The Multivariable Alexander Polynomial - Revision history

Drorbn: /* Links with Vanishing Multivariable Alexander Polynomial */

2013-08-29T23:56:59Z

Links with Vanishing Multivariable Alexander Polynomial

← Older revision		Revision as of 19:56, 29 August 2013
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	{{Knot Image Quadruple\|L9n27\|gif\|L10n32\|gif\|L10n36\|gif\|L10n107\|gif}}		{{Knot Image Quadruple\|L9n27\|gif\|L10n32\|gif\|L10n36\|gif\|L10n107\|gif}}
	{{Knot Image Quadruple\|L11n244\|gif\|L11n247\|gif\|L11n334\|gif\|L11n381\|gif}}		{{Knot Image Quadruple\|L11n244\|gif\|L11n247\|gif\|L11n334\|gif\|L11n381\|gif}}
	{{Knot Image ~~Pair~~\|L11n396\|gif\|L11n404\|gif\|L11n406\|gif}}		{{Knot Image Triple\|L11n396\|gif\|L11n404\|gif\|L11n406\|gif}}

	[[User:Drorbn\|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. (Though see the [[Talk:The Multivariable Alexander Polynomial\|Talk Page]]).		[[User:Drorbn\|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. (Though see the [[Talk:The Multivariable Alexander Polynomial\|Talk Page]]).

Drorbn: /* Links with Vanishing Multivariable Alexander Polynomial */

2013-08-29T23:56:19Z

Links with Vanishing Multivariable Alexander Polynomial

← Older revision		Revision as of 19:56, 29 August 2013
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	{{Knot Image Quadruple\|L9n27\|gif\|L10n32\|gif\|L10n36\|gif\|L10n107\|gif}}		{{Knot Image Quadruple\|L9n27\|gif\|L10n32\|gif\|L10n36\|gif\|L10n107\|gif}}
	{{Knot Image Quadruple\|L11n244\|gif\|L11n247\|gif\|L11n334\|gif\|L11n381\|gif}}		{{Knot Image Quadruple\|L11n244\|gif\|L11n247\|gif\|L11n334\|gif\|L11n381\|gif}}
	{{Knot Image ~~Triple~~\|L11n396\|gif\|L11n404\|gif\|L11n406\|gif}}		{{Knot Image Pair\|L11n396\|gif\|L11n404\|gif\|L11n406\|gif}}

	[[User:Drorbn\|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. (Though see the [[Talk:The Multivariable Alexander Polynomial\|Talk Page]]).		[[User:Drorbn\|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. (Though see the [[Talk:The Multivariable Alexander Polynomial\|Talk Page]]).

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

2009-05-27T21:27:06Z

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

← Older revision		Revision as of 17:27, 27 May 2009
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	http://www.texterzelv.com
	{{Manual TOC Sidebar}}		{{Manual TOC Sidebar}}

VarmoNsitv at 09:59, 22 May 2009

2009-05-22T09:59:47Z

← Older revision		Revision as of 05:59, 22 May 2009
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			http://www.texterzelv.com
	{{Manual TOC Sidebar}}		{{Manual TOC Sidebar}}

Drorbn at 15:51, 28 April 2009

2009-04-28T15:51:07Z

← Older revision		Revision as of 11:51, 28 April 2009
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	<!--END-->		<!--END-->

	But notice the funny labelling of the components! At ~~the~~ ~~moment~~ ~~there~~ is no ~~way~~ to ~~tell~~ <code>~~MultivariableAlexander~~</code> ~~which~~ ~~variable~~ is to be ~~associated~~ ~~with~~ ~~what~~ ~~variable~~ so <code>~~MultivariableAlexander~~</code> ~~chooses~~ an ~~arbitrary~~ ~~ordering~~ ~~of tha variables~~. Hence we had to rename <code>t[3]</code> to be <code>t4</code> and <code>t[4]</code> to be <code>t3</code>.		But notice the funny labelling of the components! The program <code>MultivariableAlexander</code> orders the variables in its output (typically denoted <code>t[i]</code>) in the same order as the order of the components of a link <code>L</code> as they appear within <code>Skeleton[L]</code>. Hence we had to rename <code>t[3]</code> to be <code>t4</code> and <code>t[4]</code> to be <code>t3</code>.

	(To be precise, <code>MultivariableAlexander</code> orders the components so that its output would be lexicographically minimal among all possible orderings. This way it is at least guaranteed that different presentations for the same link will yield the same output for <code>MultivariableAlexander</code>.)

