Talk:Alexander polynomial

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Where does this particular method for computation of the Alexander polynomial come from? I believe I followed the method correctly, but depending on the two columns eliminated, I get different determinants.

This is the method given by Alexander in his paper, except the article neglects to mention that one should "renormalize" at the end (something Alexander obviously realized), since there is ambiguity (depending on columns eliminated, as you've seen, the result can differ by multiplication by \pm x^n. Anyway, I will add the statement. --C S (Talk) 00:31, 13 April 2006 (UTC)

[edit] Example of skein relation computation for Conway poly

Moved from article:

(reader's note: This is a terrible example. The article would be much improved by a picture of a simple 3-4 crossing knot and then the step-by-step calculation of the Alexander polynomial! ) --—The preceding unsigned comment was added by 128.101.126.230 (talk • contribs).

This comment is referring to the section on the Conway polynomial which ends by saying see skein relation for an example of a computation. The point is well taken. I will try to get to it sometime. --C S (Talk) 11:29, 9 May 2006 (UTC)

[edit] Explicit computation

I think it would be nice if there was an explicit computation with pictures. I'm having trouble visualising it. Maybe even two explicit computations of the same knot looking very different. —Preceding unsigned comment added by Eigenlambda (talk • contribs) 18:46, 8 April 2008 (UTC)


[edit] Skein Relationship -- Definition used here

I was curious about why the particular definition of the Alexander-Conway skein relation used here was chosen. In his The Knot Book, Colin Adams uses the relation \nabla_{L_+}(z) - \nabla_{L_-}(z) + z \nabla_{L_0}(z) = 0 (with the conversion to the Alexander polynomials given by z = t1 / 2 − t − 1 / 2). Charles Livingston also uses this as the relation in Knot Theory published by the Mathematical Association of America. Peter Cromwell uses the relation used here in his book Knots and Links, but I've found a few major errors in that section of his book. Does anyone have access to Alexander's original paper or Conway's paper from Computational Problems in Abstract Algebra? I think he (Conway) used the same one that Adams and Livingston used listed here but can't remember. N Vale (talk) 05:46, 22 April 2008 (UTC)

People use the t-variable when considering the Alexander polynomial to be describing the order ideal of H_1 of the universal abelian cover of the knot complement. Because in this setting t represents the generating covering transformation. z has no such interpretation -- z is basically a notation designed to "compress" the information in the Alexander polynomial in order to remove the redundnacies created from the symmetry condition. The z is due to Conway, the t to Alexander. Rybu (talk) 17:07, 29 April 2008 (UTC)
I think the question is not meant to be as deep as that (perhaps I'm wrong). I think he's asking about the sign in the Conway skein relation, why was that chosen instead of the other way. First note that Adams does not use "z" at all. His skein relation with t^{1/2}, is the same, for example, as Lickorish's, but in Lickorish's book the substitution for z is made so that the skein relation for z is the same as in this article. In Conway's article is opposite the one in the article but that's because he considers a positive crossing to be a left-handed crossing, contrary to the usual convention. Kawauchi and Kauffman both use the convention listed in the article. --C S (talk) 23:06, 10 May 2008 (UTC)