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

TL;DR

The paper develops lower bounds for distances in several admissible multi-curve complexes defined on splitting surfaces of 3-manifolds and shows these bounds depend only on curve-complex distances and the number of components in each vertex. It then uses these bounds to define splitting-distance-based invariants for closed 3-manifolds (via Hatcher-Thurston cut systems) and for handlebody-knots (via the dual curve complex), proving that these invariants converge under stabilizations to nontrivial limits. A key technical contribution is establishing that splitting distances satisfy universal lower bounds, which implies stability and invariance of the limiting quantities. The results connect 2D surface-complex geometry with 3-manifold topology, enabling robust, stabilization-insensitive invariants that capture the complexity of splittings and knotted handlebodies in 3-manifolds.

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.

Figures (12)

• Figure 1: Two elementary moves
• Figure 2: Distance $3$ on genus $2$ surface
• Figure 3: Distance $6$ on genus $3$ surface
• Figure 4: Given $P$, constructing $P'$ such that $P$ and $P'$ fill $S$.
• Figure 5: A simple move

Theorems & Definitions (34)

• Remark 2.1
• Theorem 3.1
• proof
• Lemma 3.2
• proof : Proof of Lemma \ref{['lemma k <= N+1']}
• Proposition 3.3
• proof
• Remark 3.4
• Definition 3.5
• Theorem 3.6