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

Abstract

We determine the length of the Mochizuki 3-cocycle of the 5-dihedral quandle. This induces that the 2-twist-spun figure-eight knot and the 2-twist-spun .2; 5/-torus knot have the triple point number eight. 57Q45; 57Q35Dedicated to Professor Taizo Kanenobu on the occasion of his 60 th birthday

Quandles with one non-trivial column

“…In their seminal article defining quandle cohomology [5], Carter et al defined the quandle cocycle invariant and used it to show that the 2-twist-spun trefoil is not isotopic to its orientation reverse. Notable use of the invariant also includes [8,14,20,[27][28][29]. Using a quandle 2-cocycle, this invariant can be defined for classical links.…”

“…Quandle cocycles have been used to study classical links as well as surface-links. The quandle cocycle invariant and weights of quandle cocycles have been used to calculate the triple point number of surface-links [9,[26][27][28][29]. This section computes several second and third quandle cohomology groups of P σ n since cocycles of this order can be used to define knot invariants.…”

Quandles versus symmetric quandles for oriented links

“…There are several studies using quandles in knot theory. Especially quandle cocycle invariants [2] (see also [3]), introduced by J. S. Carter et al in 2003, are very useful for studies of oriented links and oriented surfacelinks, refer to [8,14,16,19,20,21,22,23,24] for example. Here, we note that in order to define quandle cocycle invariants, it is essential that links or surfacelinks are oriented.…”