Partially Multiplicative Biquandles and Handlebody-Knots

TL;DR

This paper introduces new algebraic structures related to handlebody-knots, enabling the creation of computable invariants and polynomial invariants for classifying handlebody-knots and links.

Contribution

It defines partially multiplicative biquandles and related structures, extending biquandle theory to handlebody-knots and providing new tools for their invariants.

Findings

Defined new algebraic structures for handlebody-knots

Developed coloring invariants for spatial graphs

Constructed polynomial invariants using group enhancements

Abstract

We introduce several algebraic structures related to handlebody-knots, including $G$-families of biquandles, partially multiplicative biquandles and group decomposable biquandles. These structures can be used to color the semiarcs in $Y$-oriented spatial trivalent graph diagrams representing $S^1$-oriented handlebody-knots to obtain computable invariants for handlebody-knots and handlebody-links. In the case of $G$-families of biquandles, we enhance the counting invariant using the group $G$ to obtain a polynomial invariant of handlebody-knots.