Estimating distances in simplicial complexes with applications to 3-manifolds and handlebody-knots

TL;DR

This paper investigates distance relations in simplicial complexes related to low-dimensional manifolds, introduces new invariants for 3-manifolds and handlebody-knots based on splitting distances, and proves their stability under stabilizations.

Contribution

It provides bounds on distances in complexes associated with manifolds and introduces new invariants that converge under stabilizations, advancing understanding of manifold topology.

Findings

Bounds on distances in simplicial complexes in terms of components and curve complex distance

Definition of new invariants for 3-manifolds and handlebody-knots using splitting distances

Proof that these invariants stabilize and converge under stabilizations

Abstract

We study distance relations in various simplicial complexes associated with low-dimensional manifolds. In particular, complexes satisfying certain topological conditions with vertices as simple multi-curves. We obtain bounds on the distances in such complexes in terms of number of components in the vertices and distance in the curve complex. We then define new invariants for closed 3-manifolds and handlebody-knots. These are defined using the splitting distance which is calculated using the distance in a simplicial complex associated with the splitting surface arising from the Heegard decompositions of the 3-manifold. We prove that the splitting distances in each case is bounded from below under stabilizations and as a result the associated invariants converge to a non-trivial limit under stabilizations.