Spectral geometry, link complements and surgery diagrams

TL;DR

This paper establishes bounds on spectral properties of hyperbolic 3-manifolds using surgery diagrams, linking geometric, topological, and combinatorial aspects, with implications for expansion, complexity, and knot theory.

Contribution

It introduces new bounds on spectral invariants based on surgery data and demonstrates the sharpness of existing theorems through explicit examples.

Findings

Hyperbolic alternating link complements are expanding iff they have bounded volume.

Examples show some hyperbolic 3-manifolds require complex surgery diagrams.

New upper bound on the bridge number of certain knots in terms of diagram twist number.

Abstract

We provide an upper bound on the Cheeger constant and first eigenvalue of the Laplacian of a finite-volume hyperbolic 3-manifold M, in terms of data from any surgery diagram for M. This has several consequences. We prove that a family of hyperbolic alternating link complements is expanding if and only if they have bounded volume. We also provide examples of hyperbolic 3-manifolds which require 'complicated' surgery diagrams, thereby proving that a recent theorem of Constantino and Thurston is sharp. Along the way, we find a new upper bound on the bridge number of a knot that is not tangle composite, in terms of the twist number of any diagram of the knot. The proofs rely on a theorem of Lipton and Tarjan on planar graphs, and also the relationship between many different notions of width for knots and 3-manifolds.