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Math Help - number of colorings of a square knot???

  1. #1
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    number of colorings of a square knot???

    Dear math helper

    I need you help in the following:

    The square knot is a connected sum of tow trefoil knots, one must be left-handed and the other right-handed. It is well-known that the number of colourings is a knot invariant. I know that we can colour trefoil by 9 distinct colourings, so the number of colourings of trefoil is 9. My question is what is the number of colourings of the square knot. I guess it is 18 but I am not sure.

    Please help me and I am very appreciated.

    Thank you in advance.
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  2. #2
    Senior Member vincisonfire's Avatar
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    You have 9 possible 3-coloring. There is a formula that says col_n(L_1) col_n(L_2) = n \times col_n(L_1 \cdot L_2) where \cdot means connected sum.
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  3. #3
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    Re: number of colorings of a square knot???

    That means we can color the square knot as well as the granny knot by 27 possible 3- colorings. I have not seen this formula before. Thank you so much vincisonfire
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