Measuring Complexity: From Simplicial Distances to 3-Manifold Invariants

Sayantika Mondal (CUNY)

Abstract: In this talk, we explore distance relations among various simplicial complexes associated with a surface, including the well-known curve and pants complexes and some other variants. We show how distances in these complexes can be bounded in terms of their combinatorial complexity—such as the number of components—and the distance in the classical curve complex. Building on these estimates, we apply them to 3-manifold topology: given a Heegaard or handlebody-knot decomposition, we define a new splitting distance invariant based on the associated simplicial complex of the splitting surface. We show that this invariant is bounded below under stabilization and therefore converges, in a suitable sense, to a non-trivial limit. This provides new tools for distinguishing 3-manifolds and handlebody-knots through invariants derived from simplicial complex distances. I will conclude with some open questions and future directions.