Meridional rank and bridge numbers of knotted surfaces and welded knots

Jason Joseph (Rice University)

The meridional rank conjecture (MRC) posits that the meridional rank of a classical knot is equal to its bridge number. In this talk we investigate whether or not this is a reasonable conjecture for knotted surfaces and welded knots. In particular, we find criteria to establish the equality of these values for several large families. On the flip side, we examine the behavior of meridional rank of knotted spheres under connected sum, and, using examples first studied by Kanenobu, show that any value between the theoretical limits can be achieved. This means that either the MRC is false for knotted spheres, or that their bridge number fails to be (-1)-additive. This is joint work with Puttipong Pongtanapaisan.

Flattening Knotted Surfaces

Eva Horvat (University of Ljubljana)

A knotted surface \mathcal{K} in the 4-sphere admits a projection to a 2-sphere, whose set of critical points coincides with a hyperbolic diagram of \mathcal{K}. We apply such projections, called flattenings, to define three invariants of embedded surfaces: the width, the trunk and the partition number. These invariants are studied for some families of embedded surfaces.