Elucidation of diagrammatic properties of surface-knots and construction of new invariants of surface-knots

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A surface-knot is a closed surface embedded in 4-space, and a diagram of a surface-knots is its projection image into 3-space(whose singularity set is equipped with height information). In this research, we investigate a sheet number of a surface-knot, which is one of its diagramatic properties, by using quandles. Precisely speaking, we show that a sheet number of a 11-colorable 2-knot is at least 7. As an application, we show that spun 6_2 knot has sheet number 7.