	====Links with Vanishing Multivariable Alexander Polynomial====		====Links with Vanishing Multivariable Alexander Polynomial====
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	<!--END-->		<!--END-->

	We don't know whether our mystery link appears in the link table as is, or as a mirror, or with its two components switched. Hence we let <tt>AllPossibilities</tt> contain the multivariable Alexander polynomials of all those possibilities:		We don't know whether our mystery link appears in the link table as is, or as a mirror, or with its two components switched. Hence we let <code>AllPossibilities</code> contain the multivariable Alexander polynomials of all those possibilities:

	<!--$$AllPossibilities = Union[Flatten[		<!--$$AllPossibilities = Union[Flatten[

DrorsRobot at 15:45, 28 April 2009

2009-04-28T15:45:45Z

← Older revision		Revision as of 11:45, 28 April 2009
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	<!--Robot Land, no human edits to "END"-->		<!--Robot Land, no human edits to "END"-->
	{{InOut\|		{{InOut\|
	n = 7 \|		n = 8 \|
	in = <nowiki>~~mva~~ = ~~MultivariableAlexander~~[~~L = PD~~[		in = <nowiki>AllPossibilities = Union[Flatten[
	~~X[1~~, ~~16,~~ 2, ~~17]~~, X[3, ~~15,~~ ~~4, 14~~], X[5, 8, ~~6, 9~~],		{mva, -mva} /. {{}, {t[1] -> t[2], t[2] -> t[1]}}
		⚫	]]</nowiki> \|
	X[7, 21, 8, 20], X[9, 22, 10, 13], X[11, 2, 12, 3],
			out= <nowiki> 2
	X[13, 18, 14, 19], X[15, 12, 16, 1], X[17, 11, 18, 10],
	X[~~19,~~ 4, ~~20,~~ 5], X[~~21,~~ 7, ~~22,~~ 6]		{-(((-1 + t[1]) (-1 + t[2]) (1 - 2 t[1] + t[1] - 2 t[2] +

⚫	]~~][t~~]</nowiki> \|
	~~out=~~ ~~<nowiki>~~ 2		2 2 2 2 2
	~~-(((-1~~ + t[1]~~) (-1 +~~ t[2]~~) (1~~ - 2 t[1] + t[1] - 2 t[2] + 2 t[1] t[2] -		2 t[1] t[2] - 2 t[1] t[2] + t[2] - 2 t[1] t[2] + t[1] t[2]

			3/2 3/2
			)) / (t[1] t[2] )),

			2
			((-1 + t[1]) (-1 + t[2]) (1 - 2 t[1] + t[1] - 2 t[2] + 2 t[1] t[2] -

	2 2 2 2 2		2 2 2 2 2
	2 t[1] t[2] + t[2] - 2 t[1] t[2] + t[1] t[2] )) /		2 t[1] t[2] + t[2] - 2 t[1] t[2] + t[1] t[2] )) /

	3/2 3/2		3/2 3/2
	(t[1] t[2] ))</nowiki>}}		(t[1] t[2] )}</nowiki>}}
	<!--END-->		<!--END-->

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	<!--Robot Land, no human edits to "END"-->		<!--Robot Land, no human edits to "END"-->
	{{InOut\|		{{InOut\|
	n = 8 \|		n = 9 \|
			in = <nowiki>Select[
	in = <nowiki>Select[AllLinks[], (MultivariableAlexander[#][t] == mva \|\| MultivariableAlexander[#][t] == -mva) &]</nowiki> \|
			AllLinks[],
			MemberQ[AllPossibilities, MultivariableAlexander[#][t]] &
			]</nowiki> \|
	out= <nowiki>{Link[11, Alternating, 289]}</nowiki>}}		out= <nowiki>{Link[11, Alternating, 289]}</nowiki>}}
	<!--END-->		<!--END-->
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	<!--Robot Land, no human edits to "END"-->		<!--Robot Land, no human edits to "END"-->
	{{InOut\|		{{InOut\|
	n = 9 \|		n = 10 \|
	in = <nowiki>{Jones[L][q], Jones[Link[11, Alternating, 289]][q]}</nowiki> \|		in = <nowiki>{Jones[L][q], Jones[Link[11, Alternating, 289]][q]}</nowiki> \|
	out= <nowiki> -(17/2) 4 8 12 16 18 17 15		out= <nowiki> -(17/2) 4 8 12 16 18 17 15

Drorbn: /* Detecting a Link Using the Multivariable Alexander Polynomial */

2009-04-28T15:45:29Z

Detecting a Link Using the Multivariable Alexander Polynomial

@@ Line 119: / Line 119: @@
 <!--END-->
 <!--Robot Land, no human edits to "END"-->
 {{InOut|

DrorsRobot at 15:34, 21 April 2009

2009-04-21T15:34:05Z

← Older revision		Revision as of 11:34, 21 April 2009
Line 123:		Line 123:
	{{InOut\|		{{InOut\|
	n = 8 \|		n = 8 \|
	in = <nowiki>Select[AllLinks[], (MultivariableAlexander[#][t] == mva) &]</nowiki> \|		in = <nowiki>Select[AllLinks[], (MultivariableAlexander[#][t] == mva \|\| MultivariableAlexander[#][t] == -mva) &]</nowiki> \|
	out= <nowiki>{}</nowiki>}}		out= <nowiki>{Link[11, Alternating, 289]}</nowiki>}}
	<!--END-->		<!--END-->

