Biquandles with structures related to virtual links and twisted links

TL;DR

This paper introduces v-structures and t-structures on biquandles to create invariants for virtual and twisted links, providing new tools for distinguishing link types.

Contribution

It develops a method to construct biquandles with these structures and applies them to prove certain virtual links are not equivalent to classical links.

Findings

Bourgoin's twofoil is not a virtual link.

Non-orientable virtual m-foils are distinguished from virtual links.

New biquandle invariants for virtual and twisted links.

Abstract

We introduce two kinds of structures, called v-structures and t-structures, on biquandles. These structures are used for colorings of diagrams of virtual links and twisted links such that the numbers of colorings are invariants. Given a biquandle or a quandle, we give a method of constructing a biquandle with these structures. Using the numbers of colorings, we show that Bourgoin's twofoil and non-orientable virtual $m$-foils do not represent virtual links.