The nonorientable four-genus of knots

Patrick M. Gilmer; Charles Livingston

doi:10.1112/jlms/jdr024

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arXiv · 1005.5473

The nonorientable four-genus of knots

Patrick M. Gilmer·Charles Livingston

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Abstract

We develop obstructions to a knot K in the 3-sphere bounding a smooth punctured Klein bottle in the 4-ball. The simplest of these is based on the linking form of the 2-fold branched cover of the 3-sphere branched over K. Stronger obstructions are based on the Ozsvath-Szabo correction term in Heegaard-Floer homology, along with the G-signature theorem and the Guillou-Marin generalization of Rokhlin's theorem. We also apply Casson-Gordon theory to show that for every n greater than one there exists a knot that does not bound a topologically embedded nonorientable ribbon surface F in the 4-ball with first Betti number less than n.

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Patrick M. Gilmer, Charles Livingston. 2011-02-06. The nonorientable four-genus of knots. https://doi.org/10.1112/jlms%2Fjdr024

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