The nonorientable four-genus of knots

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.

20 pages; expository changes

Cited by in corpus (10)

• The reduced knot Floer complex
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• The non-orientable 4-genus for knots with 8 or 9 crossings
• Concordance of knots in $S^1\times S^2$
• On a Nonorientable Analogue of the Milnor Conjecture
• Chiral smoothings of knots
• Non-orientable genus of a knot in punctured $\mathbb{C}{P}^2$
• On the nonorientable four-ball genus of torus knots
• Non-orientable 4-genus of torus knots
• Non-orientable genus of knots in punctured Spin 4-manifolds