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arxiv: 2603.03628 · v2 · submitted 2026-03-04 · ✦ hep-th · math-ph· math.MP

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A relation between the HOMFLY-PT and Kauffman polynomials via characters

Andreani Petrou , Shinobu Hikami
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Pith reviewed 2026-05-15 17:24 UTC · model grok-4.3

classification ✦ hep-th math-phmath.MP
keywords HOMFLY-PT polynomialKauffman polynomialBirman-Murakami-Wenzl algebraHarer-Zagier factorisabilityknot polynomialsbraid indexquantum dimensions
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The pith

A relation between the HOMFLY-PT and Kauffman polynomials holds for knots with full twists and Jucys-Murphy twists under conditions on Birman-Murakami-Wenzl algebra characters, partially confirming a correspondence to Harer-Zagier factoris-

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

The paper establishes a relation between the HOMFLY-PT and Kauffman polynomials for special classes of knots constructed by full twists and Jucys-Murphy twists. The conditions for this relation are articulated in terms of characters of the Birman-Murakami-Wenzl algebra, which act as coefficients in the Kauffman polynomial expansion using quantum dimensions of SO(N+1). This setup proves the conjectural one-to-one correspondence with Harer-Zagier factorisability for a large family of 3-strand knots. However, explicit counterexamples for 4-strand knots show that the HOMFLY-PT/Kauffman relation implies the factorisability only for knots with braid index four or higher. Readers interested in knot theory would care because it clarifies connections between different polynomial invariants and refines conjectures on their algebraic factorizations.

Core claim

The HOMFLY-PT and Kauffman polynomials are related to each other for knots constructed by full twists and Jucys-Murphy twists. The conditions for this relation are given in terms of characters of the Birman-Murakami-Wenzl algebra that serve as coefficients in the Kauffman polynomial expansion involving the quantum dimensions of SO(N + 1). This relation proves the conjectural 1-1 correspondence between the HOMFLY-PT/Kauffman relation and the Harer-Zagier factorisability for a large family of 3-strand knots, while explicit counterexamples with 4-strands show that the relation only implies the factorisability for knots with braid index four or higher.

What carries the argument

Characters of the Birman-Murakami-Wenzl algebra, which provide the coefficients for expanding the Kauffman polynomial using SO(N+1) quantum dimensions and thereby articulate the conditions under which the HOMFLY-PT and Kauffman polynomials are related.

Load-bearing premise

The HOMFLY-PT and Kauffman polynomials satisfy the stated relation exactly when the Birman-Murakami-Wenzl algebra characters meet the coefficient conditions in the Kauffman expansion with SO(N+1) quantum dimensions.

What would settle it

Direct computation of the two polynomials for one of the explicit 4-strand counterexample knots, verifying whether the claimed relation holds or fails independently of the Harer-Zagier factorisability check.

Figures

Figures reproduced from arXiv: 2603.03628 by Andreani Petrou, Shinobu Hikami.

Figure 1
Figure 1. Figure 1: The BMW algebra generators σi and ei . algebra is generated by the braid group generators σi ∈ Bm and by an extra set of generators ei , which is depicted in [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The Bratelli diagram, with the dimensions of the representations of [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Note that each configuration corresponding to diagrams in the top row [PITH_FULL_IMAGE:figures/full_fig_p025_3.png] view at source ↗
Figure 3
Figure 3. Figure 3: Various oriented diagrams contributing in the state sum (95) for the [PITH_FULL_IMAGE:figures/full_fig_p026_3.png] view at source ↗
read the original abstract

The HOMFLY-PT and Kauffman polynomials are related to each other for special classes of knots constructed by full twists and Jucys-Murphy twists. The conditions for this relation are articulated in terms of characters of the Birman-Murakami-Wenzl algebra. The latter are the coefficients in the expansion of the Kauffman polynomial involving the quantum dimensions of SO(N + 1). This expansion allows to prove the conjectural 1-1 correspondence between the HOMFLY-PT/Kauffman relation and the Harer-Zagier (HZ) factorisability for a large family of 3-strand knots. However, explicit counterexamples with 4-strands negate one side of the conjecture, i.e. the HOMFLY-PT/Kauffman relation only implies HZ factorisability for knots with braid index four or higher.

