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.

A bridge trisection of a smooth surface in S 4 is a decomposition analogous to a bridge splitting of a link in S 3 . The Kirby-Thompson invariant of a bridge trisection measures its complexity in terms of distances between disc sets in the pants complex of the trisection surface. We give the first significant bounds for the Kirby-Thompson invariant of spun knots. In particular, we show that the Kirby-Thompson invariant of the spun trefoil is 15.

The Meridional Rank Conjecture is an important open problem in the theory of knots and links in S 3 , and asks whether the bridge number of a knot is equal to the minimal number of meridians needed to generate the fundamental group of its complement. In this paper we investigate the extent to which this is a good conjecture for other knotted objects, namely knotted surfaces in S 4 and virtual and welded knots. We develop criteria using tailored quotients of knot groups and the Wirtinger number to establish the equality of bridge number and meridional rank for several large families of examples. On the other hand, we show that the meridional rank of a connected sum of knotted spheres can achieve any value between the theoretical limits, so that either bridge number also collapses, or meridional rank is not equal to bridge number. We conclude with some applications of the Wirtinger number of virtual knots to the classical MRC and to hyperbolic volume of knots in S 3 .

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For any virtual link L = S ∪ T that may be decomposed into a pair of oriented n-tangles S and T , an oriented local move of type T → T ′ is a replacement of T with the n-tangle T ′ in a way that preserves the orientation of L. After developing a general decomposition for the Jones polynomial of the virtual link L = S ∪ T in terms of various (modified) closures of T , we analyze the Jones polynomials of virtual links L 1 , L 2 that differ via a local move of type T → T ′ . Succinct divisibility conditions on V (L 1 ) − V (L 2 ) are derived for broad classes of local moves that include the ∆-move and the double-∆-move as special cases. As a consequence of our divisibility result for the double-∆-move, we introduce a necessary condition for any pair of classical knots to be S-equivalent.