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I would be very grateful if you help me to find such knots. Or to find a knot atlas, where this invariant is included. I tried to find, but did not sucseed(

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I'm sorry, I was looking for the definition of tricoloring number of a knot on the page But I didn't find the definition of number, can you give it to us? Thank you! – Giovanni De Gaetano Jun 7 '11 at 20:31
Here's a nice article, where you would find link – Amira_Shikhi Jun 7 '11 at 20:55
Ok, so the tricoloring number of a knot counts how many different tri-colorations you have on a knot (as defined on wikipedia). And it is always a power of 3. An example for k=3^1 is given by the trivial knot and an example for k=3^2 is given by the trefoil knot, as explained in the article you cited. We are looking for an example with k=3^3. I apologize for the repetition but I want to clarify the question. – Giovanni De Gaetano Jun 7 '11 at 21:18
I'm sorry, since an example of such a knot is pretty easy to find (see the square knot $3_1 # 3_1$ for example) I think your intent was to classify all the possible knots with such tricoloring number. – Giovanni De Gaetano Jun 7 '11 at 21:25

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