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.

We thank Lucia Karageorghis of Durham University, who was supported by an LMS summer undergraduate research fellowship, for help finding the band moves for the knots in Proposition 4.2, as part of her study of the doubly slice genus of the prime knots up to 12 crossings. She was aided by the Kirby calculator/KLO; we are grateful to Frank Swenton for creating this excellent tool. We are also grateful for the existence of KnotInfo; we thank Chuck Livingston for help interpreting it and for his insightful comments during the preparation of this article. Finally we thank the referee for several useful suggestions that helped us improve the article.