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

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  • 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.