Biquandle Virtual Brackets
Sam Nelson, Kanako Oshiro, Ayaka Shimizu, Yoshiro Yaguchi

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 using virtual crossings as smoothings, these invariants take the form of multisets of elements of 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.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsGeometric and Algebraic Topology · Advanced Combinatorial Mathematics · Algebraic structures and combinatorial models
