Unknotted Curves on Seifert Surfaces

TL;DR

This paper characterizes unknotted and slice curves on genus one Seifert surfaces of certain knots, revealing infinite unknotted curves for the figure eight knot and exploring their properties in Whitehead doubles, with implications for 3-manifold topology.

Contribution

It provides a complete characterization of unknotted curves on Seifert surfaces for most twist knots and analyzes their properties in Whitehead doubles, including enumeration and existence results.

Findings

Figure eight knot has infinitely many unknotted essential curves, counted by Fibonacci numbers.

Many twist knots have essential curves that are not braid closures.

Existence of a slice but not unknotted essential curve in certain cases.

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 further investigate our study of unknotted essential curves for arbitrary Whitehead doubles of non-trivial knots, and obtain that there is a precisely one unknotted essential simple closed curve in the interior of the doubles' standard genus one Seifert surface. As a consequence of all these we obtain many new examples of 3-manifolds that bound contractible 4-manifolds.