On slice alternating 3-braid closures and Stein-fillable genus one open books

Brejevs, Vitalijs (2021) On slice alternating 3-braid closures and Stein-fillable genus one open books. PhD thesis, University of Glasgow.

Abstract

This thesis consists of two parts, each concerning a different question about the relationship of 3- and 4-manifolds. The first part is devoted to a generalisation of the slice–ribbon conjecture to non-zero determinant links that can be obtained as closures of alternating 3-braids, while the second investigates the connection between Stein fillings of contact manifolds and positive monodromy factorisations of supporting open books.

In the first part, we obtain a conditional classification of alternating 3-braid closures whose double branched covers are unobstructed from bounding rational homology 4-balls by Donaldson’s theorem. This result has been independently superseded by Jonathan Simone who provided an unconditional classification that consists of five infinite families. Based on Simone’s work, we confirm the generalised slice–ribbon conjecture for four of these families by explicitly constructing ribbon surfaces via band moves. We also show that the remaining family contains infinitely many ribbon links and employ twisted Alexander polynomials to conclude the enumeration of smoothly slice knots that can be given as closures of alternating 3-braids with at most 20 crossings.

In the second part, we apply the theory of surface mapping class groups to the study of symplectic fillings of contact manifolds. For a contact manifold supported by a genus zero open book, there exists a correspondence between Stein fillings of the manifold and factorisations of the monodromy of the open book into positive Dehn twists; this correspondence is known to fail in genera greater than one. We present the results of joint work with Andy Wand in which we exhibit an infinite family of Steinfillable contact manifolds supported by open books with two-holed torus pages whose monodromies do not admit positive factorisations. These are the first known such examples of genus one, and their existence implies that the above correspondence only holds in the planar case. Our proof crucially relies on transverse contact surgery tools developed by James Conway and observations about lantern relations in the mapping class group of the two-holed torus.

• Item Type: Thesis (PhD)
• Qualification Level: Doctoral
• Colleges/Schools: College of Science and Engineering > School of Mathematics and Statistics
• Supervisor's Name: Wand, Dr. Andy and Owens, Dr. Brendan
• Date of Award: 2021
• Depositing User: Theses Team
• Unique ID: glathesis:2021-82659
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 01 Feb 2022 12:22
• Last Modified: 08 Apr 2022 16:54
• Thesis DOI: 10.5525/gla.thesis.82659
• URI: https://theses.gla.ac.uk/id/eprint/82659