Work

Examples of topologically unknotted tori

Year: 2024

Type: article

Abstract: We show that certain smooth tori with group Z \mathbb {Z} in S 4 S^4 have exteriors with standard equivariant intersection forms, and so are topologically unknotted. These include the turned 1-twist-spun tori in the 4-sphere constructed by Boyle, the union of the genus one Seifert surface of Cochran and Davis that has no slice derivative with a ribbon disc, and tori with precisely four critical points whose middle level set is a 2-component link with vanishing Alexander polynomial. This gives evidence towards the conjecture that all Z \mathbb {Z} -surfaces in S 4 S^4 are topologically unknotted, which is open for genus one and two. It is unclear whether these tori are smoothly unknotted, except for tori with four critical points whose middle level set is a split link. The double cover of S 4 S^4 branched along any of these surfaces is a potentially exotic copy of S 2 × S 2 S^2 \times S^2 , and, in the case of turned twisted tori, we show they cannot be distinguished from S 2 × S 2 S^2 \times S^2 using Seiberg–Witten invariants.

Source: Transactions of the American Mathematical Society Series B

DOI: 10.1090/btran/202

Authors: András Juhász, Mark Powell

Institutions: University of Oxford, University of Glasgow

Language: English

Topic: Geometric and Algebraic Topology

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: gold

Estimated APC paid: $2,750

Funder: Engineering and Physical Sciences Research Council