A lower bound for the doubly slice genus from signatures

Abstract

The doubly slice genus of a knot in the 3-sphere is the minimal genus among unknotted orientable surfaces in the 4-sphere for which the knot arises as a cross-section. We use the classical signature function of the knot to give a new lower bound for the doubly slice genus. We combine this with an upper bound due to C. McDonald to prove that for every nonnegative integer $N$ there is a knot where the difference between the slice and doubly slice genus is exactly $N$, refining a result of W. Chen which says this difference can be arbitrarily large.