  To each unit complex number with positive imaginary part there is defined a
Tristram-Levine knot signature function. The set of all such signature
functions is linearly independent as a set of functions defined on the set of
all knots. The set of averaged signature functions forms a linearly independent
set of homomorophisms on the knot concordance group. However, for each unit
root of an Alexander polynomial, there is a slice knot with nonvanishing
signature at that root and its conjugate, and nowhere else. These results hold
for knots in all odd dimension.
