Links on incompressible surfaces and volumes

TL;DR

This paper demonstrates that for certain classes of links, specifically weakly generalised alternating and fully augmented links on incompressible surfaces, there are no universal upper bounds on hyperbolic volume based on diagram twist number.

Contribution

The authors prove the existence of infinite families of WGA and FAL links on incompressible surfaces with unbounded volume, answering a previously open question.

Findings

No upper volume bounds for WGA links on incompressible surfaces.

No upper volume bounds for FALs on incompressible surfaces.

Counterexamples to the conjecture that volume bounds depend on twist number.

Abstract

We consider volumes of two families of links that have been the focus of recent results on geometry, namely weakly generalised alternating (WGA) links and fully augmented links (FAL). Both have known lower bounds on hyperbolic volume in terms of their diagram combinatorics, but less is known about upper bounds. In fact, Kalfagianni and Purcell recently found a family of WGA knots on a compressible surface for which there can be no upper bounds on volume in terms of twist number. They asked if upper volume bounds always exist on incompressible surfaces. We show the answer is no: we find infinite families of WGA and FALs on incompressible surfaces with no upper bound on volume in terms of twist number.