# Biquandle Module Invariants of Oriented Surface-Links

## TL;DR

This paper introduces biquandle module invariants for oriented surface-links, enhancing existing invariants with algebraic structures that are preserved under Yoshikawa moves, providing new tools for distinguishing surface-links.

## Contribution

It defines biquandle modules as algebraic structures for surface-link invariants, extending biquandle counting invariants and demonstrating their effectiveness.

## Key findings

- Invariants are preserved under Yoshikawa moves.
- Examples show invariants distinguish links beyond Alexander ideals.
- Invariants are computable and provide new distinguishing power.

## Abstract

We define invariants of oriented surface-links by enhancing the biquandle counting invariant using \textit{biquandle modules}, algebraic structures defined in terms of biquandle actions on commutative rings analogous to Alexander biquandles. We show that bead colorings of marked graph diagrams are preserved by Yoshikawa moves and hence define enhancements of the biquandle counting invariant for surface links. We provide examples illustrating the computation of the invariant and demonstrate that these invariants are not determined by the first and second Alexander elementary ideals and characteristic polynomials.