Trisected Rainbows and Braids

TL;DR

This work introduces rainbow diagrams as a flexible, intermediate representation tying together triplane diagrams, braid charts, and braided-band unlink movies for knotted surfaces in $\mathbb{R}^4$. It proves a fundamental chain of inequalities linking braid, rainbow, and bridge indices, and develops three robust procedures (Alexander’s method, Morton’s shadow threading, and Yamada’s Seifert-circle trading) to convert triplane diagrams into weak rainbows and vice versa. The authors extend the framework to satellites and spun knots, establishing exact equalities in favorable cases (e.g., BB knots) and deriving the rainbow-number behavior under satellite operations, including cables. They also provide explicit algorithmic bridges between rainbows and braid charts, enabling passage from one representation to another and back, and illustrate the approach with concrete examples such as Fox’s Example 12 and the $2$-twist spun trefoil, ultimately verifying a rainbow number of $4$ for that classic example. The results offer new computational tools and conceptual links between 4D knot theory, symplectic topology, and diagrammatic methods that may inform both theory and applications in higher-dimensional topology.

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.

Figures (60)

• Figure 1: A rainbow diagram for the standard unknotted torus.
• Figure 2: How the various diagrammatic descriptions of knotted surfaces are interconnected.
• Figure 3: Local model for perturbation/deperturbation of triplane diagrams.
• Figure 4: Local models of braided-perturbations. Notice how the resulting rainbow diagram can be chosen so that the new strand goes under or over all the old strands. The integer $m$ is the number of crossings involved in the local model before perturbation. For $m>0$, all the over-crossings in the left diagrams can also be chosen to be under-crossings. The particular case $m=0$ is drawn on the right.
• Figure 5: (Top) Rainbow version of an exchange move. (Bottom) A sequence of rainbow moves that adds up to an exchange move.

Theorems & Definitions (71)

• Theorem 1.1
• Definition 1.2
• Definition 2.1: MZ17Trans
• Definition 2.2
• Remark 2.3: Warning on 0-sector perturbations
• Lemma 2.4
• proof
• Definition 3.1
• Remark 3.2
• Lemma 3.4