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

Yuji Terashima; Yoshikazu Yamaguchi

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arXiv · 2605.19460

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

Yuji Terashima·Yoshikazu Yamaguchi

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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.

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BibTeX RIS

Yuji Terashima, Yoshikazu Yamaguchi. 2026-06-08. Gang-Kim-Yoon integrality conjectures on adjoint Reidemeister torsions for torus knots. https://arxiv.org/abs/2605.19460

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