Categorification of Biquandle Arrow Weight Invariants via Quivers

TL;DR

This work categorifies the biquandle arrow weight invariant by associating to each oriented knot a biquandle coloring quiver $BCB_{X,S}(K)$, whose vertices correspond to $X$-colorings and edges arise from endomorphisms in $S$. Weighting the quiver with the arrow-sum $\Sigma_D$ yields a Reidemeister-invariant $\mathcal{Q}_{X,W,S}(K)$, whose decategorifications produce new polynomial invariants such as $\Phi_{X,W,S}^{\mathrm{deg}_+}(K)$ and $\Phi_{X,W,S}^{2}(K)$; a quotient quiver $\mathcal{Q}^Q_{X,W,S}(K)$ gives the loop polynomial $\Phi_{X,W,S}^{Q,L}(K)$; the constructions are invariants under Reidemeister moves and depend on the data $(X,A,W,S)$, giving an infinite family of knot invariants. The paper provides examples showing the enhanced discriminative power over unweighted invariants and discusses the potential for larger biquandles and coefficient groups to unlock the full power of this approach. It also raises questions about connections to cocycle quivers, the algebraic meaning of arrow weights, and the scope for additional decategorifications and functorial enhancements.

Abstract

Introduced in arXiv:2211.12606, biquandle arrow weight invariants are enhancements of the biquandle counting invariant for oriented virtual and classical knots defined from biquandle-colored Gauss diagrams using a tensor over an abelian group satisfying certain properties. In this paper we categorify the biquandle arrow weight polynomial invariant using biquandle coloring quivers, obtaining new infinite families of polynomial invariants of oriented virtual and classical knots.

Theorems & Definitions (18)

• Theorem 1
• Example 1
• Proposition 2
• proof
• Definition 1
• Proposition 3
• proof
• Definition 2
• Proposition 4
• proof