Geometric triangulations of a family of hyperbolic 3-braids

TL;DR

This paper constructs and proves the geometric nature of triangulations for hyperbolic 3-braid complements, providing a direct proof of their hyperbolicity, which was previously shown indirectly.

Contribution

It offers a direct proof that certain hyperbolic 3-braid complements admit geometric triangulations, extending previous indirect results.

Findings

Constructed topological triangulations for $(-2,3,n)$-pretzel knot complements with n≥7.

Proved the triangulations are geometric using Casson and Rivin's theorem.

Provided a direct proof of hyperbolicity for these braids.

Abstract

We construct topological triangulations for complements of $(-2,3,n)$-pretzel knots and links with $n\ge7$. Following a procedure outlined by Futer and Gu\'eritaud, we use a theorem of Casson and Rivin to prove the constructed triangulations are geometric. Futer, Kalfagianni, and Purcell have shown (indirectly) that such braids are hyperbolic. The new result here is a direct proof.