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Divides and hyperbolic volumes

Ryoga Furutani, Yuya Koda

Abstract

A divide is the image of a proper and generic immersion of a compact $1$-manifold into the $2$-disk. Due to A'Campo's theory, each divide is associated with a link in the 3-sphere. In this paper, we reveal a hidden hyperbolic structure in the theory of links of divides. More precisely, we show that the complement of the link of a divide can be obtained by Dehn filling a hyperbolic $3$-manifold that admits a decomposition into several ideal regular tetrahedra, octahedra and cuboctahedra, where the number of each of those three polyhedra is determined by types of the double points of the divide. This immediately gives an upper bound of the hyperbolic volume of the links of divides, which is shown to be asymptotically sharp. An idea from the theory of Turaev's shadows plays an important role here.

Divides and hyperbolic volumes

Abstract

A divide is the image of a proper and generic immersion of a compact -manifold into the -disk. Due to A'Campo's theory, each divide is associated with a link in the 3-sphere. In this paper, we reveal a hidden hyperbolic structure in the theory of links of divides. More precisely, we show that the complement of the link of a divide can be obtained by Dehn filling a hyperbolic -manifold that admits a decomposition into several ideal regular tetrahedra, octahedra and cuboctahedra, where the number of each of those three polyhedra is determined by types of the double points of the divide. This immediately gives an upper bound of the hyperbolic volume of the links of divides, which is shown to be asymptotically sharp. An idea from the theory of Turaev's shadows plays an important role here.
Paper Structure (5 sections, 12 theorems, 9 equations, 41 figures)

This paper contains 5 sections, 12 theorems, 9 equations, 41 figures.

Table of Contents

  1. Preliminaries
  2. Divides
  3. Shadows
  4. The complements of links associated to divides
  5. Ideal polyhedral decomposition of the 3-manifold $\boldsymbol{ \operatorname{Int} M_P }$

Key Result

Theorem 1

Let $P \subset D$ be a connected prime divide with at least one double point. If $P$ has a double point of Type $6$-$3$, then $L_P$ is the Hopf link. Otherwise, let $n_1$, $n_2$, $n_3$, $n_4$ and $n_5$ be the number of its double points of Types $1$, $2$, $3$, $4$-$2$ and $5$-$3$, respectively. Then Here, $v_{\mathrm{tet}} = 1.014 \ldots$, $v_{\mathrm{oct}} = 3.663 \ldots$ and $v_{\mathrm{cuboct}}

Figures (41)

  • Figure 1: (Left) A perturbation of $P$. (Middle) The preimage $h^{-1} (L_P)$. (Right) A diagram of $L_P$.
  • Figure 2: Local models of a simple polyhedron.
  • Figure 3: A piece $A_2$ of a shadow.
  • Figure 4: A piece $C_{1}$ and its $3$-thickening $M_{C_{1}}$.
  • Figure 5: A polyhedral decomposition of $\pi^{-1} (C_1)$.
  • ...and 36 more figures

Theorems & Definitions (32)

  • Theorem 1
  • Corollary 2
  • Example 1
  • Definition
  • Theorem 1.1: Costantino-Thurston CT08
  • Example 2: Furutani-Koda FK21
  • Definition
  • Definition
  • Definition
  • Proposition 2.1
  • ...and 22 more