Holonomy braidings, biquandles and quantum invariants of links with $SL_2(\mathbb C)$ flat connections

TL;DR

This paper develops a new framework using quandles and biquandles to construct quantum invariants of links with flat $SL_2(C)$ connections, extending previous holonomy braiding concepts.

Contribution

It introduces a general theory for Reshetikhin-Turaev functors with quandle representations, applicable to $U_qsl_2$ and link invariants.

Findings

Constructs invariants of links with quandle representations

Extends holonomy braiding to a broader class of quantum invariants

Provides a unified approach for quantum link invariants with flat connections

Abstract

R. Kashaev and N. Reshetikhin introduced the notion of holonomy braiding extending V. Turaev's homotopy braiding to describe the behavior of cyclic representations of the unrestricted quantum group $U_qsl_2$ at root of unity. In this paper, using quandles and biquandles we develop a general theory for Reshetikhin-Turaev ribbon type functor for tangles with quandle representations. This theory applies to the unrestricted quantum group $U_qsl_2$ and produces an invariant of links with a gauge class of quandle representations.