VasCalc Results - ChordMod4T - Revision history

Drorbn at 11:35, 10 October 2006

2006-10-10T11:35:11Z

← Older revision		Revision as of 07:35, 10 October 2006
Line 1:		Line 1:
	This page documents some computational results obtained by using the <b>ChordsMod4T</b> software component of the [[VasCalc]] project. We denote by <math>{\mathcal A}(l,m,n)</math> the formal (rational) vector space generated by all chord diagrams with <math>n</math> chords on a skeleton of <math>l</math> lines and <math>m</math> circles, modulo the <math>4T</math> relations. The algorithm is straightforward: first all such diagrams are constructed, then relations are introduced iteratively over all possible terms in a <math>4T</math> relation. See the [[VasCalc Documentation - ChordsMod4T\| documentation]] for further information about the program and how these results were generated.		This page documents some computational results obtained by using the <b>ChordsMod4T</b> software component of the [[VasCalc]] project. We denote by <math>{\mathcal A}_n(l,m)</math> the formal (rational) vector space generated by all chord diagrams with <math>n</math> chords on a skeleton of <math>l</math> lines and <math>m</math> circles, modulo the <math>4T</math> relations. The algorithm is straightforward: first all such diagrams are constructed, then relations are introduced iteratively over all possible terms in a <math>4T</math> relation. See the [[VasCalc Documentation - ChordsMod4T\| documentation]] for further information about the program and how these results were generated.

	===Some Dimensions of <math>{\mathcal A}(l,m,n)</math>===		===Some Dimensions of <math>{\mathcal A}_n(l,m)</math>===

	The tables below give the dimensions of <math>{\mathcal A}_n(l,m)</math> for various values of <math>l</math> ~~(lines)~~, <math>m</math> ~~(circles)~~ and <math>n</math> ~~(chords)~~. Each table corresponds to a fixed value of <math>n</math> - that is, the number of chords.		The tables below give the dimensions of <math>{\mathcal A}_n(l,m)</math> for various values of <math>l</math>, <math>m</math> and <math>n</math>. Each table corresponds to a fixed value of <math>n</math> - that is, the number of chords.

	'''This is work in progress - these numbers are not reliable yet!!!'''		'''This is work in progress - these numbers are not reliable yet!!!'''

Drorbn: /* Some Dimensions of ${\mathcal A}(l,m,n)$ */

2006-10-10T11:33:12Z

Some Dimensions of <math>{\mathcal A}(l,m,n)</math>

← Older revision		Revision as of 07:33, 10 October 2006
Line 3:		Line 3:
	===Some Dimensions of <math>{\mathcal A}(l,m,n)</math>===		===Some Dimensions of <math>{\mathcal A}(l,m,n)</math>===

	The tables below give the dimensions of <math>{\mathcal A}(l,m,n)</math> for various values of <math>l</math>, <math>m</math> and <math>n</math>. Each table corresponds to a fixed value of <math>n</math> - that is, the number of chords.		The tables below give the dimensions of <math>{\mathcal A}_n(l,m)</math> for various values of <math>l</math> (lines), <math>m</math> (circles) and <math>n</math> (chords). Each table corresponds to a fixed value of <math>n</math> - that is, the number of chords.

	'''This is work in progress - these numbers are not reliable yet!!!'''		'''This is work in progress - these numbers are not reliable yet!!!'''
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			<table width=100%><tr>

			<td>
	{\| border="1" align=center cellspacing=0		{\| border="1" align=center cellspacing=0
	\|+ Dimensions of chord diagram spaces with 3 chords		\|+ Dimensions of chord diagram spaces with 3 chords
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	!colspan="7"\|Number of Circles		!colspan="7"\|Number of Circles
	\|- align=right		\|- align=right

	\|0\|\|1\|\|2\|\|3\|\|4\|\|5\|\|6		\|0\|\|1\|\|2\|\|3\|\|4\|\|5\|\|6
	\|- align=right		\|- align=right
Line 57:		Line 57:
	\|5\|\|1130\|\|2418\|\|4944\|\|9600\|\|17705\|\|31125		\|5\|\|1130\|\|2418\|\|4944\|\|9600\|\|17705\|\|31125
	\|}		\|}
			</td><td>


	{\| border="1" align=center cellspacing=0		{\| border="1" align=center cellspacing=0
	\|+ Dimensions of chord diagram spaces with 4 chords		\|+ Dimensions of chord diagram spaces with 4 chords
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	!colspan="6"\|Number of Circles		!colspan="6"\|Number of Circles
	\|- align=right		\|- align=right

