Wirtinger Numbers for Virtual Links

TL;DR

This paper introduces the Wirtinger number for virtual links, proving its equivalence to the virtual bridge number, and demonstrates its effectiveness in computing bounds and invariants for virtual knots.

Contribution

It establishes the Wirtinger number as an algorithmically computable invariant equal to the virtual bridge number, enabling practical calculations and new knot examples.

Findings

Wirtinger number equals virtual bridge number for virtual links

Computed bounds for virtual bridge numbers and quandle invariants of knots with ≤6 crossings

Discovered new nontrivial virtual bridge number one knots

Abstract

The Wirtinger number of a virtual link is the minimum number of generators of the link group over all meridional presentations in which every relation is an iterated Wirtinger relation arising in a diagram. We prove that the Wirtinger number of a virtual link equals its virtual bridge number. Since the Wirtinger number is algorithmically computable, it gives a more effective way to calculate an upper bound for the virtual bridge number from a virtual link diagram. As an application, we compute upper bounds for the virtual bridge numbers and the quandle counting invariants of virtual knots with 6 or fewer crossings. In particular, we found new examples of nontrivial virtual bridge number one knots, and by applying Satoh's Tube map to these knots we can obtain nontrivial weakly superslice links.