Trisected Rainbows and Braids

Abstract

New explicit procedures for passing among triplane diagrams, braid movies, and braid charts for knotted surfaces in $\mathbb{R}^4$ are presented. To this end, rainbow diagrams, which lie between braid charts and triplanes, are introduced. Inequalities relating the braid index and the bridge index of 2-knots are obtained via these procedures. Another consequence is a 4-dimensional version of the classical result that ``the minimal number of Seifert circles equals the braid index of a link'' due to Yamada. The procedures are exemplified for the spun trefoil, the 2-twist spun trefoil, and other related examples. Of independent interest, an appendix is included that describes a procedure for drawing a triplane diagram for a satellite surface with a 2-sphere companion. Thus, larger families of surfaces for which we know specific triplane diagrams are obtained.