Complexity of virtual 3-manifolds

TL;DR

This paper introduces a new measure called complexity for virtual 3-manifolds, analyzes its values for specific classes, and determines exact complexities for certain hyperbolic 3-manifolds with geodesic boundaries.

Contribution

It defines the complexity of virtual 3-manifolds and computes its exact values for a broad class of hyperbolic 3-manifolds with totally geodesic boundaries.

Findings

Complexity values for virtual 3-manifolds with one or two 2-components.

Exact complexity values for a wide class of hyperbolic 3-manifolds.

Establishment of the complexity measure as a tool for classifying virtual 3-manifolds.

Abstract

Virtual $3$-manifolds were introduced by S.V. Matveev in 2009 as natural generalizations of the classical $3$-manifolds. In this paper, we introduce a notion of complexity of a virtual $3$-manifold. We investigate the values of the complexity for virtual 3-manifolds presented by special polyhedra with one or two $2$-components. On the basis of these results, we establish the exact values of the complexity for a wide class of hyperbolic $3$-manifolds with totally geodesic boundary.