Region colorings of surfaces in 4-space

TL;DR

This work extends Niebrzydowski's region-coloring invariant to 4-dimensional surface-links using multiple diagrammatic languages (movies, marked vertex diagrams, triplane and multiplane diagrams). It proves invariance of the coloring count $Col^{R}_X(F)$ under the corresponding moves, and derives bounds linking colorings to topological data such as saddle counts and Euler characteristic, with explicit formulas in the abelian case. It shows that for spun knots and abelian tribrackets, $Col^{R}_{X_A}(F)=|A|^{|F|+1}$ and $Col^{R}_X(K)=Col^{R}_X( ext{S}(K))$, respectively. Using concrete computations with a 3-element tribracket $X_3$, the paper demonstrates the non-invertibility of Yoshikawa 2-knots $9_1$ and $10_3$, illustrating the method's power in distinguishing 2-knots and enriching the toolkit for 4-dimensional knot theory. The results provide a bridge between algebraic colorings and geometric/topological properties, enabling new obstructions and invariants for surface-links in $S^4$.

Abstract

Niebrzydowski introduced a theory of region colorings for surface links. In this paper, we translate the coloring invariant to the context of triplane diagrams and movies of knots. We provide inequalities between the number of region colorings and topological quantities of $F$, such as the number of saddles in a movie and the bridge index of a triplane diagram of $F$. As an application, we show that Yoshikawa's 2-knots $9_1$ and $10_2$ are non-invertible; that is $F\not=-F$.

Figures (23)

• Figure 1: Reidemeister III move and the tribracket equations.
• Figure 2: Reidemeister moves.
• Figure 3: Tribracket equations at a crossing.
• Figure 4: Local models for double points, triple points, and branch points in a broken surface diagram.
• Figure 5: Region colorings near double points.

Theorems & Definitions (41)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1: Lem 3.10 of Niebrzydowski_knotted_surfaces
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof