3d-3d correspondence and abelian flat connection

TL;DR

This paper connects homological blocks of knot complements with 3d supersymmetric theories, providing new methods to compute invariants like the colored Jones polynomial through integral expressions and examples.

Contribution

It introduces a novel realization of homological blocks as half-indices of 3d $ ext{N}=2$ theories using inverted Habiro series, extending to general knots.

Findings

Homological blocks expressed as half-indices for specific knots.

Colored Jones polynomial obtained from integral expressions.

Potential extension of methods to all knots.

Abstract

We realize a homological block of a knot complement in $S^3$ for $G_{\mathbb{C}}=SL(2,\mathbb{C})$ as a half-index of a 3d $\mathcal{N}=2$ theory via an expression of the homological block as an inverted Habiro series by working out some examples, which we expect to extend to general knots. Also, by choosing a certain set of poles in the integral expression of the half-index, we obtain the colored Jones polynomial.