Extensions of Colorings

Let $X$ be a quandle, and for a given abelian coefficient group $A$, take a quandle $2$-cocycle $\phi$. Let $E=A \times X$ and define a binary operation by $(a_1, x_1)*(a_2, x_2)=(a_1 + \phi(x_1, x_2), x_1 * x_2)$. It was shown in [CKS03] that $(E, *)$ defines a quandle, called an abelian extension, and is denoted by $E=E(X, A, \phi)$. There is a natural quandle homomorphism $p: E=A \times X \rightarrow X$ defined by the projection to the second factor. (This is in parallel to central extensions of groups, see Chapter IV of [Brw82].) Examples are found in [CENS01].

In [CENS01], the interpretation of the cocycle invariant was given as an obstruction to extending a quandle coloring to another coloring by a larger quandle, which is an abelian extension of the original quandle. Specifically, let $E=E(X, A, \phi)$ be the abelian extension defined above, $D$ be a diagram of a knot $K$, and ${\mathcal C}\in {\rm Col}_X(D)$. A coloring ${\mathcal C}' \in {\rm Col}_E(D)$ is an extension of ${\mathcal C}$ (or ${\mathcal C}$ extends to ${\mathcal C}'$) if $p({\mathcal C}'(\alpha)={\mathcal C}(\alpha)$ holds for any arc $\alpha$ of $D$, where $p: E \rightarrow X$ is the projection defined above. It was proved in [CENS01] that the contribution $\prod_{\tau} B({\mathcal C}, \tau)$ is trivial (the identity element of $A$) if and only if there is an extension ${\mathcal C}'$ of ${\mathcal C}$. In particular, every coloring ${\mathcal C}\in {\rm Col}_X(D)$ extends to some ${\mathcal C}' \in {\rm Col}_E(D)$ if and only if the cocycle invariant $\Phi_{\phi} (K)$ is trivial.

The proof is easily seen by walking along a knot diagram starting from a base point, and trying to extend a given coloring as one goes through each under-arc. If the source colors at a crossing $\tau$ is $(x_\tau, y_\tau)$, and the color $x_\tau$ is extended to $(x_\tau, a_\tau)$ inductively, then after going under the over-arc colored by $y_\tau$, the color of the other under-arc is extended to $(a+ \epsilon (\tau) \phi(x_\tau, y_\tau), x_\tau * y_\tau )$, whose change in the first factor is exactly the weight at this crossing. Thus after going through the diagram once, the discrepancy in the first factor is the contribution of this coloring.