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Links on incompressible surfaces and volumes

Corbin Reid

TL;DR

The paper addresses whether hyperbolic volume of link complements can be bounded above by twist number when links project to incompressible surfaces, and shows the answer is negative. It develops a generalised projection framework on surfaces, leveraging fully augmented links and annular Dehn filling to generate weakly generalised alternating links from FALs in ambient manifolds $M=\Sigma\times S^1$ or $(\Sigma\times I)/\phi$. The main construction yields infinite families $\{J_n\}$ with fixed crossing-circle count $c$ for which $\mathrm{Vol}(M\setminus J_n)\to\infty$, thereby refuting linear (or any) upper volume bounds in terms of twist number for incompressible projections; the argument uses layered curves and annular Dehn fillings to control crossings while driving volume. A bounded-case contrast shows that in the trivial mapping torus, a linear bound is possible, highlighting the role of the monodromy $\phi$ (with the genus-2 hyperelliptic involution remaining an open edge case) in determining when such bounds can hold.

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.

Links on incompressible surfaces and volumes

TL;DR

The paper addresses whether hyperbolic volume of link complements can be bounded above by twist number when links project to incompressible surfaces, and shows the answer is negative. It develops a generalised projection framework on surfaces, leveraging fully augmented links and annular Dehn filling to generate weakly generalised alternating links from FALs in ambient manifolds or . The main construction yields infinite families with fixed crossing-circle count for which , thereby refuting linear (or any) upper volume bounds in terms of twist number for incompressible projections; the argument uses layered curves and annular Dehn fillings to control crossings while driving volume. A bounded-case contrast shows that in the trivial mapping torus, a linear bound is possible, highlighting the role of the monodromy (with the genus-2 hyperelliptic involution remaining an open edge case) in determining when such bounds can hold.

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.

Paper Structure

This paper contains 6 sections, 15 theorems, 3 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Fully Augmented Links in General Surfaces
  3. Constructions
  4. Hyperbolicity
  5. Main Results
  6. Bounded Case and Open Questions

Key Result

Lemma 2.8

Let $L$ be a fully augmented link lying in the neighbourhood of an incompressible surface $\Sigma$ in a 3-manifold $M$. Suppose $M\setminus L$ is hyperbolic. Then there exists a projection $\pi(L)$ of $L$ onto $\Sigma$ that is weakly prime.

Figures (7)

  • Figure 1: Crossing circles (red) shown with and without a half-twist in the projection components (blue).
  • Figure 2: Converting a crossing circle to a twist region preserves checkerboard-colourability. It may be seen that the colouring is consistent with or without the half-twist.
  • Figure 3: Ambient isotopy of a pair of curves about handles of an example surface of genus $2$ to lie in a neighbourhood of the surface.
  • Figure 4: Adding curves about a surface such that they bound annuli through the surface.
  • Figure 5: The two constructions of $M$, represented schematically; $L$ and $L'$ in $\Sigma\times S^1$ at left, and $L$ in $(\Sigma\times I)/\phi$ at right.
  • ...and 2 more figures

Theorems & Definitions (43)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Lemma 2.8
  • proof
  • Theorem 2.9
  • ...and 33 more