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Some common representations of knots do not directly give the sign/orientation of each crossing. For instance, the trefoil knot has Gauss code

-1, 3, -2, 1, -3, 2

and Dowker-Thistlethwaite code

4 6 2

It is well known that this leaves an ambiguity in the chirality of the knot (i.e. both chiral trefoils have the same notation), because the orientation of each crossing (one of L+ or L- in this image) is not recorded.

My problem is a little different; how should I algorithmically construct any consistent set of crossing orientations given just a Gauss code or DT code? Knowing these is necessary for the calculation of many knot invariants. For instance, the trefoil crossings may all be positive or all be negative.

It seems quite easy to sit and draw a consistent planar knot diagram given these notations, but I've not been able to describe a clear algorithm for this such that I can automate the process (trying to do it in terms of actual drawings is hard). Is such an algorithm well known, or is there any trick I might use to devise one?

For reference, a harder example is the knot 6_3, which has Gauss code

1, -6, 2, -1, 4, -5, 6, -2, 3, -4, 5, -3

and DT code

4 8 10 2 12 6 

This knot has a mixture of positive and negative crossing signs.

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  • $\begingroup$ If you are still working on this, this paper by Kauffman gives you an algorithm for regaining orientation from Gauss codes. $\endgroup$ – lefttrefoil Sep 30 '17 at 10:22
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It's easy to construct a 4-valent graph from the DT notation. The hard part is then to find an embedding of that graph in the plane/sphere. There are several algorithms that take an abstract graph and find a planar embedding if there is one: https://en.wikipedia.org/wiki/Planarity_testing

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  • $\begingroup$ Thank you, this worked very nicely. $\endgroup$ – inclement Sep 16 '15 at 20:21

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