ADO invariants directly from partial traces of homological representations

TL;DR

This paper presents a direct homological formula for ADO invariants, expressing them as sums of partial traces of Lawrence type representations, simplifying previous approaches that involved truncations.

Contribution

It introduces a new homological formula for ADO invariants, avoiding the need for truncations and providing a more direct computational method.

Findings

Derived a homological formula for ADO invariants

Expressed invariants as sums of partial traces of Lawrence type representations

Simplified the computation of ADO invariants

Abstract

The ADO invariants are a sequence of non-semisimple quantum invariants coming from the representation theory of the quantum group $U_q(sl(2))$ at roots of unity. Ito showed that these invariants are sums of traces of quotients of homological representations of braid groups (truncated Lawrence representations). In this paper we show a direct homological formula for the ADO invariants, as sums of partial traces of Lawrence type representations, without further truncations.