element of the braid a 0 0 0 other without ‘breaking’ the knot, or equivalently, if they are related to another group of n strands B , with presentation 0 b c 0 n R =   . by a sequence of Reidemeister moves (RI, RII, RIII): 0 d e 0 generators : σ1, σ2, ..., σn−1 0 0 0 f From the Yang-Baxter equation, if b = 0 and e 6= 0, after normalising the relations : σiσj = σjσi, |i − j| ≥ 2, RI RII RIII entries in terms of t we obtain the R matrices ←−−→ ←−−→ ←−−→ σiσi+1σi = σi+1σiσi+1, t1/2 0 0 0  t−1/2 0 0 0  where σ denotes a positive crossing between the i-th and (i + 1)-th i  0 0 t 0   0 0 1 0  strands. The closure of a braid can be formed by connecting the R =   ,   .  0 t t1/2 − t3/2 0   0 1 t−1/2 − t1/2 0  A assigns a value to a knot such that if two knots are isotopic, they ends (i, 1, 0) and (i, 1, 1); by Alexander’s theorem, any (oriented) 0 0 0 t1/2 0 0 0 −t1/2 have the same value under the invariant. We study the Jones polynomial knot link is isotopic to the closure of a braid. For example, the closure of −1 −1 invariant and outline a construction of the Jones polynomial via representations the braid σ1 σ2σ1 σ2 is isotopic to the figure-eight knot as shown: which correspond to the Jones and Alexander polynomials respectively. of braid groups. By taking the trace of ψn(b) where b is a braid, we try to construct an isotopy invariant of a link, which requires tr(ψn(b)) to be invariant under 1 2 3 the Markov moves: tr(ψn(ab)) = tr(ψn(ba)), isotopic to −−−−→closure −−−−−−→ ±1 The Kauffman bracket and Jones polynomial tr(ψn(b)) = tr(ψn+1(bσn )) ±1 −1 However, this fails, since we require tr2R = idV but tr2R 6= idV . By 1 2 3 t−1/2 0  using h = , and using the properties R · (h ⊗ h) = (h ⊗ h) · R For a link diagram D, the Kauffman bracket hDi is a Laurent polynomial in A 0 t1/2 ±1 defined recursively by the relations and tr2((idV ⊗ h) · R ) = idV obtained by construction of R, we obtain the The closures of braids a and b are isotopic to each other if and only invariant tr(h⊗n · ψ (b)), since −1 ` 2 −2 n h i = A h i +A h i h D i= (−A − A )hDi if braids a and b are related to each other by the Markov moves , , ⊗n ⊗n ⊗n (MI, MII): tr(h · ψn(ab)) = tr(h · ψn(a)ψn(b)) = tr(ψn(b) · h · ψn(a)) ±1 ⊗n ⊗n h ∅ i = 1 MI : ab ←→ ba, MII : b ←→ bσ , = tr(h · ψn(b)ψn(a)) = tr(h · ψn(ba)) and , n ±1 where the diagrams in the first relation correspond to links identical everywhere where a, b ∈ Bn and bσn ∈ Bn+1. ⊗(n+1) ±1 ⊗(n+1) ⊗(n−1) ±1 tr(h · ψn+1(σn b)) = tr(h · (id ⊗ R ) · ψn+1(b)) except around a crossing, and the diagram of the second relation corresponds to V = tr(h⊗n · ψ (b)). the nonintersecting union of an and D. n By orienting each component of a link D, we define the sign of a crossing and The of this invariant is the same as that of the Jones polyno- the w(D) of an oriented link D as follows: mial since −1 −1 −1/2 1/2 Representations of the braid group t R − tR = (t − t )idV ⊗V −1 = positive crossing, = negative crossing, and R, R and idV ⊗V correspond to the three types of crossings. Since the value of the unknot under this invariant is t1/2 +t−1/2, its value is t1/2 +t−1/2 For a vector space V over , we consider the representation ψ : w(D) = (#positive crossings) − (#negative crossings). C n times the Jones polynomial V (t). This gives an alternate definition of the B → End(V ⊗n) given by ψ (σ ) = (id )⊗(i−1) ⊗ R ⊗ (id )⊗(n−i−1), L n n i V V Jones polynomial in terms of a braid b and its closure L: Then by modifying the Kauffman bracket to be invariant under the Reidemeister where R ∈ End(V ⊗ V ). From the presentation of the braid group, 3 −w(D) moves, we obtain the Kauffman bracket invariant (−A ) hDi for an oriented ⊗n 1/2 −1/2 ψn must satisfy tr(h · ψn(b)) = (t + t )VL(t) link. The Jones polynomial VL(t) can be defined in terms of the Kauffman 2 −2 bracket by dividing the value of the Kauffman bracket invariant by (−A − A ) ψn(σi)ψn(σj) = ψn(σj)ψn(σi), |i − j| ≥ 2, and substituting A2 = t−1/2. The skein relation of the Jones polynomial is ψn(σi)ψn(σi+1)ψn(σi) = ψn(σi+1)ψn(σi)ψn(σi+1). From these relations, R must satisfy the Yang-Baxter equation t−1V (t) − tV (t) = (t1/2 − t−1/2)V (t) L+ L− L0 References (idV ⊗ R)(R ⊗ idV )(idV ⊗ R) = (R ⊗ idV )(idV ⊗ R)(R ⊗ idV ). where L+, L− and L0 are link diagrams identical everywhere except for an area Ohtsuki, T. (2002). Quantum Invariants: A Study of Knots, 3-Manifolds, and their Sets. World Scientific. around a crossing, replaced by a positive crossing, negative crossing and no Jones, V.F.R. (1987). Hecke algebra representations of braid groups and link polynomials. Annals of Mathemat- crossing respectively. ics. 126 (2):335-388. Alexander, J. (1923). A lemma on a system of knotted curves. Proceedings of the National Academy of Sciences of the United States of America. 9 (3):93-95. Murasugi, K. (2008). and its Applications. Reprint of the 1996 edition. Birkh¨auser.

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