Invariants of Handlebody-Knots via Yokota's Invariants

TL;DR

This paper introduces quantum invariants for handlebody-knots in the 3-sphere, constructed as sums of Yokota's invariants, and demonstrates their relation to Witten-Reshetikhin-Turaev invariants through explicit calculations.

Contribution

The paper develops new quantum invariants for handlebody-knots using Yokota's invariants and establishes their connection to Witten-Reshetikhin-Turaev invariants.

Findings

Computed invariants for genus 2 handlebody-knots up to six crossings

Showed invariants are special cases of Witten-Reshetikhin-Turaev invariants

Provided a table of calculated invariants for specific handlebody-knots

Abstract

We construct quantum $\mathcal{U}_q(\mathfrak{sl}_{\,2})$ type invariants for handlebody-knots in the 3-sphere $S^3$. A handlebody-knot is an embedding of a handlebody in a 3-manifold. These invariants are linear sums of Yokota's invariants for colored spatial graphs which are defined by using the Kauffman bracket. We give a table of calculations of our invariants for genus 2 handlebody-knots up to six crossings. We also show our invariants are identified with special cases of the Witten-Reshetikhin-Turaev invariants.