Can two different prime knots have a Dowker-Thistlethwaite code in common?

Question

I was thinking about knot invariants and whether we could define an equivalence class on the set of all Dowker-Thistlethwaite codes for a knot, and whether said equivalence classes, combined with some indicator of chirality, would be a complete knot invariant for the prime knots, when it occurred to me that I have never seen a proof that a DT code generates a unique prime knot (up to chirality). I asked Professor Google but couldn't seem to turn anything up. Tait, of course, posed the ménage problem after following a similar line of logic, trying to put an upper bound on the number of knots of each crossing number, but I haven't seen an explicit proof from Tait or Dowker or Thistlethwaite or anyone else that their notation is uniquely decipherable for the prime knots up to chirality. Is the algorithm for deciphering a knot from a DT code technically a constructive proof of such uniqueness, or is there ambiguity in the algorithm such that, for some DT code somewhere, 2 different prime knots could be constructed from it?

Answer 1

A Dowker-Thistlethwaite code does not necessarily uniquely determine a knot. The counter-example below is from Colin Adams' "The Knot Book." The code $(4, 6, 2, 10 , 12, 8)$ corresponds to the two different knots appearing below.

The knots above are composite. You can see this in the DT-code because the first three terms are a shuffling of the set $\{2,4,6\}$ while the last three terms are a shuffling of the set $\{8, 10, 12\}$. When the permutation given by the DT-code can be broken into two two sub-permutations, the knot diagram is composite.

If we restrict to prime knots, then Adams states that the DT-code determines a knot up to mirror images. The code $(8, 6, 10, 2, 4)$ can equivalently describe both the knots below (which are mirror images of one another).

Edit. The proof that Dowker-Thistlethwaite codes for prime diagrams determine the diagram up to mirror imaging is Theorem 1 in the following article:

• C. H. Dowker, Morwen B. Thistlethwaite. Classification of knot projections. Topology Appl. 16 (1983), no. 1, 19–31.

Here's what their Theorem 1 states. Let $G$ be the sphere embedding of the 4-regular graph obtained from the knot diagram by considering the crossings as vertices and the segments between crossings as edges. Theorem 1 states that any two sphere embeddings corresponding to the same prime code are equivalent as embeddings on the sphere. Thus if information is added to the code that determines signs of crossings, then you can uniquely determine a prime knot diagram from this modified Dowker-Thistlethwaite code.

I believe the entire linked paper may be of interest to you.