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 is extended to many new examples in [AS03*,CEGS05,Iwa04].

The higher genus surfaces (called stabilized surfaces) obtained from $K$ 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 applied right away to stabilized surfaces is characteristic and advantage of the cocycle invariant.