Yokota type invariants derived from non-integral highest weight representations of $U_q(sl_2)$

TL;DR

This paper introduces Yokota type invariants for colored spatial graphs derived from non-integral highest weight representations of quantum groups, proposing a volume conjecture linking these invariants to hyperbolic polyhedra volumes.

Contribution

It generalizes CM invariants to define Yokota type invariants and formulates a volume conjecture relating these invariants to hyperbolic geometry.

Findings

Numerical verification of the volume conjecture for specific pyramids.

Extension of invariants to non-integral highest weight representations.

Proposal of a new geometric-topological relationship.

Abstract

We define invariants for colored oriented spatial graphs by generalizing CM invariants, which were defined via non-integral highest weight representations of $U_q(sl_2)$. We apply the same method to define Yokota's invariants, and we call these invariants Yokota type invariants. Then we propose a volume conjecture of the Yokota type invariants of plane graphs which relates to volumes of hyperbolic polyhedra corresponding to the graphs, and check it numerically for some square pyramids and pentagonal pyramids.