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.
Shin SATOH. "Sheet number and quandle-colored 2-knot." J. Math. Soc. Japan 61 (2) 579 - 606, April, 2009. https://doi.org/10.2969/jmsj/06120579