Bikei Module Invariants of Unoriented Surface-Links

TL;DR

This paper introduces bikei module invariants for unoriented surface-links, extending previous work on oriented links, and demonstrates their effectiveness in distinguishing non-orientable surface-links and orientation choices.

Contribution

It develops new bikei module invariants for unoriented surface-links, enhancing the ability to distinguish non-orientable and differently oriented links beyond existing invariants.

Findings

Bikei module invariants effectively distinguish non-orientable surface-links.

These invariants outperform bikei homset cardinality in certain cases.

Previous biquandle modules can distinguish orientation choices for links.

Abstract

We extend our previous work from arXiv:1903.06863 on biquandle module invariants of oriented surface-links to the case of unoriented surface-links using bikei modules. The resulting infinite family of enhanced invariants proves be effective at distinguishing unoriented and especially non-orientable surface-links; in particular, we show that these invariants are more effective than the bikei homset cardinality invariant alone at distinguishing non-orientable surface-links. Moreover, as another application we note that our previous biquandle modules which do not satisfy the bikei module axioms are capable of distinguishing different choices of orientation for orientable surface-links as well as classical and virtual links.