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arxiv: 2605.13291 · v1 · submitted 2026-05-13 · 🧮 math.GT

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Twisted Alexander vanishing groups of knots

Katsumi Ishikawa , Takayuki Morifuji , Masaaki Suzuki
Authors on Pith no claims yet

Pith reviewed 2026-05-14 18:54 UTC · model grok-4.3

classification 🧮 math.GT MSC 57M25
keywords twisted Alexander polynomialTAV groupknot groupfinite groupfaithful representationirreducible representationvanishing invariantknot theory
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The pith

Every faithful irreducible representation of a TAV group makes the twisted Alexander polynomial of a knot vanish.

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

The paper builds on the earlier definition of twisted Alexander vanishing groups, which are finite groups G such that the twisted Alexander polynomial vanishes for at least one knot. It examines the possible orders of these groups and supplies explicit constructions of knots for which the polynomial is zero. The central result establishes that the vanishing holds for every faithful irreducible representation of the knot group into any such TAV group. This supplies a uniform mechanism that forces the invariant to disappear once the representation is faithful and irreducible.

Core claim

A twisted Alexander vanishing group is a finite group for which the twisted Alexander polynomial of a knot vanishes. The paper shows that every faithful irreducible representation of a TAV group causes the twisted Alexander polynomial to be zero. It also discusses the orders of TAV groups and constructs knots whose twisted Alexander polynomials vanish.

What carries the argument

Twisted Alexander vanishing (TAV) group: a finite group G such that the twisted Alexander polynomial vanishes for some knot under a representation into G.

Load-bearing premise

The twisted Alexander polynomial is computed from the standard chain complex or Fox calculus applied to a faithful irreducible representation of the knot group into the finite TAV group with no further restrictions.

What would settle it

A knot group together with a faithful irreducible representation into a candidate TAV group for which the resulting twisted Alexander polynomial is nonzero.

Figures

Figures reproduced from arXiv: 2605.13291 by Katsumi Ishikawa, Masaaki Suzuki, Takayuki Morifuji.

Figure 1
Figure 1. Figure 1: Construction of Kp,q,r; (p, q, r) = (2, 3, 5). disjoint from K such that the linking number lk(K, α) is zero and f0 sends α to a generator of the commutator subgroup Cqr of G. We denote the resulting satellite knot K(α, J) by Kp,q,r; see [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗
read the original abstract

In our previous work, we introduced the notion of a twisted Alexander vanishing (TAV) group, defined as a finite group for which the corresponding twisted Alexander polynomial of a knot vanishes. In this paper, we discuss the orders of TAV groups and construct knots whose twisted Alexander polynomials vanish. Moreover, we show that every faithful irreducible representation of a TAV group causes the twisted Alexander polynomial to be zero.

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

2 major / 2 minor

Summary. The paper defines twisted Alexander vanishing (TAV) groups as finite groups G for which there exists a knot K and a representation such that the twisted Alexander polynomial vanishes. It discusses the orders of such groups, constructs explicit knots realizing vanishing twisted Alexander polynomials, and claims to prove that every faithful irreducible representation of a TAV group causes the twisted Alexander polynomial to vanish.

Significance. If the central claim holds, the result would give a group-theoretic criterion for vanishing of twisted Alexander polynomials, independent of specific knot choice, with the explicit knot constructions providing useful examples. This could aid classification efforts in knot theory and representation varieties, though the significance is tempered by the need to confirm the claim is intrinsic to the group rather than tied to the defining knot.

