Mixed-platonic 3-manifolds

TL;DR

This paper introduces mixed-platonic 3-manifolds, a new class of cusped hyperbolic 3-manifolds built from multiple types of ideal polyhedra, and explores their properties, including the absence of hidden symmetries in certain cases.

Contribution

It defines mixed-platonic 3-manifolds, establishes foundational properties, and proves the non-existence of hidden symmetries in some hyperbolic knot complements.

Findings

Mixed-platonic manifolds are composed of multiple types of ideal polyhedra.

There are no mixed-platonic hyperbolic knot complements with hidden symmetries.

Basic properties of mixed-platonic manifolds are established.

Abstract

We introduce a class of cusped hyperbolic $3$-manifolds that we call mixed-platonic, composed of regular ideal hyperbolic polyhedra of more than one type, which includes certain previously-known examples. We establish basic facts about mixed-platonic manifolds which allow us to conclude, among other things, that there is no mixed-platonic hyperbolic knot complement with hidden symmetries.