Virtual Bridge Number One Knots

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 $vu(K)\le u(K),vb(K)\le b(K).$ 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 $i\in \mathbb{N}$ there exists $K$ virtually homotopic (but not virtually isotopic) to the unknot with $vb(K)=1$ and $vu(K)=i.$

8 pages, 7 figures

References in corpus (1)

• Cobordism of knots on surfaces