Unknotted curves on genus-one Seifert surfaces of Whitehead doubles

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

Abstract

We consider homologically essential simple closed curves on Seifert surfaces of genus-one knots in S3, and in particular those that are unknotted or slice in S3. 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.

Citation

Dey, S., King, V., Shaw, C. T., Tosun, B., & Trace, B. (2024). Unknotted curves on genus-one Seifert surfaces of Whitehead doubles. Pacific Journal of Mathematics, 330(1), 123-156. https://doi.org/10.2140/pjm.2024.330.123

• Journal Article Type: Article
• Acceptance Date: Jan 9, 2024
• Online Publication Date: Jul 22, 2024
• Publication Date: 2024-05
• Deposit Date: Sep 19, 2024
• Publicly Available Date: Sep 20, 2024
• Journal: Pacific Journal of Mathematics
• Electronic ISSN: 0030-8730
• Publisher: Mathematical Sciences Publishers (MSP)
• Peer Reviewed: Peer Reviewed
• Volume: 330
• Issue: 1
• Pages: 123-156
• DOI: https://doi.org/10.2140/pjm.2024.330.123
• Public URL: https://durham-repository.worktribe.com/output/2869813