Timeline for Relation between the Alexander module of a link and intermediate free abelian covers
Current License: CC BY-SA 3.0
6 events
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| May 8, 2014 at 15:39 | comment | added | Daniel Copeland | Thanks for that insight. As you point out, the statement in my question that says "if the Alexander ideal is functorial, then the gcd is also" is incorrect. In fact, McMullen only uses $(\Delta_{\phi}) = I_{\phi} = \phi_*(I)$ (these are the ideals generated by the minors). | |
| May 8, 2014 at 6:45 | comment | added | Stefan Friedl | the point of considering the Alexander module rel a point is to get to a square presentation matrix, where the issue with the gcd's does not arise. It's a slightly awkward trick, for example it doesn't work for closed 3-manifolds. | |
| May 8, 2014 at 6:43 | comment | added | Stefan Friedl | you have to be careful: you are right that Fox derivatives give you a presentation matrix, and you can get the multivariable and the onevariable Alexander polynomial out of it. But the problem is that as a presentation matrix for the Alexander module it is not a square matrix, so you have to take the minors and then the gcd. Taking minors is functorial, but taking gcd's isn't. For example, if the presentation matrix is $(x-1,y-1)$ for two different meridians, then the order is $gcd(x-1,y-1)=1$, but if you abelianize the presentation matrix is $(t-1,t-1)$, so the gcd of the minors is $t-1$. | |
| May 7, 2014 at 1:49 | comment | added | Daniel Copeland | Wow thanks for responding! (I've been looking forward to reading your works for some time :). I think I found another route to this functoriality through the Fox calculus, since a presentation for the Alexander invariant is given by the abelianization map applied to the matrix of Fox derivatives. This also works for the intermediate maps to free abelian groups and the coefficients of this matrix factors through the abelianization. (I didn't put this as an answer because it seems to require the full proof that the Fox matrix provides the needed presentation, which is a bit unwieldy). | |
| May 7, 2014 at 1:42 | vote | accept | Daniel Copeland | ||
| May 5, 2014 at 20:36 | history | answered | Stefan Friedl | CC BY-SA 3.0 |