A Polynomial Invariant of Strongly Involutive Links

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.