Bridge numbers for virtual and welded knots

TL;DR

This paper investigates the virtual and welded bridge numbers of knots using Gauss diagrams and explores their relationships with classical invariants, addressing open questions and proposing methods to distinguish these invariants.

Contribution

It introduces new approaches to compare virtual and welded bridge numbers with classical invariants, utilizing virtual knot invariants like parity and the reduced virtual knot group.

Findings

Welded bridge number is bounded below by the meridional rank of the knot group.

Relations between virtual, welded, and classical bridge numbers are clarified.

Methods to distinguish virtual and welded bridge numbers using invariants are developed.

Abstract

Using Gauss diagrams, one can define the virtual bridge number ${\rm vb}(K)$ and the welded bridge number ${\rm wb}(K),$ invariants of virtual and welded knots with ${\rm wb}(K) \leq {\rm vb}(K).$ If $K$ is a classical knot, Chernov and Manturov showed that ${\rm vb}(K) = {\rm br}(K),$ the bridge number as a classical knot, and we ask whether the same thing is true for welded knots. The welded bridge number is bounded below by the meridional rank of the knot group $G_K$, and we use this to relate this question to a conjecture of Cappell and Shaneson. We show how to use other virtual and welded invariants to further investigate bridge numbers. Among them are Manturov's parity and the reduced virtual knot group $\overline{G}_K$, and we apply these methods to address Questions 6.1, 6.2, 6.3 and 6.5 raised by Hirasawa, Kamada and Kamada in their paper "Bridge presentation of virtual knots," J. Knot Theory Ramifications 20 (2011), no. 6, 881--893.