In [CES02], a cocycle invariant using twisted quandle
cohomology theory was defined.
Let be a quandle, and
be an Alexander quandle with a variable
(typically
for a prime
and a polynomial
).
A function
is called a twisted quandle
-cocycle if it satisfies
the twisted
-cocycle condition
Let be an oriented knot diagram with orientation normal vectors.
The underlying projection of the
diagram divides the
-space into regions.
For a crossing
,
let
be the source region
(i.e., a unique region among four regions adjacent to
such that all normal vectors of arcs near
point from
to other regions).
Take an oriented arc
from the region at infinity to
a given region
such that
intersects the
arcs (missing crossing points) of
the diagram transversely in finitely many points.
The Alexander numbering
of a region
is the number
of such intersections counted with signs.
If the orientation of
agrees with the normal vector of the arc
where
intersects, then the intersection is positive
(counted as
), otherwise negative (counted as
).
The Alexander numbering
of a crossing
is defined to be
where
is the source region of
.
For a coloring
of
a diagram
od a classical knot
by
,
the twisted
Boltzmann
weight
at
is defined by
where
is a twisted
-cocycle,
and
is the sign of the crossing
.
This is in multiplicative notation, so that the action of
on
is written by
.
The colors
assigned near
are chosen in
the same manner as in the ordinary
-cocycle invariant
(source colors, or ordered pair of colors at
)
and
is the Alexander numbering of
.
The twisted quandle
-cocycle invariant
is the state-sum