	\|0\|\|1\|\|2\|\|3\|\|4		\|0\|\|1\|\|2\|\|3\|\|4
	\|- align=right		\|- align=right
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	\|- align=right		\|- align=right
	\|2\|\|60\|\|283\|\|1160\|\|4126		\|2\|\|60\|\|283\|\|1160\|\|4126

	\|}		\|}
			</td></tr></table>

Drorbn: /* Some Dimensions of ${\mathcal A}(l,m,n)$ */

2006-10-10T11:27:22Z

Some Dimensions of <math>{\mathcal A}(l,m,n)</math>

← Older revision		Revision as of 07:27, 10 October 2006
Line 7:		Line 7:
	'''This is work in progress - these numbers are not reliable yet!!!'''		'''This is work in progress - these numbers are not reliable yet!!!'''

	{\| border="1"		{\| border="1" align=center cellspacing=0
	\|+ Dimensions of chord diagram spaces with 2 chords		\|+ Dimensions of chord diagram spaces with 2 chords
	!rowspan="2" width="100px"\|Number of Lines		!rowspan="2" width="100px"\|Number of Lines
Line 37:		Line 37:


	{\| border="1"		{\| border="1" align=center cellspacing=0
	\|+ Dimensions of chord diagram spaces with 3 chords		\|+ Dimensions of chord diagram spaces with 3 chords
	!rowspan="2" width="100px"\|Number of Lines		!rowspan="2" width="100px"\|Number of Lines
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	{\| border="1"		{\| border="1" align=center cellspacing=0
	\|+ Dimensions of chord diagram spaces with 4 chords		\|+ Dimensions of chord diagram spaces with 4 chords
	!rowspan="2" width="100px"\|Number of Lines		!rowspan="2" width="100px"\|Number of Lines

Sankaran at 16:48, 10 July 2006

2006-07-10T16:48:00Z

← Older revision		Revision as of 12:48, 10 July 2006
Line 65:		Line 65:
	\|- align=right		\|- align=right

	\|0\|\|1\|\|2\|\|3\|\|4~~\|\|5~~		\|0\|\|1\|\|2\|\|3\|\|4
	\|- align=right		\|- align=right
	\|0\|\|0\|\| 6\|\|44\|\|241\|\|1063		\|0\|\|0\|\| 6\|\|44\|\|241\|\|1063
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	\|- align=right		\|- align=right
	\|2\|\|60\|\|283\|\|1160\|\|4126		\|2\|\|60\|\|283\|\|1160\|\|4126

			\|}

Sankaran at 16:47, 10 July 2006

2006-07-10T16:47:32Z

← Older revision		Revision as of 12:47, 10 July 2006
Line 57:		Line 57:
	\|5\|\|1130\|\|2418\|\|4944\|\|9600\|\|17705\|\|31125		\|5\|\|1130\|\|2418\|\|4944\|\|9600\|\|17705\|\|31125
	\|}		\|}


			{\| border="1"
			\|+ Dimensions of chord diagram spaces with 4 chords
			!rowspan="2" width="100px"\|Number of Lines
			!colspan="6"\|Number of Circles
			\|- align=right

			\|0\|\|1\|\|2\|\|3\|\|4\|\|5
			\|- align=right
			\|0\|\|0\|\| 6\|\|44\|\|241\|\|1063
			\|- align=right
			\|1\|\|6\|\|44\|\|241\|\|1063\|\|3930
			\|- align=right
			\|2\|\|60\|\|283\|\|1160\|\|4126

Sankaran at 21:40, 7 July 2006

2006-07-07T21:40:35Z

← Older revision		Revision as of 17:40, 7 July 2006
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	\|4\|\|394\|\|954\|\|2172\|\|4618\|\|9184\|\|17189		\|4\|\|394\|\|954\|\|2172\|\|4618\|\|9184\|\|17189
	\|- align=right		\|- align=right
	\|5\|\|1130\|\|2418\|\|4944\|\|9600		\|5\|\|1130\|\|2418\|\|4944\|\|9600\|\|17705\|\|31125
	\|}		\|}

