Torus knot polynomials and susy Wilson loops

Abstract

We give, using an explicit expression obtained in [V. Jones, Ann. of Math. 126, 335 (1987)], a basic hypergeometric representation of the HOMFLY polynomial of $(n,m)$ torus knots, and present a number of equivalent expressions, all related by Heine's transformations. Using this result the $(m,n)\leftrightarrow (n,m)$ symmetry and the leading polynomial at large $N$ are explicit. We show the latter to be the Wilson loop of 2d Yang-Mills theory on the plane. In addition, after taking one winding to infinity, it becomes the Wilson loop in the zero instanton sector of the 2d Yang-Mills theory, which is known to give averages of Wilson loops in $\mathcal{N}$=4 SYM theory. We also give, using matrix models, an interpretation of the HOMFLY polynomial and the corresponding Jones-Rosso representation in terms of $q$-harmonic oscillators.

17 pages, v2: More concise (published) version; typos corrected

References in corpus (7)

• Exact results for the entanglement entropy and the energy radiated by a quark
• Supersymmetric Wilson loops on S^3
• More supersymmetric Wilson loops
• Wilson loops: From four-dimensional SYM to two-dimensional YM
• Chern-Simons matrix models and Stieltjes-Wigert polynomials
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• Harer-Zagier formulas for families of twisted hyperbolic knots