New Obstructions to Doubly Slicing Knots
نویسنده
چکیده
A knot in the 3-sphere is called doubly slice if it is a slice of an unknotted 2-sphere in the 4-sphere. We give a bi-sequence of new obstructions for a knot being doubly slice. We construct it following the idea of Cochran-Orr-Teichner’s filtration of the classical knot concordance group. This yields a bi-filtration of the monoid of knots (under the connected sum operation) indexed by pairs of half integers. Doubly slice knots lie in the intersection of this bi-filtration. We construct examples of knots which illustrate non-triviality of this bi-filtration at all levels. In particular, these are new examples of algebraically doubly slice knots that are not doubly slice, and many of these knots are slice. CheegerGromov’s von Neumann rho invariants play a key role to show non-triviality of this bi-filtration. We also show that some classical invariants are reflected at the initial levels of this bi-filtration, and obtain a bi-filtration of the double concordance group.
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