06-1350/Class Notes for Tuesday October 10 - Revision history

Drorbn at 22:35, 12 October 2006

2006-10-12T22:35:54Z

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	'''Dror's Speculation.''' If yes, it will have terrific consequences. If no, it will explain some of the misery we encounter when we deal with "associators". I would really like to understand this one.		'''Dror's Speculation.''' If yes, it will have terrific consequences. If no, it will explain some of the misery we encounter when we deal with "associators". I would really like to understand this one.

			Also see [[Some Questions About Trinions]].

Drorbn at 14:40, 10 October 2006

2006-10-10T14:40:09Z

← Older revision		Revision as of 10:40, 10 October 2006
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	'''Dror's Speculation.''' If yes, it will have terrific consequences. If no, it will explain some of the misery we encounter when we deal with "associators". I would really like to understand this one.		'''Dror's Speculation.''' If yes, it will have terrific consequences. If no, it will explain some of the misery we encounter when we deal with "associators". I would really like to understand this one.

	==Some Computations==

	The following table (taken from {{ref\|Bar-Natan_95}} and {{ref\|Kneissler_97}}) shows the number of type <math>m</math> invariants of knots and framed knots modulo type <math>m-1</math> invariants (<math>\dim{\mathcal A}_m^r</math> and <math>\dim{\mathcal A}_m</math>) and the number of multiplicative generators of the algebra <math>{\mathcal A}</math> in degree <math>m</math> (<math>\dim{\mathcal P}_m</math>) for <math>m\leq 12</math>. Some further tabulated results are in {{ref\|Bar-Natan_96}}.

	{\|align=center border=1 cellspacing=0 cellpadding=4
	\|- align=right
	\|align=center\|m
	\|0
	\|1
	\|2
	\|3
	\|4
	\|5
	\|6
	\|7
	\|8
	\|9
	\|10
	\|11
	\|12
	\|- align=right
	\|align=center\|<math>\dim{\mathcal A}_m^r</math>
	\|1
	\|0
	\|1
	\|1
	\|3
	\|4
	\|9
	\|14
	\|27
	\|44
	\|80
	\|132
	\|232
	\|- align=right
	\|align=center\|<math>\dim{\mathcal A}_m</math>
	\|1
	\|1
	\|2
	\|3
	\|6
	\|10
	\|19
	\|33
	\|60
	\|104
	\|184
	\|316
	\|548
	\|- align=right
	\|align=center\|<math>\dim{\mathcal P}_m</math>
	\|0
	\|1
	\|1
	\|1
	\|2
	\|3
	\|5
	\|8
	\|12
	\|18
	\|27
	\|39
	\|55
	\|}

	==Some Further Computations==

	Some further computations for links and tangles were made by Siddarth Sankaran and Zavosh Amir-Khosravi. See [[VasCalc Results - ChordMod4T]].

	==Some Bounds==

	Little is known about these dimensions for large <math>m</math>. There is an explicit conjecture
	in {{ref\|Broadhurst_97}} but no progress has been made in the direction of proving or disproving it. The best asymptotic bounds available are:

	'''Theorem.''' For large <math>m</math>, <math>\dim{\mathcal P}_m>e^{c\sqrt m}</math> (for any fixed <math>c<\pi\sqrt{\frac23}</math>, see {{ref\|Dasbach_00}}, {{ref\|Kontsevich}}) and <math>\dim{\mathcal A}_m<6^mm!\sqrt{m}/\pi^{2m}</math> ({{ref\|Stoimenow_98}}, {{ref\|Zagier_01}}).

	==Some References==

	{{note\|Bar-Natan_95}} D. Bar-Natan, ''On the Vassiliev knot invariants,'' Topology '''34''' (1995) 423-472.

	{{note\|Bar-Natan_96}} D. Bar-Natan, ''Some computations related to Vassiliev invariants,'' electronic publication, <tt>http://www.math.toronto.edu/~drorbn/LOP.html#Computations</tt>.

	{{note\|Broadhurst_97}} D. J. Broadhurst, ''Conjectured enumeration of Vassiliev invariants,'' preprint, September 1997, {{arXiv\|q-alg/9709031}}.

