Sheet number and quandle-colored 2-knot

Abstract

A diagram of a 2-knot consists of a finite number of compact, connected surfaces called sheets. We prove that if a 2-knot admits a non-trivial coloring by some quandle, then any diagram of the 2-knot needs at least four sheets. Moreover, if a 2-knot admits a non-trivial 5- or 7-coloring, then any diagram needs at least five or six sheets, respectively.