Geometry and Topology Seminar (1/3)

A tangle decomposition along a Conway sphere breaks a knot or link into simpler pieces, each of which is a two-string tangle. Whether the goal is direct (investigate knot invariants under mutation), indirect (investigate unknotting number), or about something else entirely (Dehn surgery), tangle decompositions can be used to address classic problems in knot theory and three-manifold topology. Numerous instances illustrate the versatility of tangle decompositions: the Montesinos correspondence of Dehn surgery with rational tangle replacement, algebraic manifestations of tangles in knot homology theories, or tangle methods for analyzing enzymatic actions in biopolymers. These themes will be highlighted, and I will discuss recent applications of tangle-based methods in the study of spatial theta curves and three-manifolds arising in my own joint work with multiple collaborators.