# The Prevalence of Persistent Tangles

## TL;DR

This paper introduces new methods for constructing persistent tangles in knot diagrams using non-trivial colorings, establishing that knots with such colorings contain persistent tangles, and discusses their non-triviality.

## Contribution

It provides novel techniques for creating persistent tangles based on non-trivial colorings and characterizes when these tangles are non-trivial.

## Key findings

- Any knot with a non-trivial coloring contains persistent tangles.
- New methods for constructing persistent tangles are introduced.
- Discussion on conditions for the non-triviality of persistent tangles.

## Abstract

This article addresses persistent tangles. These are tangles whose presence in a knot diagram forces that diagram to be knotted. We provide new methods for constructing persistent tangles. Our techniques rely mainly on the existence of non-trivial colorings for the tangles in question. Our main result in this article is that any knot admitting a non-trivial coloring gives rise to persistent tangles. Furthermore, we discuss when these persistent tangles are non-trivial.