Virtual Bridge Number One Knots

Details Virtual Bridge Number One Knots Virtual Bridge Number One Knots

2007-12-14

arxiv.org/abs/0712.2347v1

Mathematics

Geometric Topology

8 pages, 7 figures

Scientific paper

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 $vu(K)=i.$

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