G-links invariants, Markov traces and the semi-cyclic Uqsl2-modules

TL;DR

This paper develops new invariants for G-links using semi-cyclic representations of quantum groups, extending classical invariants like the Alexander polynomial through a modified Markov trace.

Contribution

It adapts and renormalizes the Reshetikhin-Turaev construction to define G-link invariants via semi-cyclic modules of non-restricted quantum sl2, introducing a novel approach.

Findings

Semi-cyclic invariants extend the Alexander polynomial.

The invariants are computed explicitly for examples.

Results suggest the invariants are actually equal.

Abstract

Kashaev and Reshetikhin proposed a generalization of the Reshetikhin-Turaev link invariant construction to tangles with a flat connection in a principal G-bundle over the complement of the tangle. The purpose of this paper is to adapt and renormalize their construction to define invariants of G-links using the semi-cyclic representations of the non-restricted quantum group associated to sl2, defined by De Concini and Kac. Our construction uses a modified Markov trace. In our main example, the semi-cyclic invariants are a natural extension of the generalized Alexander polynomial invariants defined by Akutsu, Deguchi, and Ohtsuki. Surprisingly, direct computations suggest that these invariants are actually equal.