Level-rank duality of knot and link invariants

TL;DR

This paper explores the level-rank duality in Chern-Simons theory and its implications for knot and link invariants, providing explicit examples, criteria for knot classification, and discussing symmetries and dualities in hyperbolic cases.

Contribution

It summarizes key results on level-rank duality in Chern-Simons theory and introduces a criterion to distinguish knot types based on invariants, with explicit examples and symmetry discussions.

Findings

Explicit duality relations for $SU(2)_K$ and $SU(K)_2$ invariants

A criterion to distinguish torus and hyperbolic knots

Discussion of symmetries in hyperbolic knot invariants

Abstract

A number of results for the level-rank duality of $G(N)_K$ $\leftrightarrow$ $G(K)_N$ Chern-Simons theory are summarized, with emphasis on the applications to knot and link invariants. Explicit examples for $SU(2)_K$ $\leftrightarrow$ $SU(K)_2$ illustrate general results. A criterion to distinguish torus knots and links from hyperbolic knots and links, based on tables constructed by Kaul for one and two strand invariants, is presented. Possible symmetries of hyperbolic knot and link invariants are discussed. The level-rank duality of torus knot and link invariants of minimal models is examined