The search for exotic knot traces

TL;DR

This paper introduces an algorithm to identify knots with diffeomorphic 3-manifold complements after 0-slope filling, aiding the classification of knot friends and proposing related conjectures.

Contribution

It develops a practical algorithm using SnapPy and Regina to find knot friends and creates a census of such knots, advancing understanding of their properties.

Findings

Identified several knots with diffeomorphic 3-manifold complements after 0-slope filling.

Formulated conjectures about the properties of knot friends.

Provided a comprehensive census of simple knots with friends.

Abstract

Two distinct knots are said to be friends if their complements, filled along the 0-slope, produce diffeomorphic 3-manifolds. In this article, we develop a practical algorithm, implemented using SnapPy and Regina, to search for a friend of a given knot. As an application, we construct a census of simple knots that admit friends and use these data to formulate conjectures about knot friends.