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arxiv: 2605.22308 · v1 · pith:QARMC2TInew · submitted 2026-05-21 · 🧮 math.GT

Algebraic properties of twisted Alexander polynomial and Reidemeister torsion of torus knots

Takayuki Morifuji , Anh T. Tran This is my paper

Pith reviewed 2026-05-22 02:17 UTC · model grok-4.3

classification 🧮 math.GT MSC 57M2557M27
keywords twisted Alexander polynomialReidemeister torsiontorus knotcharacter varietySL(n,C) representationSeifert fibered spacealgebraic integerTQFT
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The pith

Coefficients of twisted Alexander polynomials for torus knots with irreducible SL_n(C) representations are locally constant algebraic integer-valued functions on the character variety.

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

This paper shows that for torus knots, the coefficients of twisted Alexander polynomials built from irreducible SL_n(C) representations of the knot group act as locally constant functions that take values in the ring of algebraic integers, when viewed over the SL_n(C) character variety. The result extends to a generalization proving that Reidemeister torsions for SL_n(C) representations are algebraic integers on many Seifert fibered spaces. The authors further examine the power sums of these torsions in low-dimensional cases and observe an unexpected link to TQFT. A sympathetic reader would care because these algebraic properties suggest that certain knot invariants remain stable and discrete even as the underlying representations deform continuously.

Core claim

The paper proves that every coefficient of the twisted Alexander polynomial of a torus knot associated with an irreducible SL_n(C)-representation is an A-valued locally constant function on the SL_n(C)-character variety, where A denotes the ring of algebraic integers over C. As a generalization of earlier work, it establishes that SL_n(C)-Reidemeister torsions are algebraic integers for many Seifert fibered spaces. It also discusses power sums of Reidemeister torsions of torus knots for low-dimensional irreducible representations and their mysterious relation to TQFT.

What carries the argument

The twisted Alexander polynomial constructed from an irreducible SL_n(C)-representation of the torus knot group, regarded as a function on the SL_n(C)-character variety.

If this is right

  • The coefficients stay algebraic integers while varying in a locally constant way across connected components of the character variety.
  • Reidemeister torsions for SL_n(C) representations become algebraic integers on a broad class of Seifert fibered spaces.
  • Power sums of the torsions for low-dimensional representations of torus knots satisfy relations that connect to TQFT.
  • Local constancy implies the polynomials are constant on open sets and can change only when crossing lower-dimensional strata in the variety.

Where Pith is reading between the lines

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

  • The same algebraic-integer and local-constancy properties might hold for other knot families once their representation varieties are analyzed similarly.
  • The observed TQFT link suggests these invariants could be reinterpreted inside quantum topological frameworks.
  • Local constancy would let one stratify the character variety according to the values of these polynomials.
  • Explicit calculations on small torus knots such as the trefoil with low n could directly test the claimed stability.

Load-bearing premise

The twisted Alexander polynomial is defined in the standard way from the irreducible SL_n(C) representation of the knot group, and the character variety carries its usual topology so that local constancy is well-defined.

What would settle it

An explicit computation for the trefoil knot and some irreducible SL_2(C) representation in which one coefficient of the twisted Alexander polynomial fails to be an algebraic integer or fails to be locally constant along a continuous path in the character variety.

read the original abstract

In this paper we prove that every coefficient of twisted Alexander polynomials of torus knots associated with irreducible $\mathrm{SL}_n(\Bbb C)$-representations is an $\Bbb A$-valued locally constant function on the $\mathrm{SL}_n(\Bbb C)$-character variety, where $\Bbb A$ is the ring of all algebraic integers over $\Bbb C$. Moreover, as a generalization of a recent result of Kitano and Nozaki, we show that $\mathrm{SL}_n(\Bbb C)$-Reidemeister torsions are algebraic integers for many Seifert fibered spaces. Also, we discuss the power sums of Reidemeister torsions of torus knots for low-dimensional irreducible representations that provide a mysterious relation to TQFT.

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 / 2 minor

Summary. The manuscript proves that every coefficient of twisted Alexander polynomials of torus knots associated with irreducible SL_n(C)-representations is an A-valued locally constant function on the SL_n(C)-character variety, where A is the ring of algebraic integers. It generalizes a result of Kitano and Nozaki by showing that SL_n(C)-Reidemeister torsions are algebraic integers for many Seifert fibered spaces. The paper also discusses power sums of Reidemeister torsions of torus knots for low-dimensional irreducible representations and their relation to TQFT.

Significance. If the central claims hold, the integrality and local-constancy results supply concrete algebraic constraints on twisted Alexander polynomials over character varieties, which is useful for computations and structural questions in knot theory. The generalization to Seifert spaces extends prior work in a direct way, and the TQFT discussion, while exploratory, identifies a potential bridge between classical invariants and quantum topology.

minor comments (2)
  1. [Introduction] The introduction would benefit from a short paragraph situating the local-constancy result against existing continuity statements for Alexander polynomials of other knot families.
  2. [§6] In the discussion of power sums, a precise formulation of the 'mysterious relation' to TQFT (e.g., a conjectural equality or numerical match) would make the observation more falsifiable.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the positive assessment, including the accurate summary of our results on the algebraic integrality and local constancy of coefficients in twisted Alexander polynomials for torus knots under irreducible SL_n(C) representations, as well as the generalization of integrality results for Reidemeister torsions to many Seifert fibered spaces. We appreciate the recommendation for minor revision and are happy to incorporate any suggested improvements to clarity or presentation.

Circularity Check

0 steps flagged

No significant circularity; direct proofs from standard definitions

full rationale

The paper claims to prove algebraic properties of twisted Alexander polynomials and Reidemeister torsions for torus knots and Seifert spaces directly from the standard definitions of these invariants via irreducible SL_n(C)-representations and the topology of the character variety. The abstract and available description indicate no fitted parameters renamed as predictions, no self-definitional loops where outputs are presupposed in inputs, and no load-bearing self-citations that reduce the central claims to unverified prior results by the same authors. The derivations appear self-contained against external mathematical benchmarks such as known properties of knot groups and representation varieties.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on standard definitions and properties of twisted Alexander polynomials, Reidemeister torsion, SL_n(C) representations, and character varieties from prior literature in algebraic topology; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • standard math Standard construction of twisted Alexander polynomials via irreducible SL_n(C)-representations of the knot group
    Invoked in the definition of the polynomials whose coefficients are analyzed (abstract).
  • domain assumption The SL_n(C)-character variety carries a topology in which local constancy of functions is meaningful
    Required for the locally constant claim (abstract).

pith-pipeline@v0.9.0 · 5661 in / 1428 out tokens · 51637 ms · 2026-05-22T02:17:33.903169+00:00 · methodology

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Works this paper leans on

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