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

Sayantika Mondal; Puttipong Pongtanapaisan; Hanh Vo

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arXiv · 2505.00815

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

Sayantika Mondal·Puttipong Pongtanapaisan·Hanh Vo

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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.

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BibTeX RIS

Sayantika Mondal, Puttipong Pongtanapaisan, Hanh Vo. 2025-05-01. Estimating distances in simplicial complexes with applications to 3-manifolds and handlebody-knots. https://arxiv.org/abs/2505.00815

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