A HOMFLYPT-type invariant for pseudo links via a resolution in Hecke algebras

TL;DR

This paper introduces a new HOMFLYPT-type polynomial invariant for pseudo links, constructed via a resolution homomorphism in pseudo Hecke algebras, extending classical invariants to pseudo links with missing crossing information.

Contribution

It develops a novel algebraic framework and a resolution-based approach to define polynomial invariants for pseudo links, overcoming previous obstructions related to pseudo Reidemeister moves.

Findings

The invariant satisfies a natural pseudo skein relation.

It admits a state-sum formulation over classical resolutions.

The invariant is characterized skein-theoretically by its values on classical links.

Abstract

Pseudo links generalize classical links by allowing crossings with missing over/under information, called pre-crossings. While the pseudo braid framework provides an algebraic description of pseudo links via a Markov-type theorem, the construction of polynomial invariants using Hecke algebra techniques is obstructed by the presence of the pseudo Reidemeister 1 move. In this paper, we construct a HOMFLYPT-type invariant for oriented pseudo links via the pseudo Hecke algebra of type \(A\). The construction is based on a resolution homomorphism that maps each pseudo generator to a linear combination of a braid generator and its inverse, interpreting pre-crossings as algebraic superpositions of classical crossings. Composing this map with the Ocneanu trace and applying a suitable normalization yields an invariant satisfying a natural pseudo skein relation. We further show that the invariant admits a state-sum formulation as a weighted sum of classical HOMFLYPT-type invariants over all classical resolutions of the pseudo crossings, as well as a skein-theoretic characterization in terms of its values on classical links and the pseudo skein relation.