Birack Bracket Quivers and Framed Links
Sam Nelson, Haoqi Tom Tang
TL;DR
This work extends birack-based knot invariants by introducing birack brackets as skein-like colorings for framed knots and links, and then categorifies these into birack bracket quivers to yield richer, invariant structures. The authors define state-sum invariants over the birack homset, convert them to polynomials, and further enhance them via quivers whose vertices are colorings weighted by bracket values. Decategorifications of the quivers produce compact invariants (degrees, two-variable polynomials, maximal-path invariants) that can distinguish framings and virtual versus classical knots beyond the Jones polynomial. The paper also raises several open questions on the algebraic structure of brackets and potential functorial connections, suggesting substantial avenues for future research with larger biracks and rings.
Abstract
We introduce birack brackets, skein invariants of birack-colored framed classical and virtual knots and links with values in a commutative unital ring. The multiset of birack bracket values over the homset from a framed link's fundamental birack then forms an invariant of framed links. We then categorify this multiset to define a quiver-valued invariant of framed knots and links. From this quiver we define new polynomial invariants of framed knots and links.