Computing Matveev's complexity via crystallization theory: the boundary case

TL;DR

This paper extends a combinatorial method called Gem-Matveev complexity to 3-manifolds with boundary, showing it matches a known complexity measure and providing estimates for specific classes like Seifert manifolds and torus knot complements.

Contribution

It generalizes Gem-Matveev complexity to boundary cases and proves its equivalence with modified Heegaard complexity for certain 3-manifolds.

Findings

Gem-Matveev complexity coincides with modified Heegaard complexity for boundary-irreducible manifolds.

Provides complexity estimates for Seifert 3-manifolds with base D^2 and two exceptional fibers.

Applies to all torus knot complements.

Abstract

The notion of Gem-Matveev complexity has been introduced within crystallization theory, as a combinatorial method to estimate Matveev's complexity of closed 3-manifolds; it yielded upper bounds for interesting classes of such manifolds. In this paper we extend the definition to the case of non-empty boundary and prove that for each compact irreducible and boundary-irreducible 3-manifold it coincides with the modified Heegaard complexity introduced by Cattabriga, Mulazzani and Vesnin. Moreover, via Gem-Matveev complexity, we obtain an estimation of Matveev's complexity for all Seifert 3-manifolds with base $\mathbb D^2$ and two exceptional fibers and, therefore, for all torus knot complements.