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Given a finite quandle $Q$, for any knot $K$ one can associate an invariant, i.e. the number of proper colorings $p(K)$. Let us consider the inverse $K^{-1}$ and mirror image $K'$ of $K$. My queston is, is there any relation between $p(K^{-1})$ $(p(K'))$ and $p(K)$? In general, with an appropriate finite quandle, can we distinguish a knot from its inverse (mirror image) by counting the number of proper colorings?

It seems that the number of proper colorings is exactly the number of homomorphisms from the knot quandle of $K$ to the quandle $Q$ (trivial homomorphism corresponds to trivial coloring). Hence my question concerns the relation between the knot quandle of $K^{-1}$ $(K')$ and the knot quandle of $K$. I only know that the knot quandle is a complete invariant for unoriented knots.

Any hints are welcome.

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    $\begingroup$ A stronger question would be whether knot quandles are residually finite (distinguished by homomorphisms onto finite quandles). $\endgroup$
    – Daniel Moskovich
    Commented Mar 17, 2013 at 8:36

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Answers to some of your questions can be found in this paper: Quandle Colorings of Knots and Applications http://arxiv.org/abs/1312.3307

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