major comments (2)
  1. [Introduction and §2] Definition of TAV groups (Introduction and §2): the definition is existential over knots (a finite group G is TAV if there exists a knot whose twisted Alexander polynomial vanishes for some representation to G), yet the main theorem asserts the vanishing property for every faithful irreducible representation of G without an explicit argument that vanishing is independent of the particular knot used in the definition. This is load-bearing for the central claim.
  2. [§4] Main theorem on faithful irreps (presumably §4): the argument via the standard chain complex/Fox calculus is carried out for the constructed knots in §3, but does not contain a general step showing that if vanishing holds for one faithful irrep of one knot, it holds for all faithful irreps of any knot admitting a surjection onto the TAV group. The extension from specific constructions to the universal statement requires additional justification.
minor comments (2)
  1. [Abstract] Abstract: the phrasing 'the corresponding twisted Alexander polynomial of a knot' is ambiguous as to whether the definition is existential or requires vanishing for all knots; rephrase for precision.
  2. [Throughout] Notation: ensure consistent use of symbols for the twisted Alexander polynomial (e.g., Δ_{K,ρ}) and representations throughout; add a table summarizing the constructed knots and their TAV groups if not already present.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments, which help clarify the scope and generality of our results on twisted Alexander vanishing groups. We address each major comment below and will revise the manuscript to incorporate the necessary clarifications and additional arguments.

read point-by-point responses

  1. Referee: [Introduction and §2] Definition of TAV groups (Introduction and §2): the definition is existential over knots (a finite group G is TAV if there exists a knot whose twisted Alexander polynomial vanishes for some representation to G), yet the main theorem asserts the vanishing property for every faithful irreducible representation of G without an explicit argument that vanishing is independent of the particular knot used in the definition. This is load-bearing for the central claim.

    Authors: We agree that the definition of a TAV group is existential (there exists at least one knot and representation yielding vanishing), while the main theorem asserts a universal vanishing property for all faithful irreducible representations of G. The current manuscript demonstrates the vanishing explicitly via the chain complex for the specific knots constructed in §3. In the revision, we will add a new lemma (likely in §2 or as a preliminary to §4) proving that the vanishing is independent of the choice of knot: if vanishing holds for one faithful irreducible representation of a knot group surjecting onto G, then it holds for every faithful irreducible representation of G and for any knot admitting such a surjection. This follows from the naturality of the Fox calculus under group homomorphisms and the fact that the relevant homology modules are determined by the representation of G itself. We will revise the introduction and §2 to state this independence explicitly. revision: yes

  2. Referee: [§4] Main theorem on faithful irreps (presumably §4): the argument via the standard chain complex/Fox calculus is carried out for the constructed knots in §3, but does not contain a general step showing that if vanishing holds for one faithful irrep of one knot, it holds for all faithful irreps of any knot admitting a surjection onto the TAV group. The extension from specific constructions to the universal statement requires additional justification.

    Authors: The referee correctly notes that the explicit computation in §4 applies the twisted chain complex and Fox derivatives only to the knots constructed in §3. We will revise §4 by inserting a general proposition that extends the result: for any knot K with a surjection φ from its fundamental group onto a TAV group G, and for any faithful irreducible representation ρ of G, the twisted Alexander polynomial of K with respect to ρ ∘ φ vanishes. The argument uses the induced map on the chain complexes and shows that the determinant vanishes due to the same linear dependence in the representation module that appears in the defining case, independent of the particular knot complement beyond the existence of the surjection. This step was implicit in our reasoning but not fully written out; the revision will make the general case rigorous and self-contained. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation self-contained via standard methods and explicit constructions

full rationale

The paper introduces no new fitted parameters or self-referential equations. It references prior work only to recall the definition of TAV groups and then proceeds with explicit knot constructions together with direct applications of the standard chain complex and Fox calculus to faithful irreducible representations. These steps are independent of the original definition and do not reduce any claimed prediction or vanishing result to the input data by construction. The central statement that every faithful irreducible representation of a TAV group yields a vanishing polynomial is presented as a theorem established inside the paper rather than assumed or renamed from the definition.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The work relies on the prior definition of TAV groups and standard assumptions in algebraic topology and knot theory.

axioms (1)
  • domain assumption Twisted Alexander polynomials are well-defined invariants computed from a knot and a representation of its fundamental group via Fox calculus or twisted chain complexes.
    This is a standard background fact in knot theory invoked throughout the abstract.
invented entities (1)
  • Twisted Alexander vanishing (TAV) group no independent evidence
    purpose: Finite group for which the twisted Alexander polynomial of some knot vanishes
    Defined in the authors' previous work and used as the central object here.

pith-pipeline@v0.9.0 · 5350 in / 1304 out tokens · 58504 ms · 2026-05-14T18:54:18.751599+00:00 · methodology

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Reference graph

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