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On a volume invariant of 3-manifolds

Marc Kegel, Arunima Ray, Jonathan Spreer, Em Thompson, Stephan Tillmann

Abstract

This paper investigates a real-valued topological invariant of 3-manifolds called topological volume. For a given 3-manifold M it is defined as the smallest volume of the complement of a (possibly empty) hyperbolic link in M. Various refinements of this invariant are given, asymptotically tight upper and lower bounds are determined, and all non-hyperbolic closed 3-manifolds with topological volume of at most 3.07 are classified. Moreover, it is shown that for all but finitely many lens spaces, the volume minimiser is obtained by Dehn filling one of the cusps of the complement of the Whitehead link or its sister manifold.

On a volume invariant of 3-manifolds

Abstract

This paper investigates a real-valued topological invariant of 3-manifolds called topological volume. For a given 3-manifold M it is defined as the smallest volume of the complement of a (possibly empty) hyperbolic link in M. Various refinements of this invariant are given, asymptotically tight upper and lower bounds are determined, and all non-hyperbolic closed 3-manifolds with topological volume of at most 3.07 are classified. Moreover, it is shown that for all but finitely many lens spaces, the volume minimiser is obtained by Dehn filling one of the cusps of the complement of the Whitehead link or its sister manifold.
Paper Structure (24 sections, 18 theorems, 36 equations, 5 figures, 3 tables)

This paper contains 24 sections, 18 theorems, 36 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related work
  3. Our contributions
  4. Properties, examples and questions
  5. Definition of topological volume and its spectrum
  6. Refinements of topological volume
  7. A finiteness result
  8. Covers and branched covers
  9. Constructions of unbounded topological volume
  10. Bounds on topological volume
  11. Upper bounds from surgery descriptions or triangulations
  12. Lower bounds
  13. The minimal topological volume manifolds
  14. A census
  15. Parenthood
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $M$ be a closed, orientable 3--manifold given by surgery along a nontrivial $n$-component link $L\subset S^3$ with crossing number $c=c(L)$. Then where $v_{oct}\approx3.6638$ and $v_{tet}\approx1.01494$ are the volumes of a regular hyperbolic ideal octahedron and a regular hyperbolic ideal tetrahedron respectively.

Figures (5)

  • Figure 1: Left: The standard surgery diagram of a Seifert fibered space with genus $g$ and $N$ exceptional fibres. The surgery link is a connected sum of $g$ copies of the Borromean rings and $N$ copies of the Hopf link. Right: A hyperbolic link that contains the standard surgery diagram of a Seifert fibered space as a sublink.
  • Figure 2: The surgery on the red unknot yields $L(10,3)$. We can use SnapPy to verify that the complement of the blue knot is isometric to $m003$ and thus represents the minimiser of $L(10,3)$.
  • Figure 3: The manifold $m010$ seen as a knot in $L(2,1) \# L(3,1)$ minimises the volume.
  • Figure 4: The Whitehead link $L5a1$ (left) and its sister link $L13n5885$ (right). The unknotted blue component is the first cusp (which we fill).
  • Figure 5: A surgery diagram of $m034$

Theorems & Definitions (43)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Example 2.1
  • Proposition 2.6
  • proof
  • Example 2.7
  • Lemma 2.8
  • proof
  • Example 2.9
  • ...and 33 more