Polynomials for torus links from Chern-Simons gauge theories

Abstract

Invariant polynomials for torus links are obtained in the framework of the Chern-Simons topological gauge theory. The polynomials are computed as vacuum expectation values on the three-sphere of Wilson line operators representing the Verlinde algebra of the corresponding rational conformal field theory. In the case of the SU(2) gauge theory our results provide explicit expressions for the Jones polynomial as well as for the polynomials associated to the N-state (N > 2) vertex models (Akutsu-Wadati polynomials). By means of the Chern-Simons coset construction, the minimal unitary models are analyzed, showing that the corresponding link invariants factorize into two SU(2) polynomials. A method to obtain skein rules from the Chern-Simons knot operators is developed. This procedure yields the eigenvalues of the braiding matrix of the corresponding conformal field theory.

Polynomial Invariants for Torus Knots¶and Topological Strings

“…In particular, we will see that for gauge group SU (2) it reduces to the Akutsu-Wadati polynomials of torus knots first obtained in [17].…”

“…Given an arrangement of indices like this, with r − k indices verifying condition (I) (which will be called of type I) and k indices verifying condition (II) (which will be called of type II), a weight belonging to F l is obtained if and only if: 17) for every pair of indices i µ , i ν , verifying (I). The set of arrangements of indices selected in this way will be denoted by I (k λ ) (n), and the corresponding set of weights will be denoted…”

Torus Knots and Mirror Symmetry

“…When Q and P are not coprime, we have instead a link L Q,P with L = gcd(Q, P ) components. From the point of view of the above formalism, the operator creating such a link can be obtained [26,32] by considering the product of L torus knot operators with labels (Q/L, P/L), i.e.…”

Knots, links and branes at large N

“…Note that the parity of the power of g s in (2.18) correlates with the number of holes, and translating this to (2.8)(2.9) using Frobenius relation and 19) yields the above relation among N R,Q,s 's. There were also additional vanishing relations noted in [3] from comparing (2.18) and (2.8)(2.9) which were also in need of explanation.…”