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Reidemeister torsion of two-bridge knots and signatures of TQFT

Julien Marché, Seokbeom Yoon

TL;DR

This work connects the adjoint Reidemeister torsion of two-bridge knots to a Frobenius algebra that governs SU2-TQFT signatures, revealing a concrete algebraic underpinning for topological invariants. It provides explicit torsion formulas for the tautological and adjoint representations in terms of a Riley/Two-Variable polynomial framework and shows how the Frobenius structure encodes signature data via the element Ω. A central result is the asymptotic description of the SU2-TQFT signatures along carefully chosen root sequences, expressed through a reciprocal Frobenius algebra WM and its trace ΩWM^{g−1}, with a leading term tied to a universal zeta value and the growth rate of n. The study combines topological, algebraic, and analytic techniques, yielding exact reciprocal isomorphisms between parabolic character data and Frobenius algebra objects, and offering a precise asymptotic framework for understanding TQFT signatures in the two-bridge knot family. The results pave the way for a deeper conceptual interpretation of these relations and motivate future work on deriving WM directly from parabolic character varieties and their geometric properties.

Abstract

We establish an explicit relation between the adjoint Reidemeister torsion of the two-bridge knot $K(p,q)$ at any parabolic representation and the Frobenius algebra governing the signatures of SU$_2$-TQFT vector spaces at the root $ζ=\exp(iπq/p)$. As applications, (a) we prove that the inverse sum of torsions is constant (i.e., independent of $p$ and $q$); and (b) we show that along sequences of roots of the form $ζ_n = \exp\left(iπ\tfrac{a+bn}{c+dn}\right)$, the signatures have the same asymptotic behavior as the Verlinde formula.

Reidemeister torsion of two-bridge knots and signatures of TQFT

TL;DR

This work connects the adjoint Reidemeister torsion of two-bridge knots to a Frobenius algebra that governs SU2-TQFT signatures, revealing a concrete algebraic underpinning for topological invariants. It provides explicit torsion formulas for the tautological and adjoint representations in terms of a Riley/Two-Variable polynomial framework and shows how the Frobenius structure encodes signature data via the element Ω. A central result is the asymptotic description of the SU2-TQFT signatures along carefully chosen root sequences, expressed through a reciprocal Frobenius algebra WM and its trace ΩWM^{g−1}, with a leading term tied to a universal zeta value and the growth rate of n. The study combines topological, algebraic, and analytic techniques, yielding exact reciprocal isomorphisms between parabolic character data and Frobenius algebra objects, and offering a precise asymptotic framework for understanding TQFT signatures in the two-bridge knot family. The results pave the way for a deeper conceptual interpretation of these relations and motivate future work on deriving WM directly from parabolic character varieties and their geometric properties.

Abstract

We establish an explicit relation between the adjoint Reidemeister torsion of the two-bridge knot at any parabolic representation and the Frobenius algebra governing the signatures of SU-TQFT vector spaces at the root . As applications, (a) we prove that the inverse sum of torsions is constant (i.e., independent of and ); and (b) we show that along sequences of roots of the form , the signatures have the same asymptotic behavior as the Verlinde formula.

Paper Structure

This paper contains 23 sections, 26 theorems, 154 equations, 4 figures.

Table of Contents

  1. Introduction
  2. About two-bridge knots
  3. About signatures of TQFT
  4. Asymptotics of the signature along roots of unity
  5. Back to two-bridge knots
  6. Reidemeister torsions of two-bridge knots
  7. Topological setting
  8. Parabolic character varieties
  9. Computation of the torsion
  10. Tautological representation
  11. Simplication of $\tau_1$
  12. Adjoint representation
  13. Comparing reciprocal pairs
  14. Explicit isomorphism
  15. Cusp shape
  16. ...and 8 more sections

Key Result

Theorem A

Let $0<q<p$ be coprime odd integers and $V$ be the Frobenius algebra described above with associated element $\Omega$. Denoting by $\tau_n$ the Reidemeister torsion of $S^3\setminus K(p,q)$ with coefficients $\rho_n$, one has where $0<\ell<p$ is the unique odd integer satisfying $q\ell \equiv \pm 1 \pmod{p}$.

Figures (4)

  • Figure 1: The Conway normal form of $[a_1,a_2,\ldots,a_k]$ with a trivial component.
  • Figure 2: The quotients of $T_1$ by $\mathbb{Z}/p \mathbb{Z}$ and $D_p$.
  • Figure 3: Schubert's normal form of $K(5,3)$.
  • Figure 4: Wirtinger generators in the normal form.

Theorems & Definitions (49)

  • Theorem A: Theorems \ref{['thm.simp']}, \ref{['toradj']} and Equation \ref{['toromega']}
  • Theorem B: Theorems \ref{['thm.invtau2']} and \ref{['thm.invtau1']}
  • Theorem C: Theorem \ref{['mainsign']}
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • ...and 39 more