The cocycle invariant for classical knots [CJKLS03] using quandle -cocycles
was defined as follows.
Let
be a finite quandle and
an abelian group.
A function
is called a quandle
-cocycle
if it satisfies the
-cocycle condition
The value (an element of
) of the invariant is written as
,
so that if
, the value is written as
a polynomial
.
The fact that
is a knot invariant is proved easily by checking
Reidemeister moves [CJKLS03].
For example, in Fig.
of Chapter
,
the product of weights that appear in the LHS
is
from the top to bottom crossings,
and for the RHS it is
, and the equality obtained is exactly
the
-cocycle condition in multiplicative notation, after canceling
.
The cocycle invariant can be also written as a family (multi-set, a set with repetition allowed) of weight sums [Lop03]
The -cocycle invariant is well-defined for virtual knots, as checked in
a similar manner.
Let
, an Alexander quandle as
and an abelian group
as
, and put the source colors
, for example, at the top crossing.
This color vector goes down to the next crossing, giving
,
=
, and comes back to
, by the coloring rule, and defines a non-trivial
coloring
.
In fact any pair extends to a coloring,
and
.
Consider the function
defined by
. Then this satisfies the
-cocycle condition
by calculation (we will further discuss this method of construction
of
-cocycles later).
The contribution is computed by
We make the following notational convention.
In the above example, if we denote additive generators and
multiplicatively by
and
, then the elements of
in multiplicative notation
are written as
. Hence the invariant in the state-sum form
is written as
. In Maple calculations,
it is convenient and easier to see outputs if we use the symbols
and
for
and
. Then the additive notations
remain on superscripts and the addition is also retained in exponents
(
). With this convention the invariant value
is written as
.
We use this notation if there is no confusion.