Geometric limits of knot complements, II: Graphs determined by their complements

TL;DR

This paper investigates the geometric properties of knot complements, showing that some compact submanifolds in the 3-sphere cannot be approximated by hyperbolic knot complements, revealing limitations in geometric limits.

Contribution

It demonstrates the existence of compact submanifolds in the 3-sphere that are not homeomorphic to any geometric limit of hyperbolic knot complements, advancing understanding of geometric limits.

Findings

Certain compact submanifolds are not homeomorphic to geometric limits of hyperbolic knot complements

Limits of hyperbolic knot complements do not encompass all submanifolds in the 3-sphere

The results highlight limitations in the geometric approximation of 3-manifolds

Abstract

We prove that there are compact submanifolds of the 3-sphere whose interiors are not homeomorphic to any geometric limit of hyperbolic knot complements.