Braids, fibered knots, and concordance questions

Abstract

Given a knot in $S^3$, one can associate to it a surface diffeomorphism in two different ways. First, an arbitrary knot in $S^3$ can be represented by braids, which can be thought of as diffeomorphisms of punctured disks. Second, if the knot is fibered -- that is, if its complement fibers over $S^1$ -- one can consider the monodromy of the fibration. One can ask to what extent properties of these surface diffeomorphisms dictate topological properties of the corresponding knot. In this article we collect observations, conjectures, and questions addressing this, from both the braid perspective and the fibered knot perspective. We particularly focus on exploring whether properties of the surface diffeomorphisms relate to four-dimensional topological properties of knots such as the slice genus.

33 pages, minor changes. Accepted for publication in the Proceedings Volume of the 2019 Research Collaboration Conference for Women in Symplectic and Contact Geometry and Topology