On embeddings of $2$-string tangles into the unknot, the unlink and split links

TL;DR

This paper investigates conditions under which 2-string tangles can be embedded into unknots, unlinks, or split links, providing geometric criteria, analyzing tangle sums and colorings, and classifying prime tangles with up to seven crossings.

Contribution

It offers new geometric and combinatorial obstructions for tangle embeddings and classifies prime 2-string tangles with up to seven crossings regarding their embedding properties.

Findings

Characterized when prime 2-string tangles embed into the unknot, unlink, or split links.

Developed geometric and coloring-based obstructions for tangle embeddings.

Provided a complete classification for prime 2-string tangles with up to seven crossings.

Abstract

In this paper we study further when tangles embed into the unknot, the unlink or a split link. In particular, we study obstructions to these properties through geometric characterizations, tangle sums and colorings. As an application we determine when each prime 2-string tangle with up to seven crossings embeds into the unknot, the unlink or a split link.