Unknotted curves on genus-one Seifert surfaces of Whitehead doubles

Subhankar Dey, Veronica King, Colby T. Shaw, Bülent Tosun and Bruce Trace

Vol. 330 (2024), No. 1, 123–156

DOI: 10.2140/pjm.2024.330.123

Abstract

We consider homologically essential simple closed curves on Seifert surfaces of genus-one knots in $S^3$, and in particular those that are unknotted or slice in $S^3$. We completely characterize all such curves for most twist knots: they are either positive or negative braid closures; moreover, we determine exactly which of those are unknotted. A surprising consequence of our work is that the figure-eight knot admits infinitely many unknotted essential curves up to isotopy on its genus-one Seifert surface, and those curves are enumerated by Fibonacci numbers. On the other hand, we prove that many twist knots admit homologically essential curves that cannot be positive or negative braid closures. Indeed, among those curves, we exhibit an example of a slice but not unknotted homologically essential simple closed curve. We continue our investigation of unknotted essential curves for arbitrary Whitehead doubles of nontrivial knots, and obtain that there is precisely one unknotted essential simple closed curve in the interior of a double’s standard genus-one Seifert surface. As a consequence we obtain many new examples of 3-manifolds that bound contractible 4-manifolds.

Keywords

unknotted curves on Seifert surfaces, contractible 4-manifolds

Mathematical Subject Classification 2010

Primary: 57K33,57K43,32E20, 57K30

Secondary: 57K10

Milestones

Received: 8 August 2023

Revised: 8 January 2024

Accepted: 9 January 2024

Published: 22 July 2024