Ideal knots - Revision history

Aklotz: Added 12 crossing data files

2023-05-26T22:39:53Z

Added 12 crossing data files

← Older revision		Revision as of 18:39, 26 May 2023
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	<H1>12-Crossing Knots</H1>		<H1>12-Crossing Knots</H1>
	<P>		<P>
	The ropelength of the 2176 knots with 12 crossings are included in the the uploaded zipped CSV file, along with the space writhe and average crossing number of the tight configuration. This includes 1288 alternating knots and 888 non-alternating knots. The Cartesian coordinates of each knot may be downloaded from the [https://doi.org/10.7910/DVN/6AXP61/ Harvard Dataverse]. These configurations were computed using [https://jasoncantarella.com/wordpress/software/ridgerunner/ Ridgerunner] by Alex Klotz and Caleb Anderson, using coordinates generated by Se-Goo Kim at [https://knotinfo.math.indiana.edu/ KnotInfo]. Note that these computations use the convention that the radius of the knotted tube is 1 instead of the diameter, giving them a factor of 2 difference from the other data on this page.		The ropelength of the 2176 knots with 12 crossings are included in the the uploaded zipped CSV file, along with the space writhe and average crossing number of the tight configuration. This includes 1288 alternating knots and 888 non-alternating knots. The Cartesian coordinates of each knot may be downloaded below and from the [https://doi.org/10.7910/DVN/6AXP61/ Harvard Dataverse]. These configurations were computed using [https://jasoncantarella.com/wordpress/software/ridgerunner/ Ridgerunner] by Alex Klotz and Caleb Anderson, using coordinates generated by Se-Goo Kim at [https://knotinfo.math.indiana.edu/ KnotInfo]. Note that these computations use the convention that the radius of the knotted tube is 1 instead of the diameter, giving them a factor of 2 difference from the other data on this page. Unlike Brian Gilbert's knots, these do not have a standardized orientation.
	</P>		</P>

	<P>		<P>
	[[File:TwelveSummary.zip]]		[[File:TwelveSummary.zip]]
			</P>

			<P>
			[[File:TwelveData.zip]]
	</P>		</P>

Aklotz: Added 12-crossing section

2023-05-26T19:35:37Z

Added 12-crossing section

@@ Line 105: / Line 105: @@
 crossings non-alternating: [[Media:Ideal_11n.txt.gz]] (0.75Mb/3.99Mb)
 </P>
 <H1>Ideal Links</H1>
 <P>

BrianGilbert at 11:35, 7 November 2016

2016-11-07T11:35:46Z

← Older revision		Revision as of 07:35, 7 November 2016
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	[[IMAGE:Ideal_LD_KL_graph.JPG]]		[[IMAGE:Ideal_LD_KL_graph.JPG]]
	<P>		<P>
	As shown in the graph, for a given crossing number, n, the shortest ideal forms are the non-alternating knots		As shown in the graph, for a given crossing number, n, the shortest ideal forms are the non-alternating knots/links
	comparable to alternating knots of one or two less crossings.		comparable to alternating knots of one or two less crossings.
	</P>		</P>
	<P>		<P>
	The shortest alternating knot ~~(if n is odd)~~ is the torus knot with		The shortest alternating knot/link is the torus knot with
	<BR />		<BR />
	Conway notation <B>n</B> (L/D ~ 5.66 + 3.57×n),		Conway notation <B>n</B> (L/D ~ 5.66 + 3.57×n),
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	so this should be the asymptotic slope of these curves.		so this should be the asymptotic slope of these curves.
	<BR /><BR />		<BR /><BR />
	All knots up to 11 crossings have L/D < 4.25 + 4.20×n;		All knots/links up to 11 crossings have L/D < 4.30 + 4.19×n;
	<BR /><BR />		<BR /><BR />
	thus up to 10 crossings there is no overlap between the ranges of alternating knots;		thus up to 9 crossings there is no overlap between the ranges of alternating knots/links;
	<BR />		<BR />
	but expect increasing overlap for higher crossing numbers.		but expect increasing overlap for higher crossing numbers.

