Tug-the-hook symmetry for quantum 6j-symbols

TL;DR

The paper introduces the tug-the-hook symmetry for quantum 6j-symbols, a new symmetry applicable to all representations, supported by multiple evidences and implications for knot theory and Chern-Simons theory.

Contribution

It presents the novel tug-the-hook symmetry for quantum 6j-symbols, extending symmetry understanding to representations with multiplicities and linking it to Wilson loops in Chern-Simons theory.

Findings

Tug-the-hook symmetry applies to all representations, including those with multiplicities.

The symmetry is supported by evidence from the eigenvalue conjecture and explicit examples.

Implications extend to links in Chern-Simons Wilson loops.

Abstract

We introduce a novel symmetry for quantum 6j-symbols, which we call the tug-the-hook symmetry. Unlike other known symmetries, it is applicable for any representations, including ones with multiplicities. We provide several evidences in favour of the tug-the-hook symmetry. First, this symmetry follows from the eigenvalue conjecture. Second, it is shown by several new examples of explicit coincidence of 6j-symbols with multiplicities. Third, the tug-the-hook symmetry for Wilson loops for knots in the 3d Chern-Simons theory implies the tug-the-hook symmetry for quantum 6j-symbols. An important implication of the analysis is the generalization of the tug-the-hook symmetry for the Chern-Simons Wilson loops to the case of links.