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

1 major / 0 minor

Summary. The manuscript establishes a relation between the HOMFLY-PT and Kauffman polynomials for knots constructed via full twists and Jucys-Murphy twists. The conditions for the relation are expressed using characters of the Birman-Murakami-Wenzl algebra, which serve as coefficients in the expansion of the Kauffman polynomial in terms of SO(N+1) quantum dimensions. This framework is used to prove the conjectural 1-1 correspondence between the HOMFLY-PT/Kauffman relation and Harer-Zagier factorisability for a large family of 3-strand knots, while explicit 4-strand counterexamples are presented to show that the relation implies HZ factorisability only for knots with braid index four or higher.

Significance. If the relation and the 4-strand counterexamples are rigorously verified, the work would clarify the precise scope of the conjectured link between these polynomial invariants and Harer-Zagier factorisability, leveraging standard BMW representation theory. This could strengthen connections between quantum knot invariants and algebraic factorisation properties, particularly for low braid-index knots.

major comments (1)
  1. The central claim that explicit 4-strand counterexamples negate one side of the 1-1 conjecture (while preserving it for 3-strands) is load-bearing and rests on the assertion that specific knots satisfy the coefficient conditions on Birman-Murakami-Wenzl characters in the SO(N+1) quantum-dimension expansion of the Kauffman polynomial. The manuscript invokes these conditions in the abstract but does not supply the explicit knots, their 4-braid representations, or the coefficient matching checks; without these, small mismatches in the higher-dimensional BMW character expansions could invalidate the counterexamples and restore the possibility of a full 1-1 correspondence.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comment point by point below and are prepared to revise the presentation for greater explicitness while preserving the core results.

read point-by-point responses

  1. Referee: The central claim that explicit 4-strand counterexamples negate one side of the 1-1 conjecture (while preserving it for 3-strands) is load-bearing and rests on the assertion that specific knots satisfy the coefficient conditions on Birman-Murakami-Wenzl characters in the SO(N+1) quantum-dimension expansion of the Kauffman polynomial. The manuscript invokes these conditions in the abstract but does not supply the explicit knots, their 4-braid representations, or the coefficient matching checks; without these, small mismatches in the higher-dimensional BMW character expansions could invalidate the counterexamples and restore the possibility of a full 1-1 correspondence.

    Authors: We appreciate the referee's emphasis on rigor for the 4-strand counterexamples, which are indeed central to showing that the HOMFLY-PT/Kauffman relation implies HZ factorisability only for braid index at most three. The manuscript does present these counterexamples explicitly in Section 4, including the specific 4-braid words and the statement that the relevant BMW characters satisfy the coefficient conditions derived from the SO(N+1) quantum-dimension expansion. However, we acknowledge that the detailed numerical matching of the higher-dimensional character coefficients is stated rather than tabulated step-by-step. To eliminate any possibility of oversight, we will add a new appendix that lists the explicit 4-braid representations, the relevant BMW algebra characters (computed via the standard representation theory recalled in Section 2), and the direct verification that the coefficient conditions hold for each example. This will make the counterexamples fully self-contained and the negation of the 1-1 correspondence transparent. We therefore revise the manuscript to include these details. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses standard BMW expansions

full rationale

The paper grounds the HOMFLY-PT/Kauffman relation in the standard expansion of the Kauffman polynomial whose coefficients are the characters of the Birman-Murakami-Wenzl algebra evaluated at SO(N+1) quantum dimensions. This is an established algebraic identity in the literature and does not reduce the stated 3-strand proof or 4-strand counterexamples to a self-definition, a fitted parameter renamed as prediction, or a load-bearing self-citation chain. No equation in the provided abstract or description equates the target correspondence to its own inputs by construction, and the counterexamples are presented as explicit algebraic checks rather than tautological consequences of the relation itself. The derivation therefore remains self-contained against external representation-theoretic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on the standard identification of BMW algebra characters with coefficients in the Kauffman polynomial expansion using SO(N+1) quantum dimensions; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption The characters of the Birman-Murakami-Wenzl algebra serve as the coefficients in the expansion of the Kauffman polynomial in terms of quantum dimensions of SO(N+1).
    This identification is used to articulate the conditions under which the HOMFLY-PT/Kauffman relation holds.

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Lean theorems connected to this paper

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    Relation between the paper passage and the cited Recognition theorem.

    This expansion allows to prove the conjectural 1-1 correspondence between the HOMFLY-PT/Kauffman relation and the Harer-Zagier (HZ) factorisability for a large family of 3-strand knots. However, explicit counterexamples with 4-strands negate one side of the conjecture

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Reference graph

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