Splittings of Tangles and Spatial Graphs

TL;DR

This paper generalizes previous results on the splitting properties of tangles and spatial graphs, addressing issues in earlier work and establishing broader, more comprehensive theorems.

Contribution

It introduces new, more general splitting theorems for tangles and spatial graphs, improving upon and correcting prior results in the field.

Findings

Established broader splitting criteria for tangles and spatial graphs.

Identified and addressed issues in previous splitting theorems.

Provided new proofs and generalizations for spatial graph decompositions.

Abstract

Menasco proved the surprising result that if $G$ is a reduced, alternating, connected projection of a link $L$ and $G$ is prime then $L$ is prime. This result has been generalized to other classes of links, tangles, and spatial graphs. We draw attention to some issues with previous splitting results about tangles and spatial graphs, and obtain new more general results for tangles and spatial graphs.