A Polynomial Invariant of Strongly Involutive Links

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.