Handle Slides and Localizations of Categories

TL;DR

This paper introduces a method to evaluate certain categorifications at roots of unity through localization, resulting in handle slide invariant objects that connect to SU(2) quantum invariants at low levels.

Contribution

It presents a novel localization technique for categorifications, enabling evaluation at roots of unity and defining handle slide invariant objects linked to quantum invariants.

Findings

Objects invariant under handle slides are constructed.

Decategorification yields SU(2) quantum invariants at low levels.

The method extends categorification frameworks to roots of unity.

Abstract

We propose a means by which some categorifications can be evaluated at a root of unity. This is implemented using a suitable localization in the context of prior work by the authors on categorification of the Jones-Wenzl projectors. Within this construction we define objects, invariant under handle slides, which decategorify to the SU(2) quantum invariants at low levels.