Bridge numbers for virtual and welded knots

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.