The Montesinos trick for proper rational tangle replacement

TL;DR

This paper extends the Montesinos trick to proper rational tangle replacements, enabling the classification of certain knots with minimal unknotting operations, and provides new insights into their properties.

Contribution

It introduces a version of the Montesinos trick for proper rational tangle replacement and applies it to classify knots with proper rational unknotting number one.

Findings

Knots with proper rational unknotting number one are prime.

Classification of alternating knots with proper rational unknotting number one.

Analysis of Montesinos knots with proper rational unknotting number one.

Abstract

Recently Iltgen, Lewark and Marino introduced the concept of a proper rational tangle replacement and the corresponding notion of the proper rational unknotting number. In this note we derive a version of the Montesinos trick for proper rational tangle replacement and use it to study knots with proper rational unknotting number one. We prove that knots with proper rational unknotting number one are prime and classify the alternating knots with proper rational unknotting number one. We also study Montesinos knots with proper rational unknotting number one.