Biquandles, quivers and virtual bridge indices

TL;DR

This work connects biquandle colorings, quiver enhancements, and virtual-bridge indices $b_1(K)$ and $b_2(K)$ to produce new existence and separation results for virtual knots. By constructing the family $\kappa_{m,n}$ and analyzing colorings via $R_3$ and a tailored biquandle $T$, the authors prove that for any $m\le n$ there exists a virtual knot with $b_1(K)=m$ and $b_2(K)=n$, answering a question of Nakanishi and Satoh. They further develop enhancements, focusing on quiver invariants and column-group data, to show cases where enhancements distinguish links with identical biquandle counting invariants, and provide explicit infinite families of proper enhancements. The paper also explores generalizations to knotted surfaces and discusses comparisons with other invariants, illustrating the practical impact of enhancement-based distinctions beyond counting invariants. Overall, the results deepen understanding of how bridge indices reflect virtual knot complexity and how quiver and column-group enhancements can robustly differentiate virtual links.

Abstract

We investigate connections between biquandle colorings, quiver enhancements, and several notions of the bridge numbers $b_i(K)$ for virtual links, where $i=1,2$. We show that for any positive integers $m \leq n$, there exists a virtual link $K$ with $b_1(K) = m$ and $b_2(K) = n$, thereby answering a question posed by Nakanishi and Satoh. In some sense, this gap between the two formulations measures how far the knot is from being classical. We also use these bridge number analyses to systematically construct families of links in which quiver invariants can distinguish between links that share the same biquandle counting invariant.

Figures (10)

• Figure 1: (Left) Each crossing of negative type adds the relations $z=y\overline{\triangleright}x$ and $w=x\underline{\triangleright}y$. (Right) Each crossing of positive type adds the relations $z=y\overline{\triangleright}x$ and $w=x\underline{\triangleright}y$.
• Figure 2: A complete Wirtinger coloring sequence demonstrated on $\kappa_{2,0}$.
• Figure 3: The diagram has more than one overpasses, but one seed generates a complete Wirtinger coloring sequence, so $b_1(\kappa)=1$.
• Figure 4: A Montesinos link.
• Figure 5: A twist box means a string of crossings.

Theorems & Definitions (32)

• Theorem 1.1
• Lemma 1.1
• Definition 2.1
• Example 2.2
• Example 2.3
• Definition 2.4
• Definition 2.5
• Proposition 2.6
• proof
• Proposition 2.7