BrianGilbert at 11:15, 7 November 2016

2016-11-07T11:15:44Z

← Older revision		Revision as of 07:15, 7 November 2016
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	Nevertheless tabulating the knots sorted by L/D reveals many patterns.		Nevertheless tabulating the knots sorted by L/D reveals many patterns.
	</P>		</P>
	[[IMAGE:Ideal_LD_KL_graph.~~jpg~~]]		[[IMAGE:Ideal_LD_KL_graph.JPG]]
	<P>		<P>
	As shown in the graph, for a given crossing number, n, the shortest ideal forms are the non-alternating knots		As shown in the graph, for a given crossing number, n, the shortest ideal forms are the non-alternating knots

BrianGilbert at 11:14, 7 November 2016

2016-11-07T11:14:10Z

← Older revision		Revision as of 07:14, 7 November 2016
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	Nevertheless tabulating the knots sorted by L/D reveals many patterns.		Nevertheless tabulating the knots sorted by L/D reveals many patterns.
	</P>		</P>
	[[IMAGE:~~Ideal_LD_graph~~.jpg]]		[[IMAGE:Ideal_LD_KL_graph.jpg]]
	<P>		<P>
	As shown in the graph, for a given crossing number, n, the shortest ideal forms are the non-alternating knots		As shown in the graph, for a given crossing number, n, the shortest ideal forms are the non-alternating knots

BrianGilbert: Database files updated and 11 crossing links added, split into files <2Mb

2016-11-07T10:34:25Z

Database files updated and 11 crossing links added, split into files <2Mb

← Older revision		Revision as of 06:34, 7 November 2016
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	Database files: 3-10 crossings [[Media:Ideal.txt.gz]] (984kb/5.33Mb)		Database files: 3-10 crossings [[Media:Ideal.txt.gz]] (984kb/5.33Mb)
	<BR />		<BR />
	11 crossings: [[Media:~~Ideal_11~~.txt.gz]] (1.~~12Mb~~/4.~~84Mb~~)~~. An updated file (2.27Mb/11.89Mb) is available but too large to upload.~~		11 crossings alternating: [[Media:Ideal_11a.txt.gz]] (1.52Mb/7.90Mb)
			<BR />
			11 crossings non-alternating: [[Media:Ideal_11n.txt.gz]] (0.75Mb/3.99Mb)
	</P>		</P>
	<H1>Ideal Links</H1>		<H1>Ideal Links</H1>
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	Database files: 2-9 crossings [[Media:IdealLinks.txt.gz]] (845kb/5.45Mb)		Database files: 2-9 crossings [[Media:IdealLinks.txt.gz]] (845kb/5.45Mb)
	<BR />		<BR />
	10 crossings: [[Media:~~IdealLinks_10~~.txt.gz]] (1.~~05Mb~~/5.~~37Mb~~) ~~An updated file (2.01Mb/12.35Mb) is available but too large to upload.~~		10 crossings alternating: [[Media:IdealLinks_10a.txt.gz]] (1.20Mb/7.23Mb)
			<BR />
			10 crossings non-alternating: [[Media:IdealLinks_10n.txt.gz]] (0.81Mb/5.13Mb)
			<BR />
			11 crossings alternating L11a1 to L11a300: [[Media:IdealLinks_11a1.txt.gz]] (1.79Mb/10.02Mb)
			<BR />
			11 crossings alternating L11a301 to L11a548: [[Media:IdealLinks_11a2.txt.gz]] (1.91Mb/11.48Mb)
			<BR />
			11 crossings non-alternating L11n1 to L11n230: [[Media:IdealLinks_11n1.txt.gz]] (1.34Mb/7.70Mb)
	<BR />		<BR />
	11 crossings: alternating (3.~~07Mb/21~~.~~50Mb) and non-alternating~~ (2.~~267Mb~~/18.~~49Mb~~) ~~are available but too large to upload.~~		11 crossings non-alternating L11n231 to L11n459: [[Media:IdealLinks_11n2.txt.gz]] (1.73Mb/10.78Mb)
	<BR />		<BR />
	Brian Gilbert		Brian Gilbert

BrianGilbert: Database files updated (where possible, <2Mb) others available for 11 crossing knots, 10 and 11 crossing links.

2016-11-06T10:55:31Z

Database files updated (where possible, <2Mb) others available for 11 crossing knots, 10 and 11 crossing links.

