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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.19460 |
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| _version_ | 1866914579530907648 |
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| author | Terashima, Yuji Yamaguchi, Yoshikazu |
| author_facet | Terashima, Yuji Yamaguchi, Yoshikazu |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_19460 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Gang-Kim-Yoon integrality conjectures on adjoint Reidemeister torsions for torus knots Terashima, Yuji Yamaguchi, Yoshikazu Geometric Topology High Energy Physics - Theory Mathematical Physics 57K31, 57R56 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. |
| title | Gang-Kim-Yoon integrality conjectures on adjoint Reidemeister torsions for torus knots |
| topic | Geometric Topology High Energy Physics - Theory Mathematical Physics 57K31, 57R56 |
| url | https://arxiv.org/abs/2605.19460 |