Calculating the Alexander Polynomials

The first step towards distinguishing among inequivalent knots is the calculation of the Alexander polynomials. So long as they come out distinct, one does not need any further techniques to use. In a previous example we showed the explicit calculation for a relatively simple case. In that case however one had prepared the figure in advance; what does one do however when one is merely given a knot notation? One could of course draw the relevant figure and proceed as before, but it would be far more desirable if without using any figures, one could feed the computer a notation, and eventually obtain the polynomials.

Once the theory and the example are understood, one might think that developing the appropriate program would be fairly simple. Let a notation {(a_1,b_1),(a_2,b_2),...,(a_N,b_N)}. For each crossing one needs an appropriate equation; then one discards one of the N equations, sets one of the unknowns equal to 0, and calculates the determinant. In order now to obtain the equations, all one needs are the three strands meeting at a crossing. This is done as follows.

Since a strand is defined as a segment joining two undercrossings, first one orders the numbers b_1,b_2,...,b_N. Each of them represents the end of one strand and the beginning of another. Now let (i,j) a crossing, and j be the end of the r strand and the beginning of the r+1 strand. If i lies between

k and l, and k is the beginning of the s strand while l is its end, then i belongs to the s strand and thus the relevant strands are the ones numbered r, r+1 and s. Therefore the equation may look
x_r = t * x_s + (1-t) * x_{r+1},
or it may look
x_{r+1} = t * x_s + (1-t) * x_r.
But which one of them? This is the only true difficulty of the calculation.

This question may be answered by drawing the figure and observing whether the crossing looks as in this figure, or whether it looks as its mirror image. But it can also be answered without the use of any figure.

First, one may notice that it is possible to start working from the "shadow", assigning a sign +1 or -1 to each crossing point depending on whether the angle that starts at its segment and ends at its "partner's" segment is positive or negative. At each crossing, one of the two points is always assigned +1 and the other -1.

Second, one may also notice that by switching all assigned numbers one obtains the mirror image knot which has the same notation. Therefore one may arbitrarily choose to assign the first crossing point (which is an overcrossing) the value +1.

Therefore if at a crossing, the overcrossing is assigned -1 and the undercrossing +1, it looks as in the figure, while if the signs are inverted, it looks as its mirror image.

This means that the 2N crossing points can be divided into two classes of N points each; points in each class have the same sign with each other and opposite signs with points of the other classes. One must thus find which points belong to the same class and which to oppoiste classes. The rules to do so are slightly complicated; we will briefly summarise them here, but readers are encouraged to draw figures of their own to visualise them.

First rule, as stated before, an overcrossing and an undercrossing of the same crossing belong to opposite classes.

Second rule, let {i,j} and {k,l} be two crossings such that i < k < j < l and no number between i and k is paired to number between k and j. One then counts how many numbers between i and k are paired to numbers between j and l; if this number is even, i and k belong to different classes, if it is odd, they belong to the same class.

It is possible however that through these rules the problem is not entirely resolved. Then for crossings {i,j} and {k,l} such that i < k < j < l , one needs to use a third crossing {m,n} such that k < m < j and l < n and:
a) no number between i and k is paired to number between k and j,
b) no number between j and l is paired to number between k and j,
c) no number between k and m is paired to number between m and j.
Then one counts for how many numbers between j and n there is an odd number of i, j, k, l separating them from their pairs. An even number means that i and k belong to different classes, and odd number means they belong to the same class.

Complicated as it may seem, this is the simplest procedure I have been able to come up with in order to resolve this question. It is not guaranteed that in the end all 2N crossing points will have been separated in these two classes, in such a case one should introduce a fourth crossing.

There is one "trick" that looks simpler; since it is known that the Alexander polynomial a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 satisfies a_n = a_0 , a_{n-1} = a_1 and so on, one may try all possible combinations and keep the one that will yield an Alexander polynomial satisfying this property. There is no guarantee that just one of the combinations will satisfy this property.

This of course is merely the beginning, a method to obtain the Alexander-Conway polynomial. To see how to calculate the other Alexander polynomials, click here.

In September 1996 I replaced the calculation of Alexander polynomials with the calculation of the homflypt polynomials, which had been discovered in 1985 independently by a number of mathematicians (Hoste, Ocneanu, Millet, Freyd, Lickorish, Yetter, Przytycki, Traczyk). To see why using the homflypt polynomials is preferrable, but also the additional complications arising through the skein relations, click here.

Charilaos Aneziris,

Copyright 1995


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