# Tangle decompositions of alternating link complements

## TL;DR

This paper investigates the decomposition of prime alternating links into prime alternating tangles, refining previous results to identify when such decompositions are diagrammatically visible or correspond to pseudo-Montesinos links.

## Contribution

It refines existing theorems to characterize when prime alternating links can be decomposed into prime alternating tangles, introducing the concept of pseudo-Montesinos links.

## Key findings

- Decomposition is visible in the diagram or the link is a pseudo-Montesinos link.
- Refinement of Menasco and Thistlethwaite's results.
- Characterization of prime alternating links based on their tangle decompositions.

## Abstract

Decomposing knots and links into tangles is a useful technique for understanding their properties. The notion of prime tangles was introduced by Kirby and Lickorish in [3]; Lickorish proved [5] that by summing prime tangles one obtains a prime link. In a similar spirit, summing two prime alternating tangles will produce a prime alternating link, if summed correctly with respect to the alternating property. Given a prime alternating link, we seek to understand whether it can be decomposed into two prime tangles each of which is alternating. We refine results of Menasco and Thistlethwaite to show that if such a decomposition exists either it is visible in an alternating link diagram or the link is of a particular form, which we call a pseudo-Montesinos link.