New geometric triangulations for twist knot complements

The twist knots are the simplest family of knots in the three-sphere whose complements are hyperbolic manifolds.

Triangulating such knot complements into ideal tetrahedra is particularly convenient for computing several quantum knot invariants, such as the Teichmüller TQFT of Andersen-Kashaev, which was constructed in 2011.

In this talk I will present a new construction of ideal triangulations for the twist knot complements, based on a method of Thurston and Kashaev, with final complexity roughly half the crossing number of the knot.

Moreover, these triangulations happen to be geometric, i.e. they admit an angle structure corresponding to the complete hyperbolic metric.

We used what precedes to compute the Teichmüller TQFT and prove the associated volume conjecture for all twist knots, a part of the project I will only summarize in this talk and cover in more detail in a second talk at the Topology seminar.

Both this talk and the next one are self-contained, but can be enjoyed together to get a more complete picture of our project.

No prerequisites in quantum topology are needed.

(joint project with E. Piguet-Nakazawa and F. Guéritaud)