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arxiv: 2505.10629 · v3 · submitted 2025-05-15 · 🧮 math-ph · math.GT· math.MP

The HZ character expansion and a hyperbolic extension of torus knots

Andreani Petrou , Shinobu Hikami This is my paper

Pith reviewed 2026-05-22 14:13 UTC · model grok-4.3

classification 🧮 math-ph math.GTmath.MP
keywords HZ character expansionhyperbolic knotstorus knotsHOMFLY-PT polynomialpretzel linksYoung diagramsJucys-Murphy twistsfactorisability
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The pith

Hyperbolic knots extend torus knots while preserving HZ factorisability through specific braid operations.

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

The paper applies the Harer-Zagier transform directly to the Schur functions appearing in the character expansion of the HOMFLY-PT polynomial. This produces the HZ character expansion and shows that the resulting rational function is factorisable precisely when only hook-shaped Young diagrams give non-vanishing contributions. The authors then build an infinite family of hyperbolic knots from torus knots by repeated full twists, partial full twists, and Jucys-Murphy twists. Each of these braid operations is shown to preserve the hook-diagram restriction and therefore the factorisability. The family contains pretzel links that arise as Coxeter links for E-type Dynkin diagrams, and a decomposition conjecture for non-factorisable cases is proved when the knot has three strands.

Core claim

An infinite HZ-factorisable family of hyperbolic knots is constructed as a hyperbolic extension of torus knots. The construction proceeds by applying sequences of full twists, partial full twists and Jucys-Murphy twists, all of which preserve the property that only hook-shaped Young diagrams contribute to the HZ character expansion. Among the resulting knots are pretzel links that coincide with the Coxeter links of E-type Dynkin diagrams. When factorisability fails, the HZ function is conjectured to decompose into a sum of factorised terms; this decomposition is established for all three-strand knots and links by using symmetries of Young diagrams.

What carries the argument

The HZ character expansion obtained by applying the Harer-Zagier transform to the Schur-function expansion of the HOMFLY-PT polynomial, with factorisability controlled by the restriction to hook-shaped Young diagrams.

If this is right

  • The family includes pretzel links that are the Coxeter links for E-type Dynkin diagrams.
  • Non-factorisable HZ functions are conjectured to decompose as sums of factorised terms.
  • The decomposition into factorised terms is proved for every three-strand knot and link by using symmetries among Young diagrams.

Where Pith is reading between the lines

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

  • The construction supplies a concrete way to produce new hyperbolic knots whose rational invariants can be read off from the torus-knot data.
  • The appearance of E-type Dynkin diagrams among the pretzel links indicates a possible bridge between knot polynomials and the classification of simple Lie algebras.
  • The same preservation technique may apply to other rational transforms of knot polynomials beyond the Harer-Zagier case.

Load-bearing premise

The braid operations of full twists, partial full twists and Jucys-Murphy twists preserve the property that only hook-shaped Young diagrams contribute non-vanishing terms to the HZ character expansion.

What would settle it

An explicit computation of the HZ character expansion for any knot in the constructed family that finds a non-zero coefficient from a non-hook Young diagram would show that the claimed preservation of factorisability does not hold.

Figures

Figures reproduced from arXiv: 2505.10629 by Andreani Petrou, Shinobu Hikami.

Figure 1
Figure 1. Figure 1: Coxeter link corresponding to a star-shaped Dynkin diagram with [PITH_FULL_IMAGE:figures/full_fig_p027_1.png] view at source ↗
read the original abstract

The HOMFLY-PT polynomial is a two-parameter knot polynomial that admits a character expansion, expressed as a sum of Schur functions over Young diagrams. The Harer-Zagier (HZ) transform, which converts the HOMFLY--PT polynomial into a rational function, can be applied directly to the characters, yielding hence the HZ character expansion. This illuminates the structure of the HZ functions and articulates conditions for their factorisability, including that non-vanishing contributions should come from hook-shaped Young diagrams. An infinite HZ-factorisable family of hyperbolic knots, that can be thought of as a hyperbolic extension of torus knots, is constructed by full twists, partial full twists and Jucys-Murphy twists, which are braid operations that preserve HZ factorisability. Among them, of interest is a family of pretzel links, which are the Coxeter links for E type Dynkin diagrams. Moreover, when the HZ function is non-factorisable, which occurs for the vast majority of knots and links, we conjecture that it can be decomposed into a sum of factorised terms. In the 3-strand case, this is proven using the symmetries of Young diagrams.

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 introduces the Harer-Zagier (HZ) character expansion obtained by applying the HZ transform directly to the Schur-function character expansion of the HOMFLY-PT polynomial. It constructs an infinite family of HZ-factorisable hyperbolic knots, interpreted as a hyperbolic extension of torus knots, via three braid operations (full twists, partial full twists, and Jucys-Murphy twists) asserted to preserve the property that only hook-shaped Young diagrams contribute non-vanishing terms. The paper also identifies a family of pretzel links as Coxeter links associated to E-type Dynkin diagrams and proves a decomposition of non-factorisable HZ functions into factorised terms in the three-strand case using Young-diagram symmetries.

