number of colorings of a square knot???

• Jan 25th 2013, 06:43 AM
student2011
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.

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.