Tangle Functors from Semicyclic Representations

TL;DR

This paper constructs a tangle functor from semicyclic representations of quantum groups at roots of unity and shows it recovers Kashaev's knot invariant for certain tangles.

Contribution

It introduces a new tangle functor based on semicyclic representations and proves its equivalence to Kashaev's invariant for knots.

Findings

Constructed a tangle functor for semicyclic representations.

Proved the functor's invariants coincide with Kashaev's for knots.

Extended the understanding of quantum invariants at roots of unity.

Abstract

Let $q$ be a $2N$th root of unity where $N$ is odd. Let $U_q(sl_2)$ denote the quantum group with large center corresponding to the lie algebra $sl_2$ with generators $E,F,K$, and $K^{-1}$. A semicyclic representation of $U_q(sl_2)$ is an $N$-dimensional irreducible representation $\rho:U_q(sl_2)\rightarrow M_N(\mathbb{C})$, so that $\rho(E^N)=aId$ with $a\neq 0$, $\rho(F^N)=0$ and $\rho(K^N)=Id$. 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.