Integral positive (negative) quandle cocycle invariants are trivial for knots

TL;DR

This paper proves that for any finite quandle and 2-cocycle, the associated cocycle invariant is trivial for all knots, simplifying the understanding of these invariants in knot theory.

Contribution

It establishes that integral positive and negative quandle cocycle invariants are trivial for all knots, providing a significant simplification in the study of knot invariants.

Findings

Integral positive quandle cocycle invariants are trivial for all knots.

Integral negative quandle cocycle invariants are trivial for all knots.

The result applies to any finite quandle and 2-cocycle in Z^2_Q±(X; Z).

Abstract

In this note we prove that for any finite quandle $X$ and any 2-cocycle $\phi\in Z^2_{Q\pm}(X; \mathds{Z})$, the cocycle invariant $\Phi^{\pm}_{\phi}(K)$ is trivial for all knots $K$.