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 xy=yz where y is the generator corresponding with the overcrossing.

Now if \langle S\mid \text{relations}\rangle is a Wirtinger presentation for a knot diagram, it's natural to think of a tricolouring as a map \phi: S \rightarrow \{R,G,B\}. We'd like to try to extend this to a group homomorphism \phi:\langle S \mid \text{relations}\rangle \rightarrow \langle R,G,B \mid \text{relations} \rangle . This works if target group has the relation RG=GB along with all other relations obtained by permuting R,G,B. These relations fix the target group as D_{2\cdot 3}.

Thus, we've associated with each tricolouring a homomorphism from the fundamental group of the knot complement to D_{2\cdot 3}. 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 \phi to give a tricolouring: for every element x in \pi_1(\R^3\setminus K) whose representative as a loop in \mathbb{R}^3\setminus K has odd linking number with K, \phi(x) is an order 2 element in D_{2\cdot 3}. Hence, we can define tricolourings as certain kinds of homomorphisms from \pi_1(\R^3\setminus K) to  D_{2\cdot 3} without having to choose a diagram.