Non-isotopic Seifert surfaces in the 4-ball and new computational tools for Khovanov homology

Thesis

One of the central unsolved questions in low-dimensional topology is whether there exists an exotic smooth structure on S4 . A related problem, called the unknotting conjecture for surfaces, asks if there exists an embedded, closed, oriented surface in S4 that is exotically knotted. This thesis originates from investigating a special case of the conjecture involving Seifert surfaces. In Chapter 1, we review the background and discuss our results on this special case. The subsequent work branches into three distinct directions, unified by a recurring theme of using algorithms in algebra to prove topological theorems.

In Chapter 2, we extend the known examples of Seifert surfaces that are non-isotopic in D4 . We distinguish these genus one surfaces topologically by examining their symmetrized Seifert forms. This leads to Conway’s theory of topographs and our implementation of Conway’s algorithm to distinguish integral binary quadratic forms up to isomorphism over Z.

In Chapter 3, we describe a computer program that we developed to calculate cobordism maps induced on Khovanov homology, filling a notable gap in existing tools. We present a few applications, including the calculation of cobordism maps for Seifert surfaces for all prime knots up to 10 crossings and the smooth distinction of ribbon disks obtained through symmetries.

Finally, Chapter 4 explores 3-dimensional handlebodies with a subsurface on their boundary and examines when two such handlebodies can be glued along a homeomorphism of the subsurface to form another handlebody. This study leads to an algebraic analogue about amalgamating free groups and to the algorithm of Diao–Feighn.

Authors: Fehér, Z

• DOI: 10.5287/ora-7rrrxnv0g
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: Conway's topographs; unknotting conjecture for surfaces; non-isotopic ribbon disks; isotopy in the 4-ball; graph of groups; Whitehead's cut-vertex lemma; integral binary quadratic forms; cobordism maps in Khovanov homology; non-isotopic Seifert surfaces; Gersten representatives
• Subjects: Forms, Quadratic; Khovanov homology; Amalgams (Group theory); Mathematics; Handlebodies; Knot theory; Software; Isotopies (Topology); Low-dimensional topology; Topology