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A Polynomial Invariant of Strongly Involutive Links

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Carlo Collari, Paolo Lisca

Geometric Topology · Submitted 19 Jun 2026 · 29 references

Pith's verdict: The paper defines a new equivariant two-variable polynomial P^e for strongly involutive links via skein relations, generalizes Couture, answers a mod-2 question, and separates some alternating mutant pairs.

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arXiv ID2606.21471 (version 1)
Pith IDV53PPZBT, Pith's permanent record of this paper and its review
CitationCarlo Collari, Paolo Lisca. "A Polynomial Invariant of Strongly Involutive Links." arXiv:2606.21471 (2026). https://pith.science/paper/V53PPZBT
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The paper

The paper claims

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

The reading

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.

What carries the argument

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

Extended reading notes

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.

Load-bearing premise

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

Editorial extensions

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.

Reading between the lines

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.
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.

Figures
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 defined as follows: i(s) = | s | − n−(D) j(s) = h(s) + | s | + n+(D) − 2n−(D). Here, | s | is the number of 1-smoothings in the state underlying s; h(s) is the number of circles labelled
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
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, with corresponding outgoing thin edges e1 and e2, then σ(e) = {σ(e1), σ(e2)}; • if e and e ′ are exchanged by the reflection (x, y) 7→ (x, −y), then σ(e) = σ(e ′ ). The set of…
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
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
Figure 6. Configurations of the outgoing and incoming edges at a vertex on the right of
Figure 7. Colored simple closed curves associated to the state in
Figure 8. The local differences in the diagrams involved in Property (23)
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
Figure 10. The construction of D#vDr
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∈V (Γ′) wt(v, σ′ )   q rot(σ ′ ) = = [N − 1] X x∈XN X σ′∈Se x(Γ′)   Y v∈V (Γ′) wt(v, σ′ )   q rot(σ ′ ) = = [N − 1]⟨ ⟩
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-one correspondence between the states of Γ and the states of Γ′ , which is illustrated in
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
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, and the equality marked by (⋆) follows from the formula: X x∈XN \{y,z} q x+sign(y−x)+sign(z−x) = [N − 2] + δy,zq y , ∀y, z ∈ XN . We can now resume our computation: = [N − 2]⟨ ⟩…

Figures reproduced from arXiv:2606.21471 by the authors.

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