Biquandle Virtual Brackets

TL;DR

This paper introduces biquandle virtual brackets, a new family of quantum invariants for knots and links that generalize many existing invariants, including classical polynomials and biquandle cocycle invariants, by incorporating virtual crossings.

Contribution

It defines a broad family of invariants called biquandle virtual brackets, unifying and extending previous invariants through skein relations involving virtual crossings.

Findings

Includes all quandle and biquandle 2-cocycle invariants

Encompasses classical skein invariants like Jones and Alexander-Conway

Introduces new invariants using virtual crossings fundamentally

Abstract

We introduce an infinite family of quantum enhancements of the biquandle counting invariant we call biquandle virtual brackets. Defined in terms of skein invariants of biquandle colored oriented knot and link diagrams with values in a commutative ring $R$ using virtual crossings as smoothings, these invariants take the form of multisets of elements of $R$ and can be written in a "polynomial" form for convenience. The family of invariants defined herein includes as special cases all quandle and biquandle 2-cocycle invariants, all classical skein invariants (Alexander-Conway, Jones, HOMFLYPT and Kauffman polynomials) and all biquandle bracket invariants defined in previous work as well as new invariants defined using virtual crossings in a fundamental way, without an obvious purely classical definition.