High Distance Bridge Surfaces

TL;DR

This paper constructs 3-manifolds with links and tangles that have high-distance bridge surfaces of specified genus, intersection points, and boundary properties, advancing understanding of bridge surface complexity.

Contribution

It provides explicit constructions of manifolds with links and tangles featuring high-distance bridge surfaces with controlled genus, intersection points, and boundary conditions.

Findings

Constructed manifolds with high-distance bridge surfaces

Controlled genus and intersection points in the constructions

Applicable to links and tangles with boundary conditions

Abstract

Given integers b, c, g, and n, we construct a manifold M containing a c-component link L so that there is a bridge surface Sigma for (M,L) of genus g that intersects L in 2b points and has distance at least n. More generally, given two possibly disconnected surfaces S and S', each with some even number (possibly zero) of marked points, and integers b, c, g, and n, we construct a compact, orientable manifold M with boundary S \cup S' such that M contains a c-component tangle T with a bridge surface Sigma of genus g that separates the boundary of M into S and S', |T \cap Sigma|=2b and T intersects S and S' exactly in their marked points, and Sigma has distance at least n.