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

on wikipedia it says 7_5 is tricolorable?


Ok makse sense I will look over what I did!

on wikipedia it says 7_5 is tricolorable?


oops i misread it sorry
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