Prime Links with a Non-Prime Component - Revision history

Drorbn at 19:00, 20 October 2013

2013-10-20T19:00:25Z

← Older revision		Revision as of 15:00, 20 October 2013
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	Now, we can use the program [[SubLink.m\|<code>SubLink</code>]] that determines the PD form of a knot (or a link) made up of the selected component(s) of a certain link:		Now, we can use the program [[SubLink.m\|<code>SubLink</code>]] that determines the PD form of a knot (or a link) made up of the selected component(s) of a certain link:

	<!--$$Import["http://katlas.org/~~wiki~~/SubLink.m&action=raw"];$$-->		<!--$$Import["http://katlas.org/w/index.php?title=SubLink.m&action=raw"];$$-->
	<!--Robot Land, no human edits to "END"-->		<!--Robot Land, no human edits to "END"-->
	{{In\|		{{In\|
	n = 4 \|		n = 4 \|
	in = <nowiki>Import["http://katlas.org/~~wiki~~/SubLink.m&action=raw"];</nowiki>}}		in = <nowiki>Import["http://katlas.org/w/index.php?title=SubLink.m&action=raw"];</nowiki>}}
	<!--END-->		<!--END-->

2007-11-19T01:08:53Z

← Older revision		Revision as of 21:08, 18 November 2007
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	Now, we can use the program [[SubLink.m\|<code>SubLink</code>]] that determines the PD form of a knot (or a link) made up of the selected component(s) of a certain link:		Now, we can use the program [[SubLink.m\|<code>SubLink</code>]] that determines the PD form of a knot (or a link) made up of the selected component(s) of a certain link:

	<!--$$Import["http://katlas.org/wiki/SubLink.m&action=raw"];$$-->~~<!--END-->~~		<!--$$Import["http://katlas.org/wiki/SubLink.m&action=raw"];$$-->
			<!--Robot Land, no human edits to "END"-->
			{{In\|
			n = 4 \|
			in = <nowiki>Import["http://katlas.org/wiki/SubLink.m&action=raw"];</nowiki>}}
			<!--END-->

	Using <code>SubLink</code> and the Jones polynomials of the five composite knots mentioned above, we can find all links that have one of these as a component:		Using <code>SubLink</code> and the Jones polynomials of the five composite knots mentioned above, we can find all links that have one of these as a component:

2007-11-19T01:04:31Z

@@ Line 42: / Line 42: @@
 <!--END-->
-Now, we can use the following program that determines the PD form of a knot (or a link) made up of the selected component(s) of a certain link:
+Now, we can use the program [[SubLink.m|<code>SubLink</code>]] that determines the PD form of a knot (or a link) made up of the selected component(s) of a certain link:
-Using SubLink[ ] and the Jones polynomials of the five composite knots mentioned above, we can find all links that have one of these as a component:
+Using <code>SubLink</code> and the Jones polynomials of the five composite knots mentioned above, we can find all links that have one of these as a component:
 <!--$$NonPrimeComponentQ[L_] :=

2007-10-12T18:17:35Z

← Older revision		Revision as of 14:17, 12 October 2007
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	{{Manual TOC Sidebar}}		{{Manual TOC Sidebar}}


	Let us find all (prime!) links in the Knot Atlas that have a non-prime component. Since the links listed in the Knot Atlas have at most 11 crossings, such a component may only be the sum of exactly two knots chosen among the trefoil, the figure eight knot, and their mirror images. The figure eight knot's mirror image is itself so we have five possibilities. Computing the Jones polynomial of each, we get:		Let us find all (prime!) links in the Knot Atlas that have a non-prime component. Since the links listed in the Knot Atlas have at most 11 crossings, such a component may only be the sum of exactly two knots chosen among the trefoil, the figure eight knot, and their mirror images. The figure eight knot's mirror image is itself so we have five possibilities. Computing the Jones polynomial of each, we get:

2007-10-12T18:17:06Z

← Older revision	Revision as of 14:17, 12 October 2007
(No difference)

2007-09-22T20:19:39Z

← Older revision		Revision as of 16:19, 22 September 2007
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	S = t /. t2;		S = t /. t2;
	P = If[S != PD[] && Length[S] >= 3, S, PD[Knot[0, 1]], S]		P = If[S != PD[] && Length[S] >= 3, S, PD[Knot[0, 1]], S]
	]		];
	SubLink[pd_PD, j_] := SubLink[pd, {j}];		SubLink[pd_PD, j_] := SubLink[pd, {j}];
	SubLink[L_, js_] := SubLink[PD[L], js];</nowiki>}}		SubLink[L_, js_] := SubLink[PD[L], js];</nowiki>}}

