G-LINKS INVARIANTS, MARKOV TRACES AND THE SEMI-CYCLIC Uq𝔰𝔩(2)-MODULES

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 of 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 𝔰𝔩(2), 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.