Tangle Functors from Semicyclic Representations

Nathan Druivenga; Charles Frohman; Sanjay Kumar

paper

Tangle Functors from Semicyclic Representations

arXiv:1607.02070

Abstract

Let $q$ be a $2 N$ th root of unity where $N$ is odd. Let $U_{q} (s l_{2})$ denote the quantum group with large center corresponding to the lie algebra $s l_{2}$ with generators $E, F, K$ , and $K^{- 1}$ . A semicyclic representation of $U_{q} (s l_{2})$ is an $N$ -dimensional irreducible representation $ρ : U_{q} (s l_{2}) \to M_{N} (C)$ , so that $ρ (E^{N}) = a I d$ with $a \neq = 0$ , $ρ (F^{N}) = 0$ and $ρ (K^{N}) = I d$ . We construct a tangle functor for framed homogeneous tangles colored with semicyclic representations, and prove that for $(1, 1)$ -tangles coming from knots, the invariant defined by the tangle functor coincides with Kashaev's invariant.

18 pages, 9 figures