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arxiv: 2605.19460 · v1 · pith:MZGAOSL3new · submitted 2026-05-19 · 🧮 math.GT · hep-th· math-ph· math.MP

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

Yuji Terashima , Yoshikazu Yamaguchi This is my paper

Pith reviewed 2026-05-20 02:31 UTC · model grok-4.3

classification 🧮 math.GT hep-thmath-phmath.MP
keywords torus knotsadjoint Reidemeister torsionintegrality conjectureVerlinde numbersmodular S-matrixcharacter varietybirational model
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The pith

The sum of the (g-1)st powers of adjoint Reidemeister torsions for any torus knot is an integer for every non-negative g.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves the Gang-Kim-Yoon conjecture that the sum of the (g-1)st powers of the adjoint Reidemeister torsions of a torus knot is always an integer. It does so by introducing Verlinde numbers defined from the modular S-matrix and deriving their recursion formulas along with initial values. These recursions directly establish the integrality of the sums. The authors also construct a birational model of the character variety for a torus knot and recover the adjoint Reidemeister torsion from the Hessian of the polynomial that defines the model.

Core claim

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.

What carries the argument

Verlinde numbers defined from the modular S-matrix, whose recursion formulas and initial values establish the integrality of the torsion power sums.

If this is right

  • The integrality holds for every torus knot and every non-negative integer g.
  • The recursion relations supply an inductive method to verify integrality without direct evaluation of the torsions.
  • The birational model of the character variety yields an algebraic route to the adjoint Reidemeister torsion via the Hessian.
  • Initial values of the Verlinde numbers anchor the base cases that propagate integrality to all higher g.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Analogous Verlinde numbers defined for other knot families could extend the integrality result beyond torus knots.
  • The explicit birational model may connect classical torsion invariants to quantum invariants through the modular S-matrix.
  • Numerical checks for small torus knots and low g can test the Hessian recovery procedure independently of the recursion proof.

Load-bearing premise

The recursion formulas and initial values of the Verlinde numbers defined from the modular S-matrix are correctly derived and sufficient to establish the integrality for every torus knot.

What would settle it

Finding a torus knot and non-negative g for which the sum of the (g-1)st powers of the adjoint Reidemeister torsions is not an integer, or observing that the Verlinde numbers violate the stated recursion relations.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript proves the Gang-Kim-Yoon integrality conjecture asserting that, for any torus knot and any non-negative integer g, the sum of the (g-1)st powers of the adjoint Reidemeister torsions is an integer. The proof proceeds by defining Verlinde numbers via the modular S-matrix, establishing explicit recursion formulas together with initial values, and applying induction on g to deduce integrality of the summed powers. The authors additionally construct a birational model of the character variety of the torus knot and recover the adjoint Reidemeister torsion from the Hessian of the defining polynomial.

Significance. If the derivations hold, the work resolves the stated conjecture for the entire family of torus knots, supplying an explicit recursive mechanism and a birational model whose Hessian recovers the torsion. These concrete tools—explicit recursions, initial data, and the birational model—constitute verifiable contributions that may extend to related integrality questions in quantum topology and representation varieties.

minor comments (3)
  1. Clarify the precise identification between the summed torsion powers and the Verlinde numbers in the induction step; a short diagram or explicit low-g example would aid readability.
  2. Ensure the birational model is stated with explicit coordinates and that the Hessian computation is written out for at least one torus knot (e.g., the trefoil) to make the recovery of the torsion fully transparent.
  3. Add a brief comparison table of the new Verlinde numbers against known values of the adjoint torsions for small torus knots and small g.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive assessment of our manuscript, accurate summary of the proof, and recommendation for minor revision. We address the report below.

read point-by-point responses

  1. Referee: The manuscript proves the Gang-Kim-Yoon integrality conjecture asserting that, for any torus knot and any non-negative integer g, the sum of the (g-1)st powers of the adjoint Reidemeister torsions is an integer. The proof proceeds by defining Verlinde numbers via the modular S-matrix, establishing explicit recursion formulas together with initial values, and applying induction on g to deduce integrality of the summed powers. The authors additionally construct a birational model of the character variety of the torus knot and recover the adjoint Reidemeister torsion from the Hessian of the defining polynomial.

    Authors: We confirm that this is an accurate description of our approach and results. The Verlinde numbers are defined using the modular S-matrix, the recursion and initial values are established explicitly, induction yields the integrality, and the birational model with the Hessian recovery is constructed as stated. revision: no

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation begins with the modular S-matrix taken from established prior literature, then derives recursion formulas and initial values for Verlinde numbers that are specific to torus knots within the paper itself. These recursions are used to prove integrality of the summed (g-1)st powers of adjoint Reidemeister torsions by induction. The birational model of the character variety and the recovery of the torsion via its Hessian are likewise constructed and verified explicitly in the manuscript. None of these steps reduce by construction to a fitted input, self-definition, or load-bearing self-citation; the central integrality result follows from the newly derived recursions rather than from renaming or smuggling in prior ansatzes. The manuscript is therefore self-contained against external benchmarks once the S-matrix identification is granted.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The proof rests on defining Verlinde numbers from the modular S-matrix and deriving recursions whose validity is assumed to hold for torus knots.

axioms (1)
  • domain assumption The modular S-matrix yields well-defined Verlinde numbers for torus knots with the stated recursion formulas and initial values.
    Invoked to introduce the numbers whose recursions prove integrality.
invented entities (1)
  • Verlinde numbers for torus knots no independent evidence
    purpose: To encode the summed powers of adjoint Reidemeister torsions via recursion
    Newly introduced from the viewpoint of the modular S-matrix to prove the integrality statement.

pith-pipeline@v0.9.0 · 5677 in / 1253 out tokens · 43084 ms · 2026-05-20T02:31:13.843020+00:00 · methodology

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