← Older revision		Revision as of 06:55, 6 November 2016
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	</CODE>		</CODE>
	<BR />		<BR />
	It is even more difficult to find the global minimum for these knots ~~- so perhaps only half of the conformations are close to ideal -~~		It is even more difficult to find the global minimum for these knots.
	so this database is of limited value.
	Nevertheless tabulating the knots sorted by L/D reveals many patterns.		Nevertheless tabulating the knots sorted by L/D reveals many patterns.
	</P>		</P>
Line 84:		Line 83:
	The shortest alternating knot (if n is odd) is the torus knot with		The shortest alternating knot (if n is odd) is the torus knot with
	<BR />		<BR />
	Conway notation <B>n</B> (L/D ~ 5.63 + 3.57×n),		Conway notation <B>n</B> (L/D ~ 5.66 + 3.57×n),
	<BR />		<BR />
	the next is <B>m 2</B> (L/D ~ 6.84 + 3.58×n).		the next is <B>m 2</B> (L/D ~ 6.83 + 3.58×n).
	<BR />		<BR />
	Note that the ideal twisted pair (<SMALL>[1, chapter 2, Pieranski]</SMALL>),		Note that the ideal twisted pair (<SMALL>[1, chapter 2, Pieranski]</SMALL>),
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	<BR CLEAR="all" />		<BR CLEAR="all" />
	<P>		<P>
	Database files: 3-10 crossings [[Media:Ideal.txt.gz]] (~~492kb~~/2.~~19Mb~~)		Database files: 3-10 crossings [[Media:Ideal.txt.gz]] (984kb/5.33Mb)
	<BR />		<BR />
	11 crossings: [[Media:Ideal_11.txt.gz]] (1.12Mb/4.84Mb)		11 crossings: [[Media:Ideal_11.txt.gz]] (1.12Mb/4.84Mb). An updated file (2.27Mb/11.89Mb) is available but too large to upload.
	</P>		</P>
	<H1>Ideal Links</H1>		<H1>Ideal Links</H1>
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	several links include strings which are nearly this short.		several links include strings which are nearly this short.
	</P>		</P>
	Database files: 2-9 crossings [[Media:IdealLinks.txt.gz]] (~~434kb~~/2.~~27Mb~~)		Database files: 2-9 crossings [[Media:IdealLinks.txt.gz]] (845kb/5.45Mb)
	<BR />		<BR />
	10 crossings: [[Media:IdealLinks_10.txt.gz]] (1.05Mb/5.37Mb)		10 crossings: [[Media:IdealLinks_10.txt.gz]] (1.05Mb/5.37Mb) An updated file (2.01Mb/12.35Mb) is available but too large to upload.
			<BR />
			11 crossings: alternating (3.07Mb/21.50Mb) and non-alternating (2.267Mb/18.49Mb) are available but too large to upload.
	<BR />		<BR />
	Brian Gilbert		Brian Gilbert

BrianGilbert at 03:30, 28 August 2013

2013-08-28T03:30:43Z

← Older revision		Revision as of 23:30, 27 August 2013
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	<P>		<P>
	The database of Ideal Knots to 10-crossings gives for each knot the best conformation I have found.		The database of Ideal Knots to 10-crossings gives for each knot the best conformation I have found.
	<BR />Each is represented by a Fourier series of vectors <B>A</B>[i], <B>B</B>[i], i=1..~~100~~ (links also need <B>A</B>[0]), so that		<BR />Each is represented by a Fourier series of vectors <B>A</B>[i], <B>B</B>[i], i=1..256 (links also need <B>A</B>[0]), so that
	<BR /><BR />		<BR /><BR />
	<B>X</B>(t) = <B>A</B>[0]/2 + ∑<SUB>i</SUB> <B>A</B>[i]cos(i.t) + <B>B</B>[i]sin(i.t)		<B>X</B>(t) = <B>A</B>[0]/2 + ∑<SUB>i</SUB> <B>A</B>[i]cos(i.t) + <B>B</B>[i]sin(i.t)
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	</P>		</P>
	<P>		<P>
	Compared to Pieranski's results up to 10 crossings <SMALL>[2]</SMALL> many in this database are ~~about~~ 0.25% worse.		Compared to Pieranski's results up to 10 crossings <SMALL>[2]</SMALL> many in this database are up to 0.1% worse.
	This difference arises in the conversion from the point representation used during iteration to the Fourier version:		This difference arises in the conversion from the point representation used during iteration to the Fourier version:
	partly because of only using ~~100~~ coefficients.		partly because of only using 256 coefficients.
	Also iteration tends to leave nodes nestled equidistant from both ends of the nearest leash -		Also iteration tends to leave nodes nestled equidistant from both ends of the nearest leash -
	measuring D by node-to-node distance produces an error of about 0.1%;		measuring D by node-to-node distance produces an error of about 0.1%;
	conversion and standardisation of the Fourier form generally shifts the nodes arbitarily and reduces this error.		conversion and standardisation of the Fourier form generally shifts the nodes arbitarily and reduces this error.
	</P>
	<P>
	None of the knots are > 0.3% worse
	(N.B. I suspect Pieranski's 10<SUB>144</SUB> [3 1,2 1,2 1-] is derived from Rolfsen's wrongly drawn diagram).		(N.B. I suspect Pieranski's 10<SUB>144</SUB> [3 1,2 1,2 1-] is derived from Rolfsen's wrongly drawn diagram).
	</P>		</P>
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	The shortest alternating knot (if n is odd) is the torus knot with		The shortest alternating knot (if n is odd) is the torus knot with
	<BR />		<BR />
	Conway notation <B>n</B> (L/D ~ 5.71 + 3.57×n),		Conway notation <B>n</B> (L/D ~ 5.63 + 3.57×n),
	<BR />		<BR />
	the next is <B>m 2</B> (L/D ~ 6.76 + 3.59×n).		the next is <B>m 2</B> (L/D ~ 6.84 + 3.58×n).
	<BR />		<BR />
	Note that the ideal twisted pair (<SMALL>[1, chapter 2, Pieranski]</SMALL>),		Note that the ideal twisted pair (<SMALL>[1, chapter 2, Pieranski]</SMALL>),
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	<BR CLEAR="all" />		<BR CLEAR="all" />
	<P>		<P>
	Database files: 3-10 crossings [[Media:Ideal.txt.gz]] (~~474kb~~/2.~~09Mb~~)		Database files: 3-10 crossings [[Media:Ideal.txt.gz]] (492kb/2.19Mb)
	<BR />		<BR />
	11 crossings: [[Media:Ideal_11.txt.gz]] (1.~~07Mb~~/4.~~62Mb~~)		11 crossings: [[Media:Ideal_11.txt.gz]] (1.12Mb/4.84Mb)
	</P>		</P>
	<H1>Ideal Links</H1>		<H1>Ideal Links</H1>
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	several links include strings which are nearly this short.		several links include strings which are nearly this short.
	</P>		</P>
	Database files: 2-9 crossings [[Media:IdealLinks.txt.gz]] (~~354kb~~/1.~~77Mb~~)		Database files: 2-9 crossings [[Media:IdealLinks.txt.gz]] (434kb/2.27Mb)
	<BR />		<BR />
	10 crossings: [[Media:IdealLinks_10.txt.gz]] (~~870kb~~/4.~~24Mb~~)		10 crossings: [[Media:IdealLinks_10.txt.gz]] (1.05Mb/5.37Mb)
	<BR />		<BR />
	Brian Gilbert		Brian Gilbert

