Distinguishing surface-links with 4-charts with 2 crossings and 8 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 [11], [12], we gave an enumeration of the charts with two crossings. In particular, there are two classes for 4-charts with 2 crossings and 8 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.