You have 9 possible 3-coloring. There is a formula that says where means connected sum.
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.