THE BRAID INDEX OF SURFACE-KNOTS AND QUANDLE COLORINGS

摘要

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.