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The most complete and at the same time most concise code is P-data, which has the same form as a Dowker code, and gives a numerical code, signs of crossing points, and the relation "over-under". The program Knot 2000 (K2K) uses P-data to internally represent a KL. Let us suppose that we have already calculated for some KL its Dowker code with signs Dow, and its DT-code. For example, for the non-alternating knot 819, Dow={{8},{12,14,10,16,4,6,2,8}}, and DT={{8},{12,14,10,-16,4,6,2,-8}}. From Dow we can obtain a list of ordered pairs, where each crossing is labelled by two numbers: odd and even. In our example, that list is:
If in DT the kth number (k = 1,...,n) is negative, we need to reverse the order of the numbers of the kth ordered pair in D, leaving signs in D at their places. Because the positions of negative numbers in DT are {4,8}, after reversing the 4th and 8th ordered pair in D, we obtain the list
After sorting that list according to the first members of ordered pairs, we obtain the list
The first part of the P-data is the same as the first part of Dow, and the second part called P-word is the list of the second members of the ordered pairs from D1, so P-data={{8},{12,14,10,15,4,6,2,7}}. In fact, P-data is a kind of Dowker code with signs. The only difference is that P-data contains odd instead of the corresponding even numbers in all crossings where the relation "over-under" is disturbed with regard to the corresponding alternating KL. Dowker codes allow us to feed projections of KLs into the computer as lists of numbers. Morwen Thistlethwhaite used the Dowker notation to list all of the prime knots of 16 and fewer crossings, and Jim Hoste derived the list of all links with at the most nine crossings, representing each KL by its minimal Dowker code.
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