Inverted state sums, inverted Habiro series, and indefinite theta functions

TL;DR

This paper proves a conjecture relating the Melvin-Morton-Rozansky expansion of colored Jones polynomials to a two-variable series $F_K$, using an inverted R-matrix state sum, and explores connections to $ ext{hat}Z$ invariants and indefinite theta functions.

Contribution

It introduces an inverted R-matrix state sum approach to prove the conjecture for a broad class of links and develops regularized surgery formulas linking $F_K$ and $ ext{hat}Z$.

Findings

Proved the conjecture for all homogeneous braid links and fibered knots up to 10 crossings.

Developed a new perspective on $F_K$ via inverted Habiro series.

Derived expressions of $ ext{hat}Z$ for certain 3-manifolds using indefinite theta functions.

Abstract

S. Gukov and C. Manolescu conjectured that the Melvin-Morton-Rozansky expansion of the colored Jones polynomials can be re-summed into a two-variable series $F_K(x,q)$, which is the knot complement version of the 3-manifold invariant $\hat{Z}$ whose existence was predicted earlier by S. Gukov, P. Putrov and C. Vafa. In this paper we use an inverted version of the R-matrix state sum to prove this conjecture for a big class of links that includes all homogeneous braid links as well as all fibered knots up to 10 crossings. We also study an inverted version of Habiro's cyclotomic series that leads to a new perspective on $F_K$ and discovery of some regularized surgery formulas relating $F_K$ with $\hat{Z}$. These regularized surgery formulas are then used to deduce expressions of $\hat{Z}$ for some plumbed 3-manifolds in terms of indefinite theta functions.