# Talk:The Multivariable Alexander Polynomial

The multivariable Alexander polynomial is zero precisely when $H_1$ of the universal Abelian cover has non-zero rank (as a module over the group-ring of covering transformations). Equivalently, if $H_2$ of the universal Abelian cover is non-trivial. In the L10n36 case, $H_2$ is free on one generator, which is represented by a map of a genus 2 surface into the link complement. So far I haven't found a very appealing description of this surface, but it's there... -Ryan Budney