Threading a link by a polynomial - Revision history

Sam.panitch at 23:05, 5 August 2025

2025-08-05T23:05:44Z

← Older revision		Revision as of 19:05, 5 August 2025
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	R[Subscript[z, 1], 2]*cheb[4, Subscript[z, 2]], {1, 3}, {Subscript[		R[Subscript[z, 1], 2]*cheb[4, Subscript[z, 2]], {1, 3}, {Subscript[
	z, 1], Subscript[z, 2]}] /. {A -> a^(1/2)}]		z, 1], Subscript[z, 2]}] /. {A -> a^(1/2)}]
	Expand[(-1)^1*bracketFact[3]/bracket[1]]</nowiki> \|		Expand[(-1)^2*bracketFact[5]/bracket[1]]</nowiki> \|
	out= <nowiki>2 + 1/a^14 - 1/a^10 - 1/a^8 - 1/a^6 + 1/a^2 + a^2 - a^6 - a^8 - a^10 + a^14		out= <nowiki>2 + 1/a^14 - 1/a^10 - 1/a^8 - 1/a^6 + 1/a^2 + a^2 - a^6 - a^8 - a^10 + a^14
	2 + 1/a^14 - 1/a^10 - 1/a^8 - 1/a^6 + 1/a^2 + a^2 - a^6 - a^8 - a^10 + a^14</nowiki>}}		2 + 1/a^14 - 1/a^10 - 1/a^8 - 1/a^6 + 1/a^2 + a^2 - a^6 - a^8 - a^10 + a^14</nowiki>}}

Sam.panitch at 23:04, 5 August 2025

2025-08-05T23:04:46Z

← Older revision		Revision as of 19:04, 5 August 2025
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	bracket[n_]:=a^n-a^(-n);		bracket[n_]:=a^n-a^(-n);
	bracketFact[n_]:=Product[bracket[i],{i,1,n}];		bracketFact[n_]:=Product[bracket[i],{i,1,n}];
			lambda[n_] := A^(2n + 2) + A^(-2n - 2);
	R[z_, n_] := Product[z + lambda[2*i], {i, 0, n - 1}];		R[z_, n_] := Product[z + lambda[2*i], {i, 0, n - 1}];
	cheb[0, z_] = 1;		cheb[0, z_] = 1;

Sam.panitch at 23:04, 5 August 2025

2025-08-05T23:04:34Z

← Older revision		Revision as of 19:04, 5 August 2025
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	bracket[n_]:=a^n-a^(-n); //		bracket[n_]:=a^n-a^(-n); //
	bracketFact[n_]:=Product[bracket[i],{i,1,n}]; //		bracketFact[n_]:=Product[bracket[i],{i,1,n}]; //
			lambda[n_] := A^(2n + 2) + A^(-2n - 2); //
	R[z_, n_] := Product[z + lambda[2*i], {i, 0, n - 1}]; //		R[z_, n_] := Product[z + lambda[2*i], {i, 0, n - 1}]; //
	cheb[0, z_] = 1;		cheb[0, z_] = 1;

Sam.panitch at 22:46, 5 August 2025

2025-08-05T22:46:10Z

← Older revision		Revision as of 18:46, 5 August 2025
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	2 + 1/a^14 - 1/a^10 - 1/a^8 - 1/a^6 + 1/a^2 + a^2 - a^6 - a^8 - a^10 + a^14</nowiki>}}		2 + 1/a^14 - 1/a^10 - 1/a^8 - 1/a^6 + 1/a^2 + a^2 - a^6 - a^8 - a^10 + a^14</nowiki>}}
	<!--END-->		<!--END-->

	==References==

	{{note\|Masbaum}} Masbaum, Gregor. ''Skein-theoretical derivations of some formulas of Habiro.'' Alg. and Geo. Topology 3 (2003): 537–556. https://doi.org/10.2140/agt.2003.3.537		{{note\|Masbaum}} Masbaum, Gregor. ''Skein-theoretical derivations of some formulas of Habiro.'' Alg. and Geo. Topology 3 (2003): 537–556. https://doi.org/10.2140/agt.2003.3.537

Sam.panitch at 22:43, 5 August 2025

2025-08-05T22:43:59Z

← Older revision		Revision as of 18:43, 5 August 2025
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	<code>CableLink[link,poly,strandList,vars]</code>, whose code is available [[cableLink.m\|here]], computes the Kauffman bracket of link (given as a PD) with components L1,L2,...,Ln, cabled by the polynomial poly in the variables z1,z2,...,zn. strandList is a list of strand labels of length n, where the ith element is the first strand label corresponding to component Li.		<code>CableLink[link,poly,strandList,vars]</code>, whose code is available [[cableLink.m\|here]], computes the Kauffman bracket of link (given as a PD) with components L1,L2,...,Ln, cabled by the polynomial poly in the variables z1,z2,...,zn. strandList is a list of strand labels of length n, where the ith element is the first strand label corresponding to component Li.
	As an example, we can verify some formulas from ~~Mausbaum~~, after importing KnotTheory` and the CableLink code:		As an example, we can verify some formulas from {{ref\|Masbaum}}, after importing KnotTheory` and the CableLink code:

