The lengths of 3-cocycles of the 7-dihedral and the octahedral quandles

TL;DR

This paper calculates the lengths of specific 3-cocycles for 7-dihedral and octahedral quandles, linking these algebraic invariants to topological properties of certain spun knots.

Contribution

It provides the first explicit determination of 3-cocycle lengths for these quandles and relates them to the triple point number of particular spun knots.

Findings

Lengths of 3-cocycles for 7-dihedral and octahedral quandles are determined.

Both the 2-twist-spun 5_2-knot and the 4-twist-spun trefoil have triple point number eight.

Abstract

We determine the lengths of certain 3-cocycles of the 7-dihedral and the octahedral quandles. As a consequence, we show that both of the 2-twist-spun $5_{2}$-knot and the 4-twist-spun trefoil have the triple point number eight.