AKT-14/Navigation

2014-01-10T00:42:38Z

Jzung:

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AKT-14/Tricolourability without Diagrams

2014-01-10T00:38:07Z

Jzung: Created page with "{{AKT-14/Navigation}} 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 t..."

{{AKT-14/Navigation}}

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 <math>xy=yz</math> where <math>y</math> is the generator corresponding with the overcrossing.

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

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

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AKT-14/Tricolourability without Diagrams