Triple point numbers of surface-links and symmetric quandle cocycle invariants

Quandles and symmetric quandles for higher dimensional knots

“…can be also applied for non-orientable surface knots in R 4 (cf. [10,12,13]). Some other related researches are found in [2,6].…”

A Survey of Quandle Ideas

“…The 3-cocycles defined over symmetric quandle homology (see above) were used by Oshiro and Kamada and Oshiro 39,45 to determine lower bounds for triple point numbers of linked surfaces in which at least one component is non-orientable. We 20 found families of non-orientable connected surfaces in thickened 3-manifolds whose projections contain as many triple points as you might like (see also the material above).…”

The length of a 3–cocycle of the 5–dihedral quandle

“…We introduced the notion of the length of a cocycle of a quandle, and proved that the 3-twist-spun trefoil knot has the triple point number six [19]. Oshiro [14] used a symmetric quandle to determine the triple point numbers of some nonorientable surface-links. This paper is motivated by the study of Hatakenaka [8].…”