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On hyperbolic links associated to Eulerian cycles on ideal right-angled hyperbolic $3$-polytopes

Nikolai Erokhovets

TL;DR

The paper shows that every nonselfcrossing Eulerian cycle in the 4-valent graph of an ideal right-angled hyperbolic $3$-polytope determines a hyperbolic link whose components correspond to the polytope's vertices and whose complement decomposes into four copies of the polytope. A $2$-fold branched cover of this link yields a small cover defined by a Hamiltonian cycle on a simple $3$-polytope, and conversely, Hamiltonian cycles on compact (or finite-volume) right-angled hyperbolic $3$-polytopes give rise to such links under precise turning conditions. The work develops a machine of local transformations—edge-twists and conjugated-vertex moves—that generate all nonselfcrossing Eulerian cycles from antiprisms and relate cycles via linked-edge conjugation, connecting toric-topology constructions with hyperbolic geometry. It also specializes to antprisms, enumerating cycles for $A(3)$ and $A(4)$, and outlines how these constructions extend to broader classes of Pogorelov and almost-Pogorelov polytopes, with implications for the hyperbolic geometry of link complements and their branched covers.

Abstract

We consider Eulerian cycles without transversal selfintersections in $4$-valent planar graphs. We prove that any cycle of this type in the graph of an ideal right-angled hyperbolic $3$-polytope corresponds to a hyperbolic link such that its complement consists of $4$-copies of this polytope glued according to its checkerboard coloring. Moreover, this link consists of trivially embedded circles bijectively corresponding to the vertices of the polytope. For such cycles we prove that the $3$-antiprism $A(3)$ (octahedron) has exactly $2$ combinatorially different cycles, the $4$-antiprism $A(4)$ has exactly $7$ combinatorially different cycles, and these cycles correspond to $7$ cycles (perhaps combinatorially equivalent) on any polytope different from antiprisms, and any antiprism $A(k)$ has at least $2$ combinatorially different cycles. The $2$-fold branched covering space corresponding to our link is a small cover over some simple $3$-polytope. This small cover is defined by a Hamiltonian cycle on it. We show that any Hamiltonian cycle on a compact right-angled hyperbolic $3$-polytope arises in this way, while for a Hamiltonian cycle on a right-angled hyperbolic $3$-polytope of finite volume the necessary and sufficient condition is that at each ideal vertex it does not go straight. We introduce a transformation of a Eulerian cycle along conjugated vertices allowing to build new cycles from a given one.

On hyperbolic links associated to Eulerian cycles on ideal right-angled hyperbolic $3$-polytopes

TL;DR

The paper shows that every nonselfcrossing Eulerian cycle in the 4-valent graph of an ideal right-angled hyperbolic -polytope determines a hyperbolic link whose components correspond to the polytope's vertices and whose complement decomposes into four copies of the polytope. A -fold branched cover of this link yields a small cover defined by a Hamiltonian cycle on a simple -polytope, and conversely, Hamiltonian cycles on compact (or finite-volume) right-angled hyperbolic -polytopes give rise to such links under precise turning conditions. The work develops a machine of local transformations—edge-twists and conjugated-vertex moves—that generate all nonselfcrossing Eulerian cycles from antiprisms and relate cycles via linked-edge conjugation, connecting toric-topology constructions with hyperbolic geometry. It also specializes to antprisms, enumerating cycles for and , and outlines how these constructions extend to broader classes of Pogorelov and almost-Pogorelov polytopes, with implications for the hyperbolic geometry of link complements and their branched covers.

Abstract

We consider Eulerian cycles without transversal selfintersections in -valent planar graphs. We prove that any cycle of this type in the graph of an ideal right-angled hyperbolic -polytope corresponds to a hyperbolic link such that its complement consists of -copies of this polytope glued according to its checkerboard coloring. Moreover, this link consists of trivially embedded circles bijectively corresponding to the vertices of the polytope. For such cycles we prove that the -antiprism (octahedron) has exactly combinatorially different cycles, the -antiprism has exactly combinatorially different cycles, and these cycles correspond to cycles (perhaps combinatorially equivalent) on any polytope different from antiprisms, and any antiprism has at least combinatorially different cycles. The -fold branched covering space corresponding to our link is a small cover over some simple -polytope. This small cover is defined by a Hamiltonian cycle on it. We show that any Hamiltonian cycle on a compact right-angled hyperbolic -polytope arises in this way, while for a Hamiltonian cycle on a right-angled hyperbolic -polytope of finite volume the necessary and sufficient condition is that at each ideal vertex it does not go straight. We introduce a transformation of a Eulerian cycle along conjugated vertices allowing to build new cycles from a given one.

Paper Structure

This paper contains 7 sections, 12 theorems, 3 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Basic facts
  3. Main construction
  4. Edge-twists and the existence of nonselfcrossing Eulerian cycles
  5. Transformations of nonselfcrossing Eulerian cycles
  6. Links associated to Hamiltonian cycles on right-angled $3$-polytopes
  7. Acknowledgements

Key Result

Proposition 2.2

Nonselfcrossing Eulerian cycles in a $4$-valent planar graph $G$ correspond to Hamiltonian cycles in $M(G)$.

Figures (10)

  • Figure 1: Enumeration of nonselfcrossing Eulerian cycles on the octahedron
  • Figure 2: Hyperbolic links corresponding to nonselfcrossing Eulerian cycles on the octahedron
  • Figure 3: Nonselfcrossing Eulerian cycles on the antiprism
  • Figure 4: Enumeration of nonselfcrossing Eulerian cycles on $A(4)$
  • Figure 5: Nonselfcrossing Eulerian cycles on $A(4)$ and the corresponding polytopes
  • ...and 5 more figures

Theorems & Definitions (36)

  • Definition 2.1
  • Proposition 2.2
  • Definition 2.4
  • Definition 2.6
  • Theorem 3.2
  • proof
  • Remark 3.3
  • Remark 3.4
  • Example 3.5
  • Example 3.6
  • ...and 26 more