Cutting Knots Down to Size: The Utility of Tangle Decompositions in Topology

Knotting phenomena arise across a wide range of physical and mathematical contexts, from molecular biology to low-dimensional topology. The complexity and variation of knotted structures demands methods that allow for systematic analyses. A tangle decomposition along a planar surface cuts a knot, link, or spatial graph into simpler pieces, making otherwise intractable structures more accessible. Whether the objective is direct (such as computing a knot invariant) or indirect (such as probing the topology of three-manifolds or modeling enzymatic recombination in DNA) the strategy of tangle decomposition provides a methodical framework. Numerous instances illustrate the versatility of tangle decompositions: the Montesinos correspondence of Dehn surgery
with rational tangle replacement, classic tangle methods for analyzing enzymatic actions in biopolymers, computation strategies for knot
polynomials and three-manifold invariants, and the algebraic manifestations of tangles occurring in knot homology theories. 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 collaborators.