	<!--$$hopfLink=PD[X[3,1,4,2],X[2,4,1,3]]; //		<!--$$hopfLink=PD[X[3,1,4,2],X[2,4,1,3]]; //
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	2 + 1/a^14 - 1/a^10 - 1/a^8 - 1/a^6 + 1/a^2 + a^2 - a^6 - a^8 - a^10 + a^14</nowiki>}}		2 + 1/a^14 - 1/a^10 - 1/a^8 - 1/a^6 + 1/a^2 + a^2 - a^6 - a^8 - a^10 + a^14</nowiki>}}
	<!--END-->		<!--END-->

			==References==

			{{note\|Masbaum}} Masbaum, Gregor. ''Skein-theoretical derivations of some formulas of Habiro.'' Alg. and Geo. Topology 3 (2003): 537–556. https://doi.org/10.2140/agt.2003.3.537

Sam.panitch at 22:35, 5 August 2025

2025-08-05T22:35:49Z

← Older revision		Revision as of 18:35, 5 August 2025
Line 1:		Line 1:
	<code>CableLink[link,poly,strandList,vars]</code>, whose code is available [[cableLink.m\|here]], computes the Kauffman bracket of link (given as a PD) with components L1,L2,...,Ln, cabled by the polynomial poly in the variables z1,z2,...,zn. strandList is a list of strand labels of length n, where the ith element is the first strand label corresponding to component Li.		<code>CableLink[link,poly,strandList,vars]</code>, whose code is available [[cableLink.m\|here]], computes the Kauffman bracket of link (given as a PD) with components L1,L2,...,Ln, cabled by the polynomial poly in the variables z1,z2,...,zn. strandList is a list of strand labels of length n, where the ith element is the first strand label corresponding to component Li.
	As an example, we can verify some formulas from Mausbaum:		As an example, we can verify some formulas from Mausbaum, after importing KnotTheory` and the CableLink code:

	{{Startup Note}}
	<!--$$Import["http://katlas.org/w/index.php?title=CableLink.m&action=raw"];$$-->
	<!--$$Import["http://katlas.org/w/index.php?title=CableLink.m&action=raw"];$$-->
	<!--Robot Land, no human edits to "END"-->
	{{In\|
	n = 2 \|
	in = <nowiki>Import["http://katlas.org/w/index.php?title=CableComponent.m&action=raw"];</nowiki>}}
	<!--END-->
	<!--$$hopfLink=PD[X[3,1,4,2],X[2,4,1,3]]; //		<!--$$hopfLink=PD[X[3,1,4,2],X[2,4,1,3]]; //
	bracket[n_]:=a^n-a^(-n); //		bracket[n_]:=a^n-a^(-n); //

Sam.panitch at 22:27, 5 August 2025

2025-08-05T22:27:17Z

← Older revision		Revision as of 18:27, 5 August 2025
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	As an example, we can verify some formulas from Mausbaum:		As an example, we can verify some formulas from Mausbaum:

			{{Startup Note}}
	<!--$$Import["http://katlas.org/w/index.php?title=CableLink.m&action=raw"];$$-->		<!--$$Import["http://katlas.org/w/index.php?title=CableLink.m&action=raw"];$$-->
	<!--$$Import["http://katlas.org/w/index.php?title=CableLink.m&action=raw"];$$-->		<!--$$Import["http://katlas.org/w/index.php?title=CableLink.m&action=raw"];$$-->
	<!--Robot Land, no human edits to "END"-->		<!--Robot Land, no human edits to "END"-->
	{{In\|		{{In\|
	n = 1 \|		n = 2 \|
	in = <nowiki>Import["http://katlas.org/w/index.php?title=CableComponent.m&action=raw"];</nowiki>}}		in = <nowiki>Import["http://katlas.org/w/index.php?title=CableComponent.m&action=raw"];</nowiki>}}
	<!--END-->		<!--END-->
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	<!--Robot Land, no human edits to "END"-->		<!--Robot Land, no human edits to "END"-->
	{{In\|		{{In\|
	n = 2 \|		n = 3 \|
	in = <nowiki>hopfLink=PD[X[3,1,4,2],X[2,4,1,3]];		in = <nowiki>hopfLink=PD[X[3,1,4,2],X[2,4,1,3]];
	bracket[n_]:=a^n-a^(-n);		bracket[n_]:=a^n-a^(-n);
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	<!--Robot Land, no human edits to "END"-->		<!--Robot Land, no human edits to "END"-->
	{{InOut\|		{{InOut\|
	n = 3 \|		n = 4 \|
	in = <nowiki>Expand[CableLink[hopfLink,		in = <nowiki>Expand[CableLink[hopfLink,
	R[Subscript[z, 1], 1]*cheb[2, Subscript[z, 2]], {1, 3}, {Subscript[		R[Subscript[z, 1], 1]*cheb[2, Subscript[z, 2]], {1, 3}, {Subscript[
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	$$-->		$$-->
	{{InOut\|		{{InOut\|
	n = 4 \|		n = 5 \|
	in = <nowiki>Expand[CableLink[hopfLink,		in = <nowiki>Expand[CableLink[hopfLink,
	R[Subscript[z, 1], 2]*cheb[4, Subscript[z, 2]], {1, 3}, {Subscript[		R[Subscript[z, 1], 2]*cheb[4, Subscript[z, 2]], {1, 3}, {Subscript[

