Gang-Kim-Yoon integrality conjectures on adjoint Reidemeister torsions for torus knots

TL;DR

This paper proves the integrality of sums of powers of adjoint Reidemeister torsions for torus knots by introducing Verlinde numbers and analyzing their recursion relations.

Contribution

It establishes the conjecture for all torus knots and all non-negative g, using modular S-matrix and recursion formulas for Verlinde numbers.

Findings

Proved the conjecture for all torus knots and g ≥ 0.

Derived recursion formulas for Verlinde numbers.

Connected Reidemeister torsions to the Hessian of a polynomial in a birational model.

Abstract

We study the conjecture that a sum of the (g-1)st powers of adjoint Reidemeister torsions for a torus knot is an integer. We prove that the conjecture is true for any torus knot and all non-negative g. To prove the conjecture, we introduce the Verlinde numbers for torus knots from the viewpoint of modular S-matrix and show the recursion formulas and initial values of them. The recursion formulas of Verlinde numbers prove the integrality of the sum of the (g-1)st powers of adjoint Reidemeister torsions. Related to a modular S-matrix, we also provide a birational model of the character variety for a torus knot and show how to recover the adjoint Reidemeister torsion for a torus knot from the Hessian of the polynomial defining the birational model.