Research ArticleNo Access

Distinguishing surface-links described by 4-charts with two crossings and eight black vertices

Teruo Nagase

https://orcid.org/0009-0006-7639-9881

Tokai University, 4-1-1 Kitakaname, Hiratuka, Kanagawa 259-1292, Japan

E-mail Address: nagase@keyaki.cc.u-tokai.ac.jp

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and

Akiko Shima

https://orcid.org/0000-0001-5157-9778

Department of Mathematics, Tokai University, 4-1-1 Kitakaname, Hiratuka, Kanagawa 259-1292, Japan

E-mail Address: shima@keyaki.cc.u-tokai.ac.jp

Corresponding author.

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https://doi.org/10.1142/S021821652350092XCited by:0 (Source: Crossref)

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 (called a surface-link) embedded in 4-space. In this paper, we investigate surface-links by using charts. In [T. Nagase and A. Shima, The structure of a minimal $n$ $n$ -chart with two crossings I: Complementary domains of $Γ_{1} \cup Γ_{n - 1}$ $Γ_{1} \cup Γ_{n - 1}$ , J. Knot Theory Ramifactions27(14) (2018) 1850078; T. Nagase and A. Shima, The structure of a minimal $n$ $n$ -chart with two crossings II: Neighbourhoods of $Γ_{1} \cup Γ_{n - 1}$ $Γ_{1} \cup Γ_{n - 1}$ , Revista de la Real Academia de Ciencias Exactas, Fiskcas y Natrales. Serie A. Math.113 (2019) 1693–1738, arXiv:1709.08827v2] we gave an enumeration of the charts with two crossings. In particular, there are two classes for 4-charts with two crossings and eight black vertices. The first class represents surface-links each of which is connected. The second class represents surface-links each of which is exactly two connected components. In this paper, by using quandle colorings, we shall show that the charts in the second class represent different surface-links.

Keywords:

AMSC: 57K45, 05C10, 57M15

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