The Newton polygon of a recurrence sequence of polynomials and its role in TQFT

TL;DR

This paper proves a combinatorial theorem about the quasi-linear nature of Newton polygons in recurrence sequences of polynomials and explores its applications in classical and quantum topology, especially in the behavior of the A-polynomial and quantum invariants.

Contribution

It introduces a new combinatorial theorem on Newton polygons of recurrence polynomial sequences, combining p-adic number theory with polyhedral and tropical geometry, with applications in topology.

Findings

Newton polygons of recurrence polynomial sequences are quasi-linear

The theorem applies to the behavior of the A-polynomial under filling

Quantum invariants like the Jones polynomial exhibit predictable behavior

Abstract

The paper contains a combinatorial theorem (the sequence of Newton polygons of a reccurent sequence of polynomials is quasi-linear) and two applications of it in classical and quantum topology, namely in the behavior of the $A$-polynomial and a fixed quantum invariant (such as the Jones polynomial) under filling. Our combinatorial theorem, which complements results of Calegari-Walker \cite{CW} and the author \cite{Ga4}, occupies the bulk of the paper and its proof requires the Lech-Mahler-Skolem theorem of $p$-adic analytic number theory combined with basic principles in polyhedral and tropical geometry.