The braid index of surface-knots and quandle colorings

Abstract

The braid index of a surface-knot $F$ is the minimal number among the degrees of all simple surface braids whose closures are ambient isotopic to $F$. We prove that there exists a surface-knot with braid index $k$ for any positive integer $k$. To prove it, we use colorings of surface-knots by quandles and give lower bounds of the braid index of surface-knots.