Virtual Bridge Number One Knots

TL;DR

This paper introduces the virtual bridge number and virtual unknotting number invariants for virtual knots, exploring their properties and relationships with classical knot invariants, and demonstrating the existence of infinitely many virtual knots with specific invariant values.

Contribution

It defines new invariants for virtual knots and investigates their properties, including relationships with classical invariants and the existence of infinitely many knots with certain invariant values.

Findings

Virtual bridge number and virtual unknotting number are closely related to classical invariants.

There are infinitely many homotopy classes of virtual knots with virtual bridge number 1.

For each natural number i, there exists a virtual knot homotopic to the unknot with virtual bridge number 1 and virtual unknotting number i.

Abstract

We define the virtual bridge number $vb(K)$ and the virtual unknotting number $vu(K)$ invariants for virtual knots. For ordinary knots $K$ they are closely related to the bridge number $b(K)$ and the unknotting number $u(K)$ and we have $vu(K)\leq u(K), vb(K)\leq b(K).$ There are no ordinary knots $K$ with $b(K)=1.$ We show there are infinitely many homotopy classes of virtual knots each of which contains infinitely many isotopy classes of $K$ with $vb(K)=1.$ In fact for each $i\in \N$ there exists $K$ virtually homotopic (but not virtually isotopic) to the unknot with $vb(K)=1$ and $vu(K)=i.$