On 2-knots with total width eight

Abstract

A $2$-knot is (the isotopy class of) a $2$-sphere smoothly embedded in $4$-space. The apparent contour of a generic planar projection of a $2$-knot divides the plane into several regions, and to each such region, we associate the number of sheets covering it. The total width of a $2$-knot is defined to be the minimum of the sum of these numbers, where we take the minimum among all generic planar projections of the given $2$-knot. In this paper, we show that a $2$-knot has total width eight if and only if it is an $n$-twist spun $2$-bridge knot for some $n \neq\pm1$.