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Colored chirality of knots

For a given knot diagram $ K$ with a (Fox) $ n$-coloring $ {\mathcal C}$ (a coloring by $ R_n$), its mirror $ K^*$ has the $ n$-coloring $ {\mathcal C}^*$ (called the mirror coloring) that is the mirror image of $ {\mathcal C}$. Specifically, let $ {\mathcal A}$ be the set of (over-)arcs of a diagram $ K$, then the mirror image $ K^*$ has the set of arcs $ {\mathcal A}^*$ that is in a natural bijection with $ {\mathcal A}$, such that each arc $ a \in {\mathcal A}$ has its mirror $ a^* \in {\mathcal A}^*$, and vice-versa (for any arc $ \alpha \in {\mathcal A}^*$, there is a unique arc $ a \in {\mathcal A}$ such that $ \alpha=a^*$). Then $ {\mathcal C}^* : {\mathcal A}^* \rightarrow \mathbb{Z}_n$ is defined by $ {\mathcal C}^* (a^*)={\mathcal C} (a)$.

Figure: Is figure-eight amphicheiral with $ 5$-colorings?
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In Fig. [*] a $ 5$-colored figure-eight knot diagram and its mirror with its mirror coloring are depicted. It is, then, natural to ask if an $ n$-colored knot diagram (a pair $ (K, {\mathcal C})$ of a diagram $ K$ and a coloring $ {\mathcal C}$) is equivalent to its mirror with its mirror coloring $ (K^*, {\mathcal C}^*)$. If this is the case, we call the colored diagram $ (K, {\mathcal C})$ amphicheiral (or achiral) with (respect to) the $ n$-coloring $ {\mathcal C}$. Otherwise we say a diagram is chiral with (respect to) the $ n$-coloring.

The answer to the above question for the colored figure-eight knot is NO, and it is seen from the contributions of the $ 3$-cocycle invariant with the dihedral quandle $ R_5$, as they are distinct between the left and right of Fig. [*]. Many other achiral knots have colorings with respect to which they are chiral. On the other hand, there are many colorings of many knots that are achiral with the colorings.


next up previous contents
Next: Minimal number of type Up: Applications of Cocycle Invariants Previous: Chirality of knots and   Contents
Masahico Saito - Quandle Website 2006-09-19