  For a given knot, we study the minimal number of positive eigenvalues of the
double branched cover over spanning surfaces for the knot. The value gives a
lower bound for various genera, the dealternating number and the alternation
number of knots, and we prove that Batson's bound for the non-orientable
4-genus gives an estimate of the value. In addition, we use the value to give a
necessary condition for being quasi-alternating.
