Divides and hyperbolic volumes

TL;DR

This paper uncovers a hyperbolic structure in links of divides, showing their complements can be constructed from hyperbolic 3-manifolds with specific polyhedral decompositions, providing volume bounds that are asymptotically sharp.

Contribution

It reveals a hidden hyperbolic structure in links of divides and relates their volumes to polyhedral decompositions based on divide singularities.

Findings

The link complements can be obtained by Dehn filling a hyperbolic 3-manifold.

The hyperbolic volume bounds are asymptotically sharp.

Polyhedral decompositions depend on the types of double points of the divide.

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.