Sankaran at 21:31, 7 July 2006

2006-07-07T21:31:10Z

← Older revision		Revision as of 17:31, 7 July 2006
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	!rowspan="2" width="100px"\|Number of Lines		!rowspan="2" width="100px"\|Number of Lines
	!colspan="7"\|Number of Circles		!colspan="7"\|Number of Circles
			\|- align=right
	\|-

	\|0\|\|1\|\|2\|\|3\|\|4\|\|5\|\|6		\|0\|\|1\|\|2\|\|3\|\|4\|\|5\|\|6
	\|- align=right~~{8, 3, 19, 80, 270, 770, 1918}~~		\|- align=right
	\|0\|\|0\|\|3\|\|19\|\|80\|\|270\|\|770\|\|1918		\|0\|\|0\|\|3\|\|19\|\|80\|\|270\|\|770\|\|1918
	\|- align=right		\|- align=right

Sankaran at 21:29, 7 July 2006

2006-07-07T21:29:53Z

← Older revision		Revision as of 17:29, 7 July 2006
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	\|0\|\|1\|\|2\|\|3\|\|4\|\|5\|\|6		\|0\|\|1\|\|2\|\|3\|\|4\|\|5\|\|6
			\|- align=right{8, 3, 19, 80, 270, 770, 1918}
			\|0\|\|0\|\|3\|\|19\|\|80\|\|270\|\|770\|\|1918
	\|- align=right		\|- align=right
	\|~~0\|\|0~~\|\|3\|\|19\|\|92\|\|~~370~~\|\|~~1120~~\|\|~~2778~~		\|1\|\|3\|\|19\|\|80\|\|270\|\|770\|\|1918\|\|4284
	\|- align=right		\|- align=right
	\|1\|\|3\|\|19\|\|88\|\|~~351~~\|\|~~1096~~\|\|~~2768~~\|\|~~6083~~		\|2\|\|23\|\|88\|\|283\|\|789\|\|1944\|\|4318\|\|8803
	\|- align=right		\|- align=right
	\|2\|\|23\|\| 88\|\|~~329~~\|\|~~1053~~\|\|~~2734~~\|\|~~6073~~\|\|~~12176~~		\|3\|\|111\|\|321\|\|845\|\|2021\|\|4419\|\|8931\|\|16876
	\|- align=right		\|- align=right
	\|3\|\|~~111~~\|\|~~321~~\|\|~~991~~\|\|~~2657~~\|\|~~6029~~\|\|~~12166\|\|22689~~		\|4\|\|394\|\|954\|\|2172\|\|4618\|\|9184\|\|17189
	\|- align=right		\|- align=right
	\|4\|\|~~394~~\|\|~~954~~\|\|~~2524~~\|\|~~5908\|\|12112\|\|22679~~		\|5\|\|1130\|\|2418\|\|4944\|\|9600
	\|- align=right
	\|5\|\|1130\|\|2418\|\|5664\|\|11937\|\|22615\|\|39875
	\|- align=right
	\|6\|\|2778\|\|5424\|\|11533\|\|22376\|\|39801\|\|66793
	\|}		\|}

