Complexity of 3-manifolds obtained by Dehn filling

TL;DR

This paper develops bounds for the complexity of 3-manifolds obtained by Dehn filling, providing methods to estimate their complexity with small gaps, and applies these to specific knot complements.

Contribution

It introduces a new approach to estimate the complexity of Dehn filled 3-manifolds with explicit bounds depending on triangulation size.

Findings

Bounds on complexity of Dehn filled manifolds are established.

Method applied successfully to knot complements like figure eight and pretzel knots.

Complexity gaps are minimized to at most 10 for several families.

Abstract

Let $M$ be a compact 3--manifold with boundary a single torus. We present upper and lower complexity bounds for closed 3--manifolds obtained as even Dehn fillings of $M.$ As an application, we characterise some infinite families of even Dehn fillings of $M$ for which our method determines the complexity of its members up to an additive constant. The constant only depends on the size of a chosen triangulation of $M$, and the isotopy class of its boundary. We then show that, given a triangulation $\mathcal T$ of $M$ with $2$--triangle torus boundary, there exist infinite families of even Dehn fillings of $M$ for which we can determine the complexity of the filled manifolds with a gap between upper and lower bound of at most $13 |\mathcal T| + 7.$ This result is bootstrapped to obtain the gap as a function of the size of an ideal triangulation of the interior of $M$, or the number of crossings of a knot diagram. We also show how to compute the gap for explicit families of fillings of knot complements in the three-sphere. The practicability of our approach is demonstrated by determining the complexity up to a gap of at most 10 for several infinite families of even fillings of the figure eight knot, the pretzel knot $P(-2,3,7)$, and the trefoil.