TITLE: Two knots with same Dowker-Thistlethwaite code have isomorphic knot groups?
QUESTION [5 upvotes]: Given a (tame) knot diagram, one derives the Dowker-Thistlethwaite code by travelling around the knot  and numbering each crossing 1,2,3,.... A negative sign is given to an even number if you cross on an underpass with that even number. Each crossing gets an even-odd pair of numbers, and we sort the pairs by the odd numbers. The resulting even number sequence is the Dowker-Thistlethwaite code of the knot.
This code determines a prime knot uniquely but there may be multiple composite knots with the same code.

Question: Given any two knots with the same Dowker-Thistlethwaite code, are their knot groups isomorphic?

I believe the answer is yes because of the comment on Wikipedia that two knots with the same code will be almost the same except for a reflection or a reflection of the way the connected sum is taken in a prime knot decomposition. Therefore one can define an isomorphism of fundamental groups easily enough. However there does not seem to be a reference for this fact either, and Dowker and Thistlethwaite (at least to me) don't seem to talk about this in their paper [1]. However, I'm mainly an algebraist and my algebraic topology expertise is not as thorough as I'd like, so I would appreciate some help.
[1] Dowker, C. H.; Thistlethwaite, Morwen B. Classification of knot projections. Topology Appl. 16 (1983), no. 1, 19--31. MR0702617

REPLY [2 votes]: You are correct.  The DT code gives a composite diagram if and only if the permutation splits into two sub-permutations.  So, if the code does not split, then it specifies a prime diagram which we can reconstruct up to taking a mirror image.
This is discussed (briefly) in Section 2.2 of "The Knot Book" by Colin Adams.