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(
Knots with Fox tricoloring number tri(K)=27
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knot-theory
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1I'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. – 2011-06-07