Wirtinger numbers for virtual links

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.

Bridge numbers and meridional ranks of knotted surfaces and welded knots

“…Bridge numbers of virtual and welded knots. Virtual and welded bridge numbers have been studied in [7,8,16,31,30] and the references therein. The definitions of virtual and welded bridge numbers that have appeared in the literature generalize perspectives from Sections 2.1.2, 2.1.3, and 2.1.4 to a virtual knot diagram by ignoring the virtual crossings.…”

“…This demonstrates the effectiveness of the Wirtinger number. The main result of [31] shows that ω(K) = β O (K) for virtual knots. Using a very similar argument, one can deduce that ω(K) = β O (K) for welded knots as well.…”

“…The main result of the second author in [31] equates ω(K) with β O (K) for a virtual knot K. So if the Coxeter and Wirtinger coloring technique imply that µ…”

“…In that work they prove that the Wirtinger number is in fact equal to the bridge number. In [31], the second author extended the Wirtinger number to virtual knots and proved that it equals the virtual bridge (overpass) number. Theorem 1.3 combined with the upper bound coming from regarding the virtual bridge number as the Wirtinger number allows us to compute the exact values of the bridge numbers and meridional ranks of virtual and welded knots admitting diagrams described in Theorem 1.3, which we will call virtual BBKM knots.…”

“…We will end this paper demonstrating how the virtual bridge number is related to the classical Meridional Rank Conjecture and the hyperbolic volume of a knot in S 3 . These applications follow from the proof of the main theorem of [31].…”