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arxiv: 2606.21471 · v1 · pith:V53PPZBTnew · submitted 2026-06-19 · 🧮 math.GT

A Polynomial Invariant of Strongly Involutive Links

Carlo Collari , Paolo Lisca This is my paper

Pith reviewed 2026-06-26 12:42 UTC · model grok-4.3

classification 🧮 math.GT
keywords strongly involutive linksequivariant skein relationsHOMFLY-PT polynomialmutant knotsspectral sequenceLobb-Watson filtrationCouture invariant
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The pith

Strongly involutive links admit a two-variable polynomial invariant defined by equivariant skein relations as an analogue of the HOMFLY-PT polynomial.

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

The paper introduces the polynomial invariant P^e for strongly involutive links. It is uniquely determined by a collection of equivariant skein relations and functions as an equivariant version of the HOMFLY-PT polynomial. A specialization of P^e recovers the graded Euler characteristic of the third page of the Lobb-Watson G-filtration spectral sequence, thereby generalizing Couture's invariant. After a change of variables, P^e reduces modulo 2 to the HOMFLY-PT polynomial up to a power of the skein variable. The skein relations are then applied to distinguish infinitely many pairs of alternating mutant knots and to show that P^e is strictly stronger than the refined Lobb-Watson invariants on infinitely many strongly invertible knots.

Core claim

The authors define a two-variable polynomial P^e of strongly involutive links that is uniquely characterised by equivariant skein relations. This polynomial generalises Couture's invariant because a specialisation recovers the graded Euler characteristic of the third page of the Lobb-Watson G-filtration spectral sequence. After a change of variables, P^e reduces modulo 2 to the HOMFLY-PT polynomial up to an explicit power of the skein variable. The resulting relations distinguish infinitely many pairs of alternating mutant knots and establish that P^e is strictly stronger than the refined Lobb-Watson invariants on infinitely many strongly invertible knots.

What carries the argument

The two-variable polynomial P^e uniquely characterised by equivariant skein relations.

If this is right

  • A specialisation of P^e equals the graded Euler characteristic of the third page of the Lobb-Watson G-filtration spectral sequence.
  • After a change of variables, P^e reduces modulo 2 to the HOMFLY-PT polynomial times an explicit power of the skein variable.
  • The equivariant skein relations distinguish infinitely many pairs of alternating mutant knots.
  • P^e is strictly stronger than the refined Lobb-Watson invariants on infinitely many strongly invertible knots.

Where Pith is reading between the lines

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

  • The modulo-2 reduction may indicate that similar reductions exist for other link polynomials under involution actions.
  • The ability to separate mutant pairs suggests the invariant could be applied to classify larger families of knots with strong involutions.
  • Further computations on examples beyond the alternating mutants might reveal additional distinctions not captured by existing invariants.

Load-bearing premise

The equivariant skein relations uniquely determine the polynomial invariant P^e for strongly involutive links.

What would settle it

A strongly involutive link for which no assignment of values satisfies the stated equivariant skein relations, or for which the claimed specialisation fails to equal the graded Euler characteristic of the third page of the Lobb-Watson spectral sequence.

Figures

Figures reproduced from arXiv: 2606.21471 by Carlo Collari, Paolo Lisca.

Figure 1
Figure 1. Figure 1: A crossing and its two smoothings: the 0-smoothing (centre) and the 1-smoothing (right). smoothing for each crossing. The Khovanov chain complex CKh(D) is the bi-graded F2-vector space spanned by the enhanced states of D, where an enhanced state is a state whose circles have been labelled either by x+ or by x−. Each enhanced state s has a homological grading i(s) and a quantum grading j(s), which are defin… view at source ↗
Figure 2
Figure 2. Figure 2: The strongly invertible knots (Kn, τn) and (K′ n , τ ′ n ) follows that τn is the only strong inversion of Kn up to equivalence. Now consider the strong involution τ ′ n of K′ n := K(5, 3, 2n, 3, 5) with axis of fixed points shown in the diagram on the right-hand side of [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Examples of thick edges in the equivariant setting We now define the notion of state of Γ. Fix an integer N ≥ 2 and set XN = {−N + 1, −N + 3, . . . , N − 3, N − 1}. An equivariant N-state σ is an assignment of a subset A ⊂ XN to each edge e ∈ E(Γ), subject to the following conditions: • if e ∈ T E(Γ), then |σ(e)| = 2; • if e ∈ E(Γ) \ T E(Γ), then |σ(e)| = 1; • if e is the only thick edge at a vertex v, wit… view at source ↗
Figure 4
Figure 4. Figure 4: An equivariant state of S e 4 (Γ) where V (Γ) is the set of vertices of Γ and wt(v, σ), rot(σ) are, respectively, the weight associated to v and σ and the rotation of σ defined in [21] as follows. The weight of v and σ is given by wt(v, σ) = q 1 2 sign(x2−x1) = ( q 1/2 if x1 < x2 q −1/2 if x1 > x2 , where q is a formal variable and the outgoing/incoming edges are colored as in [PITH_FULL_IMAGE:figures/ful… view at source ↗
Figure 5
Figure 5. Figure 5: Coloring of the outgoing/incoming edges at a vertex order to define rot(σ) observe that at each thick edge we must have one of the configurations of [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Configurations of the outgoing and incoming edges at a vertex on the right of [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Colored simple closed curves associated to the state in [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The local differences in the diagrams involved in Property (23) [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Equivariant isotopy cancelling a crossing. invertible link with diagram De satisfying p on(De) = p on(D), n(De) = n(D) − 1, non(De) = n on(D) − 1 (see [PITH_FULL_IMAGE:figures/full_fig_p022_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The construction of D#vDr [PITH_FULL_IMAGE:figures/full_fig_p026_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The local assignment of an equivariant state σ ∈ Se x,y(Γ) (left) and of the corresponding σ ′ ∈ Se x (Γ′ ) (right). To conclude the proof it suffices to compare the two sides of the equation. Using the easily verified equality P y∈XN \{x} q sign(x−y)+y = [N − 1] we get ⟨ ⟩ e N = X x∈XN X y∈XN \{x} X σ∈Se x,y(Γ)   Y v∈V (Γ) wt(v, σ)   q rot(σ) = = X x∈XN X y∈XN \{x} q sign(x−y)+y X σ′∈Se x(Γ)   Y v∈… view at source ↗
Figure 12
Figure 12. Figure 12: An equivariant state σ ∈ Se (Γ) (left) and the corresponding two equivariant states σ1 (center) and σ2 (right) in S e (Γ′ ). of the local graphs. The second equality can be also proven similarly. We concentrate on the proof of the rightmost, and last, equality. Denote by Γ and Γ′ , respectively, the equivariant MOY graphs on the right, respectively on the left hand side of the equation. There is a two-to-… view at source ↗
Figure 13
Figure 13. Figure 13: The local labelling of a state of Γ (a), and the associated circles (b). Similarly, denote by Γ′ and Γ′′ the strongly involutive MOY graph on first and second sum￾mand, respectively, on the right hand side of the equality. We can partition the equivariant states in S e N (Γ′ ) and S e N (Γ′′) into subsets S ′ a and S ′′ a,c, respectively, where a, c ∈ XN – as in [PITH_FULL_IMAGE:figures/full_fig_p031_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: The local labelling in the states of Γ′ and Γ′′ . We are now ready to prove the statement: ⟨ ⟩ e N = X a, b, c ∈ XN b ̸= a, c X σ∈Sa,b,c   Y v∈V (Γ) wt(v, σ)   q rot(σ) = X a,c∈XN X b ∈ XN b ̸= a, c X σ′′∈S′′ a,c q b+sign(a−b)+sign(c−b)   Y v∈V (Γ′′) wt(v, σ′′)   q rot(σ ′′) (⋆) = X a,c∈XN ([N − 2] + δa,cq a ) X σ′′∈S′′ a,c   Y v∈V (Γ′′) wt(v, σ′′)   q rot(σ ′′) where δi,j is Kronecker’s delta… view at source ↗
read the original abstract

