Knotted surfaces

Meridional rank and bridge numbers of knotted surfaces and welded knots

| chrisman.76

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

| chrisman.76

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.

Characterization(s) of the Reduced Peripheral System

| chrisman.76

Jean-Baptiste Meilhan (Université Grenoble Alpes)

The reduced peripheral system was introduced by Milnor in the 50’s for the study of links up to link-homotopy, i.e. up to homotopies leaving distinct components disjoint. This invariant, however, fails to classify links up to link-homotopy for links of 4 or more components. The purpose of this paper is to show that the topological information which is detected by Milnor’s reduced peripheral system is actually 4-dimensional. We give a topological characterization in terms of ribbon solid tori in 4-space up to link-homotopy, using a version of Artin’s Spun construction. The proof relies heavily on an intermediate characterization, in terms of welded links up to self-virtualization, providing hence a purely topological application of the combinatorial theory of welded links.