AKT-14/Tricolourability without Diagrams

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Here are some thoughts on how to define tricolourability without choosing a diagram.

Another place in which arcs of a diagram come up naturally is in the Wirtinger presentation for the fundamental group of the knot complement. Here is how the presentation is defined: each arc of the knot diagram corresponds with a generator, and each crossing corresponds with a relation between the generators of the incident arcs of the form where is the generator corresponding with the overcrossing.

Now if is a Wirtinger presentation for a knot diagram, it's natural to think of a tricolouring as a map . We'd like to try to extend this to a group homomorphism . This works if target group has the relation along with all other relations obtained by permuting . These relations fix the target group as .

Thus, we've associated with each tricolouring a homomorphism from the fundamental group of the knot complement to . Not every such homomorphism gives a tricolouring; for example, take the trivial homomorphism. I believe that the following is a sufficient condition for a homomorphism to give a tricolouring: for every element in whose representative as a loop in has odd linking number with , is an order 2 element in . Hence, we can define tricolourings as certain kinds of homomorphisms from to without having to choose a diagram.