The Jones polynomial in terms of braid group representations Liang Tee (supervisor: Peter McNamara) Overview The braid group and Markov moves Representations of the braid group (cont.) 3 A knot 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 × f0; 1g, 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 link is a disjoint union of knots, each considered a component endpoints: one in f1; 2; :::; ng×f0g×f0; 1g and one in f1; 2; :::; ng× the form 2 3 of the link. Two knots are isotopic if one can be continously deformed into each f1g × f0; 1g. 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 60 b c 07 n R = 6 7 : by a sequence of Reidemeister moves (RI, RII, RIII): 40 d e 05 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; ji − jj ≥ 2; RI RII RIII entries in terms of t we obtain the R matrices −−! −−! −−! σiσi+1σi = σi+1σiσi+1; 2t1=2 0 0 0 3 2t−1=2 0 0 0 3 where σ denotes a positive crossing between the i-th and (i + 1)-th i 6 0 0 t 0 7 6 0 0 1 0 7 strands. The closure of a braid can be formed by connecting the R = 6 7 ; 6 7 : 4 0 t t1=2 − t3=2 0 5 4 0 1 t−1=2 − t1=2 0 5 A knot invariant 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 n(b) where b is a braid, we try to construct an isotopy invariant of a link, which requires tr( n(b)) to be invariant under 1 2 3 the Markov moves: tr( n(ab)) = tr( n(ba)); isotopic to −−−−!closure −−−−−−! ±1 The Kauffman bracket and Jones polynomial tr( n(b)) = tr( n+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 · (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 · n(ab)) = tr(h · n(a) n(b)) = tr( n(b) · h · n(a)) ±1 ⊗n ⊗n h ? i = 1 MI : ab ! ba; MII : b ! bσ ; = tr(h · n(b) n(a)) = tr(h · n(ba)) and , n ±1 where the diagrams in the first relation correspond to links identical everywhere where a; b 2 Bn and bσn 2 Bn+1. ⊗(n+1) ±1 ⊗(n+1) ⊗(n−1) ±1 tr(h · n+1(σn b)) = tr(h · (id ⊗ R ) · n+1(b)) except around a crossing, and the diagram of the second relation corresponds to V = tr(h⊗n · (b)): the nonintersecting union of an unknot and D. n By orienting each component of a link D, we define the sign of a crossing and The skein relation of this invariant is the same as that of the Jones polyno- the writhe 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 : 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 (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 2 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 n must satisfy tr(h · n(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 ) n(σi) n(σj) = n(σj) n(σi); ji − jj ≥ 2; and substituting A2 = t−1=2. The skein relation of the Jones polynomial is n(σi) n(σi+1) n(σi) = n(σi+1) n(σi) n(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). Knot Theory and its Applications. Reprint of the 1996 edition. Birkh¨auser. LATEX TikZposter.