2007-07-17T14:45:03Z

2007-07-03T19:17:38Z

← Older revision		Revision as of 15:17, 3 July 2007
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	ConnectedSum[K31, Mirror[K31]],		ConnectedSum[K31, Mirror[K31]],
	ConnectedSum[Mirror[K31], Mirror[K31]], ConnectedSum[K31, K41],		ConnectedSum[Mirror[K31], Mirror[K31]], ConnectedSum[K31, K41],
	ConnectedSum[Mirror[K31]~~, K41], ConnectedSum[K41~~, K41]}</nowiki> \|		ConnectedSum[Mirror[K31], K41]}</nowiki> \|
	out= <nowiki> -8 2 -6 2 2 -2 -3 -2 1 2 3		out= <nowiki> -8 2 -6 2 2 -2 -3 -2 1 2 3
	{q - -- + q - -- + -- + q , 3 - q + q - - - q + q - q ,		{q - -- + q - -- + -- + q , 3 - q + q - - - q + q - q ,
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	1 2 3 4 5 6		1 2 3 4 5 6
	-1 + - + 2 q - 3 q + 3 q - 2 q + 2 q - q ,		-1 + - + 2 q - 3 q + 3 q - 2 q + 2 q - q }
	q		q</nowiki>}}

	-4 2 3 4 2 3 4
	5 + q - -- + -- - - - 4 q + 3 q - 2 q + q }
	3 2 q
	q q</nowiki>}}
	<!--END-->		<!--END-->

2007-07-03T19:15:50Z

← Older revision		Revision as of 15:15, 3 July 2007
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	ConnectedSum[K31, Mirror[K31]],		ConnectedSum[K31, Mirror[K31]],
	ConnectedSum[Mirror[K31], Mirror[K31]], ConnectedSum[K31, K41],		ConnectedSum[Mirror[K31], Mirror[K31]], ConnectedSum[K31, K41],
	ConnectedSum[Mirror[K31]~~, K41], ConnectedSum[K41~~, K41]}$$-->		ConnectedSum[Mirror[K31], K41]}$$-->
	<!--Robot Land, no human edits to "END"-->		<!--Robot Land, no human edits to "END"-->
	{{InOut\|		{{InOut\|

2007-07-03T19:12:48Z

← Older revision		Revision as of 15:12, 3 July 2007
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	{{In\|		{{In\|
	n = 2 \|		n = 2 \|
	in = <nowiki>~~J31~~ = ~~Jones[~~Knot[3, 1]]; ~~J41~~ = ~~Jones[~~Knot[4, 1]];</nowiki>}}		in = <nowiki>K31 = Knot[3, 1]; K41 = Knot[4, 1];</nowiki>}}
	<!--END-->		<!--END-->

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	n = 3 \|		n = 3 \|
	in = <nowiki>CompositeJones =		in = <nowiki>CompositeJones =
	~~Expand~~[~~{J31[q~~] ~~J31~~[q], ~~J31[q]~~ ~~J31[1~~/~~q],~~ ~~J31~~[~~1/q]~~ ~~J31[1/q~~],		Jones[#][q] & /@ {ConnectedSum[K31, K31],
			ConnectedSum[K31, Mirror[K31]],
	J31[q] J41[q], J31[1/q] J41[q]}]</nowiki> \|
			ConnectedSum[Mirror[K31], Mirror[K31]], ConnectedSum[K31, K41],
			ConnectedSum[Mirror[K31], K41], ConnectedSum[K41, K41]}</nowiki> \|
	out= <nowiki> -8 2 -6 2 2 -2 -3 -2 1 2 3		out= <nowiki> -8 2 -6 2 2 -2 -3 -2 1 2 3
	{q - -- + q - -- + -- + q , 3 - q + q - - - q + q - q ,		{q - -- + q - -- + -- + q , 3 - q + q - - - q + q - q ,
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	1 2 3 4 5 6		1 2 3 4 5 6
	-1 + - + 2 q - 3 q + 3 q - 2 q + 2 q - q }		-1 + - + 2 q - 3 q + 3 q - 2 q + 2 q - q ,
	q~~</nowiki>}}~~		q

			-4 2 3 4 2 3 4
			5 + q - -- + -- - - - 4 q + 3 q - 2 q + q }
			3 2 q
			q q</nowiki>}}
	<!--END-->		<!--END-->

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	n = 4 \|		n = 4 \|
	in = <nowiki>SubLink[pd_PD, js_List] := Module[		in = <nowiki>SubLink[pd_PD, js_List] := Module[
	{k, t0, t, t1, t2, S, P~~, T~~},		{k, t0, t, t1, t2, S, P},
	t0 = Flatten[List @@@ Skeleton[pd][[js]]];		t0 = Flatten[List @@@ Skeleton[pd][[js]]];
	t = pd /. x_X :> Select[x, MemberQ[t0, #] &];		t = pd /. x_X :> Select[x, MemberQ[t0, #] &];