Knot Theory, 3-colourbility of knots

Decide the 3-colourability of each of the knots 7_5, 7_6, 7_7: In each case, either give a valid
3-colouring or prove that no 3-colouring exists.


Use the Rolfsen Knot Table


Answers can be viewed only if
  1. The questioner was satisfied and accepted the answer, or
  2. The answer was disputed, but the judge evaluated it as 100% correct.
View the answer

1 Attachment

  • Thanks I understand 7_7 however what I did for 7_5 and 7_6 is that I proved it by trying to draw the strands with only 3 colours however I said that since the under/over crossing strands must be different it must use more than 3 colours, do u think that is still a valid proof ?

  • If you formalize it correctly, it is, but I think that formalizing it correctly is essentially equivalent to doing what I did here. You can't just say "I tried and it didn't work", you have to do a case-by-case analysis, explain why forced choices are actually forced, and then make sure that your cases are exhaustive. It is doable, but you need to be careful. I find this linear algebra approach more streamlined.

  • Also under/over crossings don't have to be different, they can be the same color if the third strand in the crossing also has the same color, that's the tricky part I think.

  • Ok makse sense I will look over what I did!

  • oops i misread it sorry

The answer is accepted.
Join Matchmaticians Affiliate Marketing Program to earn up to 50% commission on every question your affiliated users ask or answer.