  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. Cheeger-Gromov'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.
