A (classical or higher dimensional) knot is called invertible (or reversible) if it is equivalent to its orientation reversed counterpart, with the orientation of the ambient space fixed. A knot is called non-invertible (or irreversible) if it is not invertible.
The cocycle invariant provided a diagrammatic method
(by a state-sum) of detecting non-invertibility
of knotted surfaces. In
[CJKLS03], it was proved that
the
-twist spun trefoil
and its orientation-reversed counterpart
have distinct cocycle invariants
and
with a cocycle of the dihedral quandle
, and therefore,
is non-invertible.
This result was the first application of
the cocycle invariant, and was
extended to infinite families of twist spun knots in [AS03*,CEGS05,Iwa04].
The higher genus surfaces (called stabilized surfaces)
obtained from the surface , whose non-invertibility is
detected by the cocycle invariant, by adding an arbitrary
number of trivial
-handles are also non-invertible, since
such handle additions do not alter the cocycle invariant.
This property that the conclusion is applied right away to stabilized surfaces
is characteristic and an advantage of the cocycle invariant.