Sankaran at 21:01, 7 July 2006

2006-07-07T21:01:10Z

← Older revision		Revision as of 17:01, 7 July 2006
Line 14:		Line 14:
	\|\|0\|\|1\|\|2\|\|3\|\|4\|\|5\|\|6\|\|7\|\|8\|\|9		\|\|0\|\|1\|\|2\|\|3\|\|4\|\|5\|\|6\|\|7\|\|8\|\|9
	\|- align=right		\|- align=right
	\|0\|\|0\|\|2\|\|8\|\|27\|\|69\|\|~~145~~\|\|~~272~~\|\|~~469~~\|\|~~758~~\|\|~~1164~~		\|0\|\|0\|\| 2\|\|8\|\|24\|\|59\|\|125\|\|237\|\|413\|\|674\|\|1044
	\|- align=right		\|- align=right
	\|1\|\|2\|\|8\|\|26\|\|68\|\|~~145~~\|\|~~272~~\|\|~~469~~\|\|~~758~~\|\|~~1164~~\|\|~~1715~~		\|1\|\|2\|\|8\|\|24\|\|59\|\|125\|\|237\|\|413\|\|674\|\|1044\|\|1550
	\|- align=right		\|- align=right
	\|2\|\|9\|\|25\|\|66\|\|~~144~~\|\|~~272~~\|\|~~469~~\|\|~~758~~\|\|~~1164~~\|\|~~1715~~\|\|~~2442~~		\|2\|\|9\|\|25\|\|60\|\|126\|\|238\|\|414\|\|675\|\|1045\|\|1551\|\|2223
	\|- align=right		\|- align=right
	\|3\|\|28\|\|63\|\|~~141~~\|\|~~271~~\|\|~~469~~\|\|~~758~~\|\|~~1164~~\|\|~~1715~~\|\|~~2442~~\|\|~~3379~~		\|3\|\|28\|\|63\|\|129\|\|241\|\|417\|\|678\|\|1048\|\|1554\|\|2226\|\|3097
	\|- align=right		\|- align=right
	\|4\|\|69\|\|135\|\|~~267~~\|\|~~468~~\|\|~~758~~\|\|~~1164~~\|\|~~1715~~\|\|~~2442~~\|\|~~3379~~\|\|~~4563~~		\|4\|\|69\|\|135\|\|247\|\|423\|\|684\|\|1054\|\|1560\|\|2232\|\|3103\|\|4209
	\|- align=right		\|- align=right
	\|5\|\|145\|\|257\|\|~~463~~\|\|~~757~~\|\|~~1164~~\|\|~~1715~~\|\|~~2442~~\|\|~~3379~~\|\|~~4563~~\|\|~~6034~~		\|5\|\|145\|\|257\|\|433\|\|694\|\|1064\|\|1570\|\|2242\|\|3113\|\|4219\|\|5599
	\|- align=right		\|- align=right
	\|6\|\|272\|\|448\|\|~~751~~\|\|~~1163~~\|\|~~1715~~\|\|~~2442~~\|\|~~3379~~\|\|~~4563~~\|\|~~6034~~\|\|~~7835~~		\|6\|\|272\|\|448\|\|709\|\|1079\|\|1585\|\|2257\|\|3128\|\|4234\|\|5614\|\|7310
	\|- align=right		\|- align=right
	\|7\|\|469\|\|730\|\|~~1156~~\|\|~~1714~~\|\|~~2442~~\|\|~~3379~~\|\|~~4563~~\|\|~~6034~~\|\|~~7835~~\|\|~~10012~~		\|7\|\|469\|\|730\|\|1100\|\|1606\|\|2278\|\|3149\|\|4255\|\|5635\|\|7331\|\|9388
			\|- align=right
			\|8\|\|758\|\|1128\|\|1634\|\|2306\|\|3177\|\|4283\|\|5663\|\|7359\|\|9416\|\|11882
			\|- align=right
			\|9\|\|1164\|\|1670\|\|2342\|\|3213\|\|4319\|\|5699\|\|7395\|\|9452\|\|11918\|\|14844
	\|}		\|}

Drorbn: /* Some Dimensions of ${\mathcal A}(l,m,n)$ */

2006-07-04T18:03:50Z

Some Dimensions of <math>{\mathcal A}(l,m,n)</math>

← Older revision		Revision as of 14:03, 4 July 2006
Line 5:		Line 5:
	The tables below give the dimensions of <math>{\mathcal A}(l,m,n)</math> for various values of <math>l</math>, <math>m</math> and <math>n</math>. Each table corresponds to a fixed value of <math>n</math> - that is, the number of chords.		The tables below give the dimensions of <math>{\mathcal A}(l,m,n)</math> for various values of <math>l</math>, <math>m</math> and <math>n</math>. Each table corresponds to a fixed value of <math>n</math> - that is, the number of chords.

			'''This is work in progress - these numbers are not reliable yet!!!'''

	{\| border="1"		{\| border="1"

VasCalc Results - ChordMod4T - Revision history

Drorbn at 11:35, 10 October 2006

Drorbn: /* Some Dimensions of {\mathcal A}(l,m,n) */

Drorbn: /* Some Dimensions of {\mathcal A}(l,m,n) */

Sankaran at 16:48, 10 July 2006

Sankaran at 16:47, 10 July 2006

Sankaran at 21:40, 7 July 2006

Sankaran at 21:31, 7 July 2006

Sankaran at 21:29, 7 July 2006

Sankaran at 21:01, 7 July 2006

Drorbn: /* Some Dimensions of {\mathcal A}(l,m,n) */

Drorbn: /* Some Dimensions of ${\mathcal A}(l,m,n)$ */

Drorbn: /* Some Dimensions of ${\mathcal A}(l,m,n)$ */

Drorbn: /* Some Dimensions of ${\mathcal A}(l,m,n)$ */