The linear minimal 4-chart with three crossings

TL;DR

This work analyzes linear minimal 4-charts with exactly three crossings, using C-moves, mal-cycles, and IO-path techniques to classify their structure. By decomposing charts into AB(Γ1) and AB(Γ3) components and applying Consecutive Triplet Lemma and pinwheel/null-mal-cycle arguments, the authors rule out all but a single configuration. They show that any such chart is lor-equivalent to the chart describing a $2$-twist spun trefoil knot after removing free edges and hoops, combining case analyses (edges terminal vs non-terminal) to reach the result. The findings advance the understanding of surface-link charts and their equivalence under local moves, with implications for recognizing spinor constructions of classical knots in 4-space.

Abstract

Charts are oriented labeled graphs in a disk. Any simple surface braid (2-dimensional braid) can be described by using a chart. Also, a chart represents an oriented closed surface embedded in 4-space. In this paper, we investigate embedded surfaces in 4-space by using charts. Let $Γ$ be a chart, and we denote by $Cross(Γ)$ the set of all the crossings of $Γ$, and we denote by $Γ_m$ the union of all the edges of label $m$. For a 4-chart $Γ$, if the closure of each connected component of the set $(Γ_1\cup Γ_3)-Cross(Γ)$ is acyclic, then $Γ$ is said to be {\it linear}. In this paper, we shall show that any linear minimal $4$-chart with three crossings is lor-equivalent (Label-Orientation-Reflection equivalent) to the chart describing a $2$-twist spun trefoil knot by omitting free edges and hoops.

Figures (34)

• Figure 1: Charts are lor-equivalent to the 4-chart describing a 2-twist spun trefoil. Here $k$ is a positive integer.
• Figure 2: (a) A black vertex. (b) A crossing. (c) A white vertex. Each arc with three transversal short arcs is a middle arc at the white vertex.
• Figure 3: For the C-III move, the edge with the black vertex does not contain a middle arc at a white vertex in the left figure.
• Figure 4: Bigons.
• Figure 5: The gray regions are disks $D$, and $m$ is a label.

Theorems & Definitions (28)

• Theorem 1.1
• Proposition 2.1
• Proposition 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 3.1
• Lemma 3.2
• Remark 4.1
• Lemma 4.2
• Lemma 4.3