BrianGilbert: New file sizes

2009-04-09T09:28:15Z

New file sizes

← Older revision		Revision as of 05:28, 9 April 2009
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	<BR /><BR />		<BR /><BR />
	<CODE>		<CODE>
	<DATA>		<DATA Title="..." Author="..." Date="...">
	<BR />		<BR />
	<AB Id="3:1:1" Conway="3" L="16.372861" D=" 1.000000">		<AB Id="3:1:1" Conway="3" L="16.372861" D=" 1.000000">
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	several links include strings which are nearly this short.		several links include strings which are nearly this short.
	</P>		</P>
	Database files: 2-9 crossings [[Media:IdealLinks.txt.gz]] (~~333kb~~/1.~~65Mb~~)		Database files: 2-9 crossings [[Media:IdealLinks.txt.gz]] (354kb/1.77Mb)
	<BR />		<BR />
	10 crossings: [[Media:IdealLinks_10.txt.gz]] (~~858kb~~/4.~~22Mb~~)		10 crossings: [[Media:IdealLinks_10.txt.gz]] (870kb/4.24Mb)
	<BR />		<BR />
	Brian Gilbert		Brian Gilbert

BrianGilbert: Add 10-crossing links

2009-02-28T23:26:17Z

Add 10-crossing links

← Older revision		Revision as of 19:26, 28 February 2009
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	A few Ideal links have strings made from straight lines and unit radius arcs ([2], [.1], [2,2,2,2--])		A few Ideal links have strings made from straight lines and unit radius arcs ([2], [.1], [2,2,2,2--])
	which were calculated directly.		which were calculated directly.
	The latter has oval strings of length π+2, which form a chain;		The latter has oval strings of length 2π+2, which form a chain;
	this does not have a unique conformation as the strings can be moved without changing shape or length.		this does not have a unique conformation as the strings can be moved without changing shape or length.
	The 12-crossing 6-string link [2,2,2,2,2,2---] is a chain with even more freedom of movement.		The 12-crossing 6-string link [2,2,2,2,2,2---] is a chain with even more freedom of movement.
	</P>		</P>
	<P>		<P>
	The oval of length π+2 is the shortest loop which has two strings threaded through it;		The oval of length 2π+2 is the shortest loop which has two strings threaded through it;
	several links include strings which are nearly this short.		several links include strings which are nearly this short.
	</P>		</P>
	Database files: 2-9 crossings [[Media:IdealLinks.txt.gz]] (333kb/1.65Mb)		Database files: 2-9 crossings [[Media:IdealLinks.txt.gz]] (333kb/1.65Mb)
			<BR />
			10 crossings: [[Media:IdealLinks_10.txt.gz]] (858kb/4.22Mb)
	<BR />		<BR />
	Brian Gilbert		Brian Gilbert