# 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 ${\displaystyle xy=yz}$ where ${\displaystyle y}$ is the generator corresponding with the overcrossing.

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

Thus, we've associated with each tricolouring a homomorphism from the fundamental group of the knot complement to ${\displaystyle D_{2\cdot 3}}$. 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 ${\displaystyle \phi }$ to give a tricolouring: for every element ${\displaystyle x}$ in ${\displaystyle \pi _{1}(\mathbb {R} ^{3}\setminus K)}$ whose representative as a loop in ${\displaystyle \mathbb {R} ^{3}\setminus K}$ has odd linking number with ${\displaystyle K}$, ${\displaystyle \phi (x)}$ is an order 2 element in ${\displaystyle D_{2\cdot 3}}$. Hence, we can define tricolourings as certain kinds of homomorphisms from ${\displaystyle \pi _{1}(\mathbb {R} ^{3}\setminus K)}$ to ${\displaystyle D_{2\cdot 3}}$ without having to choose a diagram.