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 a trefoil ($3_1$) is chiral and the figure-eight knot ($4_1$) is achiral.

In [RkSn00*], a new proof of the chirality of trefoil was given using quandle homology theory. Their result can be expressed by the difference in values of the $3$-cocycle invariant with the dihedral quandle $R_3$. Specifically, in Example , the value of the invariant for a trefoil $K$ is $\Phi_{\theta}(K)=9 + 18u$, and one computes $\Phi_{\theta}(K^*)=9+18u^2$, so that $K$ and $K^*$ are not equivalent.

Figure: Infinitely many Suzuki's $\theta_n$-curves are chiral

In [Sat04*], for infinitely many spatial graphs called Suzuki's $\theta_n$-curves, it was proved that they are chiral. The $3$-cocycle invariants of dihedral quandles $R_p$ with the Mochizuki's cocycle were used to prove this fact. In Fig. , a special family of Suzuki's $\theta_n$-curves are depicted, with a coloring by $R_3$. Suzuki's $\theta_n$-curves have the property that any proper subgraph is trivially embedded, but they are non-trivial (the embeddings are not planar although there are planar embeddings), so that any method that uses proper subgraphs can not be applied effectively.