Invariants for trivalent tangles and handlebody-tangles

TL;DR

This paper introduces a method to construct invariants for trivalent tangles and handlebody-tangles using enhanced trivalent tangles and classical knot theory, advancing the understanding of these complex topological structures.

Contribution

It provides a novel recipe for creating invariants of trivalent tangles and handlebody-tangles by combining enhanced tangles with classical knot theory techniques.

Findings

Constructed invariants for knotted trivalent graphs.

Extended invariants to handlebody-tangles and handlebody-links.

Demonstrated the effectiveness of the method in topological classification.

Abstract

An enhanced trivalent tangle is a trivalent tangle with some of its edges labeled. We use enhanced trivalent tangles and classical knot theory to provide a recipe for constructing invariants for trivalent tangles, and in particular, for knotted trivalent graphs. Our method also yields invariants of, what we refer to as, enhanced handlebody-tangles and enhanced handlebody-links.