  We use the rational Witt class of a knot in the 3-sphere as a tool for
addressing questions about its unknotting number. We apply these tools to
several low crossing knots (151 knots with 11 crossing and 100 knots with 12
crossings) and to the family of n-stranded pretzel knots for various values of
n>2. In many cases we obtain new lower bounds and in some cases explicit values
for their unknotting numbers. Our results are mainly concerned with unknotting
number one but we also address, somewhat more marginally, the case of higher
unknotting numbers.
