Talk:The Multivariable Alexander Polynomial - Revision history

Drorbn: Reverted edits by CvaroUc4tr (Talk); changed back to last version by Drorbn

2009-05-27T21:24:48Z

Reverted edits by CvaroUc4tr (Talk); changed back to last version by Drorbn

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CvaroUc4tr at 09:55, 22 May 2009

2009-05-22T09:55:15Z

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Drorbn: Reverted edits by LazelRacda (Talk); changed back to last version by Drorbn

2008-12-18T12:09:36Z

Reverted edits by LazelRacda (Talk); changed back to last version by Drorbn

← Older revision		Revision as of 08:09, 18 December 2008
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	alchiaceld
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LazelRacda: zelracch

2008-12-18T11:41:32Z

zelracch

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Drorbn at 19:50, 3 May 2007

2007-05-03T19:50:38Z

← Older revision		Revision as of 15:50, 3 May 2007
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	:Dror doesn't understand the multivariable Alexander polynomial well enough to give simple topological reasons for the vanishing of the said polynomial for these knots.		:Dror doesn't understand the multivariable Alexander polynomial well enough to give simple topological reasons for the vanishing of the said polynomial for these knots.

	The multivariable Alexander polynomial is zero precisely when H1 of the universal ~~abelian~~ cover has non-zero rank (as a module over the group-ring of covering transformations). Equivalently, if H2 of the universal ~~abelian~~ cover is non-trivial. In the L10n36 case, H2 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~~		The multivariable Alexander polynomial is zero precisely when <math>H_1</math> of the universal Abelian cover has non-zero rank (as a module over the group-ring of covering transformations). Equivalently, if <math>H_2</math> of the universal Abelian cover is non-trivial. In the [[L10n36]] case, <math>H_2</math> 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

Rybu at 18:17, 13 April 2007

2007-04-13T18:17:54Z

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	:Dror doesn't understand the multivariable Alexander polynomial well enough to give simple topological reasons for the vanishing of the said polynomial for these knots.		:Dror doesn't understand the multivariable Alexander polynomial well enough to give simple topological reasons for the vanishing of the said polynomial for these knots.

	The multivariable Alexander polynomial is zero precisely when H1 of the universal abelian cover has non-zero rank (as a module over the group-ring of covering transformations). Equivalently, if H2 of the universal abelian cover is non-trivial. In ~~this~~ case, H2 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		The multivariable Alexander polynomial is zero precisely when H1 of the universal abelian cover has non-zero rank (as a module over the group-ring of covering transformations). Equivalently, if H2 of the universal abelian cover is non-trivial. In the L10n36 case, H2 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

Rybu at 18:12, 13 April 2007

2007-04-13T18:12:22Z

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	:Dror doesn't understand the multivariable Alexander polynomial well enough to give simple topological reasons for the vanishing of the said polynomial for these knots.		:Dror doesn't understand the multivariable Alexander polynomial well enough to give simple topological reasons for the vanishing of the said polynomial for these knots.

	The multivariable Alexander polynomial is zero precisely when H1 of the universal abelian cover has non-zero rank (as a module over the group-ring of covering transformations). Equivalently, if H2 of the universal abelian cover is non-trivial. In this case, H2 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...		The multivariable Alexander polynomial is zero precisely when H1 of the universal abelian cover has non-zero rank (as a module over the group-ring of covering transformations). Equivalently, if H2 of the universal abelian cover is non-trivial. In this case, H2 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

Rybu: comment

2007-04-13T18:11:40Z

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Regarding this comment:

:Dror doesn't understand the multivariable Alexander polynomial well enough to give simple topological reasons for the vanishing of the said polynomial for these knots.

The multivariable Alexander polynomial is zero precisely when H1 of the universal abelian cover has non-zero rank (as a module over the group-ring of covering transformations). Equivalently, if H2 of the universal abelian cover is non-trivial. In this case, H2 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...