Today I’m going to be talking to the graduate students about various topics relating to coloring knots. I think I’ll leave you with a little project to play with.
First, go to Bar-Natan’s table of knots. Notice how all the diagrams seem to be made up of arcs meeting up where one strand of the knot crosses under another. Pick a knot diagram and try to color each arc either red, green, or blue, subject to the following rule: at any crossing, the three arcs that meet (two for the undercrossing strand and one for the overcrossing) must either be all the same color or all different colors.
Which knots can you color using all three colors at least once? If that’s too easy for you, how many ways can you color a given knot? If that’s too easy for you, you’ve almost surely seen this before.
To get you started, I’ve tricolored the trefoil knot using all three colors.