Significance. If the preservation of hook-only contributions under the three braid operations is established for arbitrary strand numbers and twist parameters, the construction would supply an explicit infinite family of hyperbolic knots whose HZ functions factorise, extending known torus-knot results and furnishing concrete examples for the broader conjecture on decompositions of non-factorisable cases.

major comments (2)
  1. [Section describing the braid operations and the infinite family] The central claim that full twists, partial full twists and Jucys-Murphy twists map any HZ-factorisable braid to another whose character expansion receives non-vanishing contributions exclusively from hook diagrams is load-bearing for the infinite hyperbolic family, yet the transformation law for the Schur-function sum under each operation is asserted without an explicit derivation or uniform verification that holds when the number of strands or twist parameters increase (see the construction of the family and the statement that these operations “preserve HZ factorisability”).
  2. [Proof of the three-strand decomposition and its relation to the hyperbolic family] The three-strand decomposition proof using symmetries of Young diagrams does not automatically extend to the multi-strand hyperbolic constructions; an explicit check or inductive argument is required to confirm that the factorisability property survives for the full infinite family.
minor comments (2)
  1. [Introduction and definition of the HZ character expansion] Clarify the precise definition of the HZ transform when applied to individual Schur functions and state the resulting rational function explicitly.
  2. [Examples and pretzel links] Provide at least one explicit low-strand example (e.g., a small pretzel link) with the full character expansion before and after each braid operation to illustrate preservation of the hook-only property.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive feedback. The comments identify areas where additional explicit derivations and arguments will strengthen the presentation of the infinite family and its properties. We address each major comment below and will incorporate the necessary revisions.

read point-by-point responses

  1. Referee: The central claim that full twists, partial full twists and Jucys-Murphy twists map any HZ-factorisable braid to another whose character expansion receives non-vanishing contributions exclusively from hook diagrams is load-bearing for the infinite hyperbolic family, yet the transformation law for the Schur-function sum under each operation is asserted without an explicit derivation or uniform verification that holds when the number of strands or twist parameters increase (see the construction of the family and the statement that these operations “preserve HZ factorisability”).

    Authors: We agree that the preservation property was stated without a full general derivation. In the revised manuscript we will add a dedicated subsection deriving the transformation of the Schur-function character expansion under each of the three braid operations. These derivations will be carried out uniformly for arbitrary strand number and arbitrary twist parameters, explicitly confirming that only hook diagrams survive. This will rigorously justify the construction of the infinite HZ-factorisable hyperbolic family. revision: yes

  2. Referee: The three-strand decomposition proof using symmetries of Young diagrams does not automatically extend to the multi-strand hyperbolic constructions; an explicit check or inductive argument is required to confirm that the factorisability property survives for the full infinite family.

    Authors: We acknowledge that the three-strand decomposition does not automatically carry over. Because the infinite family is generated by iterated application of the braid operations to base HZ-factorisable knots, the explicit transformation laws (to be added) allow an inductive argument: each operation maps a hook-only expansion to another hook-only expansion. We will include this inductive reasoning in the revision, together with direct verification on representative multi-strand examples, thereby confirming factorisability throughout the family. revision: yes

Circularity Check

0 steps flagged

No significant circularity; algebraic construction from Schur-function identities.

full rationale

The derivation begins from the definition of the Harer-Zagier transform applied to the character expansion of the HOMFLY-PT polynomial and proceeds by exhibiting explicit braid operations (full twists, partial full twists, Jucys-Murphy twists) whose action on Young diagrams is shown to map hook diagrams to hook diagrams. This preservation follows directly from the algebraic transformation rules for Schur functions under these operations, not from any fitted parameter or self-referential definition. The 3-strand non-factorisable conjecture is proved separately via Young-diagram symmetries, supplying an independent algebraic verification rather than a load-bearing self-citation. No step reduces a claimed prediction to an input by construction, and the infinite family is generated by iterated application of these operations without circular renaming or imported uniqueness theorems.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the algebraic properties of Schur functions under the Harer-Zagier transform and on the assertion that the listed braid operations map hook diagrams to hook diagrams; no free parameters or new postulated entities are introduced in the abstract.

axioms (2)
  • domain assumption The Harer-Zagier transform applied term-by-term to the character expansion of the HOMFLY-PT polynomial yields a well-defined rational function whose factorisation properties are controlled by the shape of the Young diagrams.
    Invoked in the opening paragraph to define the HZ character expansion.
  • domain assumption Only hook-shaped Young diagrams produce non-vanishing contributions that allow the HZ function to factorise.
    Stated as a condition for factorisability.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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

    hep-th 2026-03 conditional novelty 7.0

    HOMFLY-PT/Kauffman relation via BMW characters proves HZ factorisability for 3-strand knots but fails for 4-strand knots.

Reference graph

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