Complexity, Heegaard diagrams and generalized Dunwoody manifolds

TL;DR

This paper introduces a modified Heegaard complexity measure for 3-manifolds, providing bounds on their Matveev complexity and generalizing Dunwoody manifolds to include various knot and link coverings.

Contribution

It defines a new complexity measure for Heegaard diagrams and manifolds, and extends the class of Dunwoody manifolds to include several important knot and link coverings.

Findings

Upper bounds for Matveev complexity linearly depend on the covering order.

Lower bounds are obtained using homology arguments.

Generalization of Dunwoody manifolds to include cyclic branched coverings and specific knots.

Abstract

We deal with Matveev complexity of compact orientable 3-manifolds represented via Heegaard diagrams. This lead us to the definition of modified Heegaard complexity of Heegaard diagrams and of manifolds. We define a class of manifolds which are generalizations of Dunwoody manifolds, including cyclic branched coverings of two-bridge knots and links, torus knots, some pretzel knots, and some theta-graphs. Using modified Heegaard complexity, we obtain upper bounds for their Matveev complexity, which linearly depend on the order of the covering. Moreover, using homology arguments due to Matveev and Pervova we obtain lower bounds.