We introduce a new two-variable polynomial invariant \(P^e\) of strongly involutive links, uniquely characterised by equivariant skein relations and naturally viewed as an equivariant analogue of the HOMFLY--PT polynomial. We prove that a specialisation of \(P^e\) recovers the graded Euler characteristic of the third page of the Lobb--Watson \(\mathcal{G}\)-filtration spectral sequence, generalising Couture's polynomial invariant. We further show that, after a change of variables, \(P^e\) reduces modulo \(2\) to the HOMFLY--PT polynomial, up to an explicit power of the skein variable, thereby answering a generalized form of a question of Couture. We use the resulting skein relations to distinguish infinitely many pairs of alternating mutant knots, and show that \(P^e\) is strictly stronger than the refined Lobb--Watson invariants on infinitely many strongly invertible knots.

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 paper introduces a new two-variable polynomial invariant P^e of strongly involutive links, uniquely characterised by equivariant skein relations as an equivariant analogue of the HOMFLY-PT polynomial. It proves that a specialisation of P^e recovers the graded Euler characteristic of the third page of the Lobb-Watson G-filtration spectral sequence (generalising Couture's invariant), shows that after a change of variables P^e reduces modulo 2 to the HOMFLY-PT polynomial up to an explicit power of the skein variable (answering a generalised form of a question of Couture), and demonstrates that the resulting skein relations distinguish infinitely many pairs of alternating mutant knots while P^e is strictly stronger than the refined Lobb-Watson invariants on infinitely many strongly invertible knots.

Significance. If the results hold, the construction supplies a new equivariant polynomial invariant with direct ties to existing spectral sequences and mod-2 reductions, providing independent checks via specialisations. The applications to distinguishing mutant pairs and strengthening invariants on infinite families of strongly invertible knots indicate concrete utility beyond existing tools in the field.

minor comments (2)
  1. Ensure that the normalisation conditions and base cases for the uniqueness theorem are stated explicitly in the section establishing the characterisation of P^e, to facilitate verification of the skein relations.
  2. The change of variables in the mod-2 reduction statement should be written out in full (including the explicit power of the skein variable) in the relevant theorem statement for immediate readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the summary of results, the assessment of significance, and the recommendation to accept. There are no major comments requiring a point-by-point response.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper defines the new invariant P^e directly via equivariant skein relations (standard for HOMFLY-PT analogues) and derives its properties, specializations, and applications from those relations. Uniqueness is asserted as part of the characterizing construction with routine normalization and base cases; no equations reduce by construction to fitted inputs, no load-bearing self-citations appear, and no ansatz or renaming is smuggled in. The derivation chain is self-contained against external knot-theoretic benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The central claim rests on the definition via skein relations and the proofs of its properties, with the uniqueness being a key axiom-like assumption.

free parameters (1)
  • two variables of P^e
    The polynomial is two-variable, but variables are indeterminates, not fitted to data.
axioms (1)
  • ad hoc to paper Existence and uniqueness of polynomial satisfying equivariant skein relations for strongly involutive links
    This is the defining property stated in the abstract.
invented entities (1)
  • P^e polynomial no independent evidence
    purpose: Invariant of strongly involutive links
    Newly defined entity without external falsifiable prediction beyond the claims.

pith-pipeline@v0.9.1-grok · 5677 in / 1376 out tokens · 40462 ms · 2026-06-26T12:42:50.167450+00:00 · methodology

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