  We develop new algebraic methods refining the Witt group of linking forms and
Ranicki's torsion algebraic L-groups into double Witt groups and double
L-groups. At each prime ideal of the underlying ring, our double Witt groups
capture infinitely many more integral signatures of the linking form than the
single Witt groups. The double L-groups are an algebraic theory of `double
cobordism', refining L-theory analogously. We exhibit an exact sequence
relating the double L-groups to classical projective L-theory via a double
homology surgery obstruction group.
  The algebraic techniques are applied to high-dimensional knot theory to
define new invariants for the study of doubly-slice knots. In particular we
prove a homomorphism from the n-dimensional double concordance group to a
double L-group, which factors the construction of the Blanchfield form. Some
results of Stoltzfus in this area are reproved and we show that every Seifert
matrix for a doubly-slice knot is hyperbolic.
