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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
\begin{figure}\begin{center} \mbox{ \epsfxsize =2in \epsfbox{suzuki.eps}} \end{center}\end{figure}

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.

next up previous contents
Next: Colored chirality of knots Up: Applications of Cocycle Invariants Previous: Extensions of Colorings   Contents
Masahico Saito - Quandle Website 2006-09-19