Gordian Distance and Complete Alexander Neighbors

Authors

Ana Wright, University of Nebraska-LincolnFollow

First Advisor

Mark Brittenham

Second Advisor

Alex Zupan

Date of this Version

Spring 5-4-2023

Document Type

Article

Comments

A DISSERTATION Presented to the Faculty of The Graduate College at the University of Nebraska In Partial Fulfilment of Requirements For the Degree of Doctor of Philosophy, Major: Mathematics, Under the Supervision of Professor Mark Brittenham and Professor Alex Zupan. Lincoln, Nebraska: May, 2023

Copyright © 2023 Ana Wright

Abstract

We call a knot K a complete Alexander neighbor if every possible Alexander polynomial is realized by a knot one crossing change away from K. It is unknown whether there exists a complete Alexander neighbor with nontrivial Alexander polynomial. We eliminate infinite families of knots with nontrivial Alexander polynomial from having this property and discuss possible strategies for unresolved cases.

Additionally, we use a condition on determinants of knots one crossing change away from unknotting number one knots to improve KnotInfo’s unknotting number data on 11 and 12 crossing knots. Lickorish introduced an obstruction to unknotting number one, which proves the same result. However, we show that Lickorish’s obstruction does not subsume the obstruction coming from the condition on determinants.

Adviser: Professor Mark Brittenham and Professor Alex Zupan