Non-invertibility of knotted surfaces

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 $2$-twist spun trefoil $K$ and its orientation-reversed counterpart $-K$ have distinct cocycle invariants $(6 +12u$ and $6+12 u^2)$ with a cocycle of the dihedral quandle $R_3$, and therefore, $K$ 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 $K$, whose non-invertibility is detected by the cocycle invariant, by adding an arbitrary number of trivial $1$-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.