Expansion joints in hyperbolic manifolds

TL;DR

This paper studies deformations of hyperbolic manifolds with cone singularities, introducing a method to interpolate between cusped and conformally bounded hyperbolic manifolds, with applications to knot theory and augmented links.

Contribution

It introduces a novel technique for deforming hyperbolic manifolds via cone singularities, expanding the understanding of their geometric structures and applications in knot theory.

Findings

Cone deformations can interpolate between cusped and conformal boundary hyperbolic manifolds.

Upper unknotting tunnels in twisted 2-bridge links can be drilled out via cone deformations.

Structures naturally arise in fully augmented links, providing new examples.

Abstract

Deformations of hyperbolic manifolds through metrics with cone singularities along closed loops were first studied by Thurston as continuous realisations of Dehn fillings. Instead of gluing singular solid tori into rank $2$ cusps, we glue singular $2$-handles into rank $1$ cusps. Our method is to find substructures within which the hyperbolic metric can be `fractured' in a controlled way by direct manipulation of a fundamental polyhedron, changing the cone angle around an ideal arc to interpolate between cusped hyperbolic manifolds and hyperbolic manifolds with conformal surfaces on the visual boundary. As an application, we use cone deformations of a family of arithmetic manifolds derived from the Borromean rings to show that the upper unknotting tunnels of highly twisted $2$-bridge links can be drilled out by cone deformations. We also show that our structures arise naturally in fully augmented links, providing a large family of examples.