Drorbn: /* Detecting a Link Using the Multivariable Alexander Polynomial */

2009-04-21T15:33:49Z

Detecting a Link Using the Multivariable Alexander Polynomial

← Older revision		Revision as of 11:33, 21 April 2009
Line 119:		Line 119:
	<!--END-->		<!--END-->

	<!--$$Select[AllLinks[], (MultivariableAlexander[#][t] == mva) &]$$-->		<!--$$Select[AllLinks[], (MultivariableAlexander[#][t] == mva \|\| MultivariableAlexander[#][t] == -mva) &]$$-->
	<!--Robot Land, no human edits to "END"-->		<!--Robot Land, no human edits to "END"-->
	{{InOut\|		{{InOut\|

DrorsRobot at 15:18, 21 April 2009

2009-04-21T15:18:14Z

← Older revision		Revision as of 11:18, 21 April 2009
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	{{InOut\|		{{InOut\|
	n = 5 \|		n = 5 \|
	in = <nowiki>mva - (mva /. {t1->t2, t2->t1})</nowiki> \|		in = <nowiki>Simplify[mva - (mva /. {t1->t2, t2->t1})]</nowiki> \|
	out= <nowiki>~~-((-t1~~ - t2 + t1 t2 - ~~t3 +~~ t1 ~~t3 + 2 t2 t3~~ - t1 t2 t3 - t4 ~~+ 2 t1 t4 +~~		out= <nowiki> (t1 - t2) (t3 - t4)
			-----------------------------------

		⚫	Sqrt[t1] Sqrt[t2] Sqrt[t3] Sqrt[t4]</nowiki>}}
	t2 t4 - t1 t2 t4 + t3 t4 - t1 t3 t4 - t2 t3 t4) /

	(Sqrt[t1] Sqrt[t2] Sqrt[t3] Sqrt[t4])) +

	(-t1 - t2 + t1 t2 - t3 + 2 t1 t3 + t2 t3 - t1 t2 t3 - t4 + t1 t4 +

	2 t2 t4 - t1 t2 t4 + t3 t4 - t1 t3 t4 - t2 t3 t4) /

⚫	(Sqrt[t1] Sqrt[t2] Sqrt[t3] Sqrt[t4])</nowiki>}}
	<!--END-->		<!--END-->

← Older revision		Revision as of 19:56, 29 August 2013
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	{{Knot Image Quadruple\|L9n27\|gif\|L10n32\|gif\|L10n36\|gif\|L10n107\|gif}}		{{Knot Image Quadruple\|L9n27\|gif\|L10n32\|gif\|L10n36\|gif\|L10n107\|gif}}
	{{Knot Image Quadruple\|L11n244\|gif\|L11n247\|gif\|L11n334\|gif\|L11n381\|gif}}		{{Knot Image Quadruple\|L11n244\|gif\|L11n247\|gif\|L11n334\|gif\|L11n381\|gif}}
	{{Knot Image ~~Pair~~\|L11n396\|gif\|L11n404\|gif\|L11n406\|gif}}		{{Knot Image Triple\|L11n396\|gif\|L11n404\|gif\|L11n406\|gif}}

	[[User:Drorbn\|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. (Though see the [[Talk:The Multivariable Alexander Polynomial\|Talk Page]]).		[[User:Drorbn\|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. (Though see the [[Talk:The Multivariable Alexander Polynomial\|Talk Page]]).

← Older revision		Revision as of 19:56, 29 August 2013
Line 84:		Line 84:
	{{Knot Image Quadruple\|L9n27\|gif\|L10n32\|gif\|L10n36\|gif\|L10n107\|gif}}		{{Knot Image Quadruple\|L9n27\|gif\|L10n32\|gif\|L10n36\|gif\|L10n107\|gif}}
	{{Knot Image Quadruple\|L11n244\|gif\|L11n247\|gif\|L11n334\|gif\|L11n381\|gif}}		{{Knot Image Quadruple\|L11n244\|gif\|L11n247\|gif\|L11n334\|gif\|L11n381\|gif}}
	{{Knot Image ~~Triple~~\|L11n396\|gif\|L11n404\|gif\|L11n406\|gif}}		{{Knot Image Pair\|L11n396\|gif\|L11n404\|gif\|L11n406\|gif}}

	[[User:Drorbn\|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. (Though see the [[Talk:The Multivariable Alexander Polynomial\|Talk Page]]).		[[User:Drorbn\|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. (Though see the [[Talk:The Multivariable Alexander Polynomial\|Talk Page]]).