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

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https://hdl.handle.net/20.500.11811/10178

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##### Citation

Ruppik, B. M. (2022). Casson-Whitney unknotting, Deep slice knots and Group trisections of knotted surface type. PhD Thesis, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn.

Cite as: https://hdl.handle.net/21.11116/0000-000C-7410-F

##### Abstract

In this thesis, we study knotted surfaces in 4-dimensional manifolds from three different but interconnected perspectives.

In the first part, we introduce a measure of 'distance', or rather 'length', between regularly homotopic knotted surfaces in 4-manifolds via the minimal length of a regular homotopy. For knotted surfaces in the 4-sphere, it is natural to consider the smallest number of finger moves and Whitney moves in such a regular homotopy to an unknotted surface, which is defined to be the Casson-Whitney number. This number is closely related, but not equal, to the stabilization number of a surface. We give lower bounds on this unknotting number coming from the fundamental group of the complements and construct explicit regular homotopies for ribbon surfaces. The Casson-Whitney number of surfaces arising from rim surgery is related to the classical unknotting number of the pattern knot. We close the section by considering the Casson-Whitney number of a family of knotted surfaces, each with trefoil knot group, that was originally constructed by Suciu.

In the second part, we move to the relative setting and study surfaces with knotted boundaries properly embedded in non-closed 4-manifolds. We then distinguish such knots in the boundary which have slice disks embedded in a collar (so called shallow slice knots) from knots which are slice in the 4-manifold, but the slice disks always need to go far away from the 3-manifold boundary (so called deep slice knots). In 4-dimensional 1-handlebodies, every slice knot is automatically shallow slice. On the other hand, every 4-dimensional 2-handlebody which is not the 4-ball contains deep slice knots in the boundary. In addition, we compare topological and smooth shallow and deep sliceness. We construct infinitely many non-local null-homotopic knots in the boundary of the (-1)-trace of the left-handed trefoil which are topologically shallow slice, but smoothly deep slice. For this result, topological constructions employing Casson towers and a result of Freedman are contrasted with smooth sliceness obstructions in products of a 3-manifold with an interval. We recall the immersed curve language for bordered Heegaard Floer homology, and use it to reprove special cases of the formulas for the tau-invariants for Whitehead doubles and cables. Various generalizations to links are discussed as well, in particular constructing examples of links for which every proper sublink is shallow slice, but the link as a whole is deep slice. Then we consider 4-manifolds where every knot in the boundary bounds an embedded disk in the interior. On the contrary, we show that every compact oriented topological 4-manifold V with boundary a 3-sphere contains a knot in its boundary that does not bound a null-homologous embedded disk in V, i.e., there exists a knot which is not topologically H-slice in V.

In the third part, we define group trisection of knotted surface type for bridge trisected surfaces in the 4-sphere. This is a decomposition of the fundamental group of the complement of the surface into three group epimorphisms from a punctured sphere group onto three free groups, with a pairwise compatibility condition. This datum not only allows recovering the group, but also the whole bridge trisection of the knotted surface, and we give many examples together with a computer implementation.

In the first part, we introduce a measure of 'distance', or rather 'length', between regularly homotopic knotted surfaces in 4-manifolds via the minimal length of a regular homotopy. For knotted surfaces in the 4-sphere, it is natural to consider the smallest number of finger moves and Whitney moves in such a regular homotopy to an unknotted surface, which is defined to be the Casson-Whitney number. This number is closely related, but not equal, to the stabilization number of a surface. We give lower bounds on this unknotting number coming from the fundamental group of the complements and construct explicit regular homotopies for ribbon surfaces. The Casson-Whitney number of surfaces arising from rim surgery is related to the classical unknotting number of the pattern knot. We close the section by considering the Casson-Whitney number of a family of knotted surfaces, each with trefoil knot group, that was originally constructed by Suciu.

In the second part, we move to the relative setting and study surfaces with knotted boundaries properly embedded in non-closed 4-manifolds. We then distinguish such knots in the boundary which have slice disks embedded in a collar (so called shallow slice knots) from knots which are slice in the 4-manifold, but the slice disks always need to go far away from the 3-manifold boundary (so called deep slice knots). In 4-dimensional 1-handlebodies, every slice knot is automatically shallow slice. On the other hand, every 4-dimensional 2-handlebody which is not the 4-ball contains deep slice knots in the boundary. In addition, we compare topological and smooth shallow and deep sliceness. We construct infinitely many non-local null-homotopic knots in the boundary of the (-1)-trace of the left-handed trefoil which are topologically shallow slice, but smoothly deep slice. For this result, topological constructions employing Casson towers and a result of Freedman are contrasted with smooth sliceness obstructions in products of a 3-manifold with an interval. We recall the immersed curve language for bordered Heegaard Floer homology, and use it to reprove special cases of the formulas for the tau-invariants for Whitehead doubles and cables. Various generalizations to links are discussed as well, in particular constructing examples of links for which every proper sublink is shallow slice, but the link as a whole is deep slice. Then we consider 4-manifolds where every knot in the boundary bounds an embedded disk in the interior. On the contrary, we show that every compact oriented topological 4-manifold V with boundary a 3-sphere contains a knot in its boundary that does not bound a null-homologous embedded disk in V, i.e., there exists a knot which is not topologically H-slice in V.

In the third part, we define group trisection of knotted surface type for bridge trisected surfaces in the 4-sphere. This is a decomposition of the fundamental group of the complement of the surface into three group epimorphisms from a punctured sphere group onto three free groups, with a pairwise compatibility condition. This datum not only allows recovering the group, but also the whole bridge trisection of the knotted surface, and we give many examples together with a computer implementation.