Complexity of 3-manifolds obtained by Dehn filling

Jaco, William¹ ; Hyam Rubinstein, Joachim² ; Spreer, Jonathan³ ; Tillmann, Stephan³

¹Department of Mathematics, Oklahoma State University, Stillwater, OK, United States
²School of Mathematics and Statistics, The University of Melbourne, Melbourne VIC, Australia
³School of Mathematics and Statistics, The University of Sydney, Sydney NSW, Australia

Algebraic and Geometric Topology, Tome 25 (2025) no. 1, pp. 301-327

Résumé

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 their 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 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 bounds of at most 13|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 3-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.

Détail

DOI : 10.2140/agt.2025.25.301

Keywords: 3-manifold, minimal triangulation, layered triangulation, complexity, Farey tessellation, slope norm

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