Sam.panitch at 22:26, 5 August 2025

2025-08-05T22:26:38Z

← Older revision		Revision as of 18:26, 5 August 2025
Line 1:		Line 1:
	<code>CableLink[link,poly,strandList,vars]</code>, whose code is available [[cableLink.m\|here]], computes the Kauffman bracket of link (given as a PD) with components L1,L2,...,Ln, cabled by the polynomial poly in the variables z1,z2,...,zn. strandList is a list of strand labels of length n, where the ith element is the first strand label corresponding to component Li.		<code>CableLink[link,poly,strandList,vars]</code>, whose code is available [[cableLink.m\|here]], computes the Kauffman bracket of link (given as a PD) with components L1,L2,...,Ln, cabled by the polynomial poly in the variables z1,z2,...,zn. strandList is a list of strand labels of length n, where the ith element is the first strand label corresponding to component Li.
	As an example, we can verify some formulas from Mausbaum:		As an example, we can verify some formulas from Mausbaum:

	<!--$$Import["http://katlas.org/w/index.php?title=CableLink.m&action=raw"];$$-->		<!--$$Import["http://katlas.org/w/index.php?title=CableLink.m&action=raw"];$$-->
	<!--$$Import["http://katlas.org/w/index.php?title=CableLink.m&action=raw"];$$-->		<!--$$Import["http://katlas.org/w/index.php?title=CableLink.m&action=raw"];$$-->

Sam.panitch at 22:26, 5 August 2025

2025-08-05T22:26:19Z

@@ Line 2: / Line 2: @@
 As an example, we can verify some formulas from Mausbaum:
 <!--$$Import["http://katlas.org/w/index.php?title=CableLink.m&action=raw"];$$-->
 <!--END-->
 <!--$$hopfLink=PD[X[3,1,4,2],X[2,4,1,3]]; //
@@ Line 11: / Line 16: @@
 cheb[n_, z_] := cheb[n, z] = z*cheb[n - 1, z] - cheb[n - 2, z];
 $$-->
 <!--END-->
 <!--$$Expand[CableLink[hopfLink,
    R[Subscript[z, 1], 1]*cheb[2, Subscript[z, 2]], {1, 3}, {Subscript[
@@ Line 17: / Line 34: @@
 Expand[(-1)^1*bracketFact[3]/bracket[1]]
 $$-->
 <!--END-->
 <!--$$Expand[CableLink[hopfLink,
@@ Line 23: / Line 49: @@
 Expand[(-1)^2*bracketFact[5]/bracket[1]] //
 $$-->
 <!--END-->

Sam.panitch at 21:29, 5 August 2025

2025-08-05T21:29:05Z

← Older revision		Revision as of 17:29, 5 August 2025
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	$$-->		$$-->
	<!--END-->		<!--END-->
	<!--$$Expand[~~cableLink~~[hopfLink,		<!--$$Expand[CableLink[hopfLink,
	R[Subscript[z, 1], 1]*cheb[2, Subscript[z, 2]], {1, 3}, {Subscript[		R[Subscript[z, 1], 1]*cheb[2, Subscript[z, 2]], {1, 3}, {Subscript[
	z, 1], Subscript[z, 2]}] /. {A -> a^(1/2)}] //		z, 1], Subscript[z, 2]}] /. {A -> a^(1/2)}] //
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	$$-->		$$-->
	<!--END-->		<!--END-->
	<!--$$Expand[~~cableLink~~[hopfLink,		<!--$$Expand[CableLink[hopfLink,
	R[Subscript[z, 1], 2]*cheb[4, Subscript[z, 2]], {1, 3}, {Subscript[		R[Subscript[z, 1], 2]*cheb[4, Subscript[z, 2]], {1, 3}, {Subscript[
	z, 1], Subscript[z, 2]}] /. {A -> a^(1/2)}]		z, 1], Subscript[z, 2]}] /. {A -> a^(1/2)}]