Bridge numbers and meridional ranks of knotted surfaces and welded knots

Abstract

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 .

Bounds for Kirby-Thompson invariants of knotted surfaces

“…It was shown in [11] that pb; b, c 2 , c 3 q-bridge trisections are completely decomposable. The same was proven for pb; b´1, c 2 , c 3 q-bridge trisections by Joseph, Meier, Miller, and Zupan in [7]. A result of [4] states that if a knotted surface has L-invariant equal to zero, then the bridge trisection is completely decomposable.…”

“…Let F be the k-twist spun of a p2, 2k ´1q-torus knot. By Theorem 1.2 of [7], the meridional rank of F is 2. Moreover, Meier and Zupan were able to produce p4, 2q-bridge trisection diagrams of F (see Figure 22 of [11]), which implies that bpF q " 4.…”