Casson-Whitney unknotting, Deep slice knots and Group trisections of knotted surface type

Advisors: Arunima Ray and Peter Teichner.

PhD thesis published at University of Bonn.

• Published version available from bonndoc
• Mirror hosted on GitHub (.pdf)

Main new results:

• An explicit construction of regular homotopies between rim surgeries (which subsumes some of the previous applications to twist-spun knots).
• A generalization of the fusion number upper bound for the Casson-Whitney number to higher genus ribbon surfaces in the 4-sphere.
• The stabilization number of Suciu's ribbon 2-knots \(R_{k}\) is equal to 1.
• The Casson-Whitney distance of the standard unknotted \(\mathbb{RP}^{2}\) and Suciu's associated \(\mathbb{RP}^{2}\)-knots \(R_{k} \# \mathbb{RP}^{2}\) is equal to 1.
• Partial results of the algebraic Casson-Whitney number of Suciu's knots.
• The (+1)-trace on the right handed trefoil knot contains infinitely many non-local knots in its boundary which exhibit the difference between shallow topological sliceness and smooth deep sliceness. This is an interplay between Freedman's construction of a topological slice for 3-fold iterated positively clasped Whitehead doubles on the one hand, and the Heegaard-Floer \(\tau\)-invariant giving a smooth obstruction for sliceness in \(\partial X \times [0,1]\).
• Group trisections of knotted surface type of the trefoil group, which comes from a bridge trisection of Suciu's ribbon 2-knots.