Let be a quandle, and for a given abelian coefficient group
, take
a quandle
-cocycle
.
Let
and define a binary operation by
.
It was shown in [CKS03] that
defines a quandle, called
an abelian extension, and is
denoted by
.
There is a natural quandle homomorphism
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
be the abelian extension defined above,
be a diagram of a knot
, and
.
A coloring
is an extension of
(or
extends to
) if
holds for any arc
of
,
where
is the projection defined above.
It was proved in [CENS01] that
the contribution
is trivial (the identity element of
)
if and only if there is an extension
of
.
In particular, every coloring
extends to some
if and only if
the cocycle invariant
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 is
,
and the color
is extended to
inductively,
then after going under the over-arc colored by
,
the color of the other under-arc is extended to
,
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.