Chirality of knots and graphs

A knot or a spatial graph $K$ is called achiral or amphichaeral if it is equivalent to its mirror image $K^*$. Otherwise $K$ is called chiral. It is well known, for example, that trefoil ($3_1$) is chiral and the figure-eight knot ($4_1$) is achiral.

In [RouSand00*], a new proof of the chirality of trefoil was given using quandle homology theory. This can be detected by the difference in values of the $3$-cocycle invariant with the dihedral quandle $R_3$.

In [Sat04*], infinitely many spatial graphs called Suzuki's $\theta_n$-curves were proven to be chiral using $3$-cocycle invariants of dihedral quandles $R_p$. Suzuki's $\theta_n$-curves has the property that any proper subgraph is trivially embedded, but itself is non-trivial, so that any method that uses proper subgraphs can not be applied effectively.