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

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 [Formula: see text]-chart with two crossings I: Complementary domains of [Formula: see text], J. Knot Theory Ramifactions 27(14) (2018) 1850078; T. Nagase and A. Shima, The structure of a minimal [Formula: see text]-chart with two crossings II: Neighbourhoods of [Formula: see text], 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.