On the Bonahon--Wong--Yang invariants of pseudo-Anosov maps

TL;DR

This paper explores the relationship between Bonahon--Wong--Yang invariants and 1-loop invariants for pseudo-Anosov maps, providing proofs, computational methods, and conjectures about their equivalence and transformations at roots of unity.

Contribution

It proves the conjecture for once-punctured torus bundles, introduces efficient computational techniques, and proposes a new Fourier transform relation for descendant invariants.

Findings

BWY invariants coincide with 1-loop invariants at roots of unity for certain cases

Numerical methods enable computation of invariants and asymptotics to high order

Conjecture of Fourier transform relation between descendant invariants

Abstract

We conjecture (and prove for once-punctured torus bundles) that the Bonahon--Wong--Yang invariants of pseudo-Anosov homeomorphisms of a punctured surface at roots of unity coincide with the 1-loop invariant of their mapping torus at roots of unity. This explains the topological invariance of the BWY invariants and how their volume conjecture, to all orders, and with exponentially small terms included, follows from the quantum modularity conjecture. Using the numerical methods of Zagier and the first author, we illustrate how to efficiently compute the invariants and their asymptotics to arbitrary order in perturbation theory, using as examples the $LR$ and the $LLR$ pseudo-Anosov monodromies of the once-punctured torus. Finally, we introduce descendant versions of the 1-loop and BWY invariants and conjecture (and numerically check for pseudo-Anosov monodromies of $L/R$-length at most 5) that they are related by a Fourier transform. This edition includes statements and proofs for roots of unity of all order, even and odd.