On gluing and splitting series invariants of plumbed 3-manifolds

TL;DR

This paper introduces series invariants for plumbed 3-manifolds and knot complements, demonstrating their gluing and splitting properties and providing explicit descriptions for specific cases.

Contribution

It generalizes existing invariants to broader classes of 3-manifolds and establishes their behavior under topological operations.

Findings

Series invariants recover recent key results in the field.

Invariants satisfy gluing and splitting properties.

Explicit descriptions provided for lens spaces and Brieskorn spheres.

Abstract

We study series invariants for plumbed 3-manifolds and knot complements twisted by a root lattice. Our series recover recent results of Gukov-Pei-Putrov-Vafa, Gukov-Manolescu, Park, and Ri and apply more generally to 3-manifolds which are not necessarily negative definite. We show that our series verify certain gluing and splitting properties related to the corresponding operations on 3-manifolds. We conclude with an explicit description of the case of lens spaces and Brieskorn spheres.