	{{note\|Dasbach_00}} O. T. Dasbach, ''On the combinatorial structure of primitive Vassiliev invariants III - a lower bound,'' Comm. in Cont. Math. '''2-4''' (2000) 579-590, {{arXiv\|math.GT/9806086}}.

	{{note\|Kneissler_97}} J. A. Kneissler, ''The number of primitive Vassiliev invariants up to degree twelve,'' preprint, June 1997, {{arXiv\|q-alg/9706022}}.

	{{note\|Kontsevich}} M. Kontsevich, unpublished.

	{{note\|Stoimenow_98}} A. Stoimenow, ''Enumeration of chord diagrams and an upper bound for Vassiliev invariants,'' Jour. of Knot Theory and its Ramifications '''7(1)''' (1998) 94-114.

	{{note\|Zagier_01}} D. Zagier, ''Vassiliev invariants and a strange identity related to the Dedekind eta-function,'' Topology '''40(5)''' (2001) 945-960.

Drorbn at 12:40, 10 October 2006

2006-10-10T12:40:54Z

← Older revision		Revision as of 08:40, 10 October 2006
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	in {{ref\|Broadhurst_97}} but no progress has been made in the direction of proving or disproving it. The best asymptotic bounds available are:		in {{ref\|Broadhurst_97}} but no progress has been made in the direction of proving or disproving it. The best asymptotic bounds available are:

	'''Theorem.''' For large <math>m</math>, <math>\dim{\mathcal P}_m>e^{c\sqrt m}</math> (for any fixed <math>c<\pi\sqrt{\frac23}</math>, see {{ref\|Dasbach_00}}, {{ref\|~~Kontsevich_93~~}}) and <math>\dim{\mathcal A}_m<6^mm!\sqrt{m}/\pi^{2m}</math> ({{ref\|Stoimenow_98}}, {{ref\|Zagier_01}}).		'''Theorem.''' For large <math>m</math>, <math>\dim{\mathcal P}_m>e^{c\sqrt m}</math> (for any fixed <math>c<\pi\sqrt{\frac23}</math>, see {{ref\|Dasbach_00}}, {{ref\|Kontsevich}}) and <math>\dim{\mathcal A}_m<6^mm!\sqrt{m}/\pi^{2m}</math> ({{ref\|Stoimenow_98}}, {{ref\|Zagier_01}}).

	==Some References==		==Some References==
Line 112:		Line 112:
	{{note\|Kneissler_97}} J. A. Kneissler, ''The number of primitive Vassiliev invariants up to degree twelve,'' preprint, June 1997, {{arXiv\|q-alg/9706022}}.		{{note\|Kneissler_97}} J. A. Kneissler, ''The number of primitive Vassiliev invariants up to degree twelve,'' preprint, June 1997, {{arXiv\|q-alg/9706022}}.

	{{note\|~~Kontsevich_93~~}} M. Kontsevich, ~~''Vassiliev's knot invariants,'' Adv. in Sov. Math., '''16(2)''' (1993) 137-150~~.		{{note\|Kontsevich}} M. Kontsevich, unpublished.

	{{note\|Stoimenow_98}} A. Stoimenow, ''Enumeration of chord diagrams and an upper bound for Vassiliev invariants,'' Jour. of Knot Theory and its Ramifications '''7(1)''' (1998) 94-114.		{{note\|Stoimenow_98}} A. Stoimenow, ''Enumeration of chord diagrams and an upper bound for Vassiliev invariants,'' Jour. of Knot Theory and its Ramifications '''7(1)''' (1998) 94-114.

Drorbn at 12:39, 10 October 2006

2006-10-10T12:39:22Z

← Older revision		Revision as of 08:39, 10 October 2006
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	\|55		\|55
	\|}		\|}

			==Some Further Computations==

			Some further computations for links and tangles were made by Siddarth Sankaran and Zavosh Amir-Khosravi. See [[VasCalc Results - ChordMod4T]].

	==Some Bounds==		==Some Bounds==

Drorbn at 12:34, 10 October 2006

2006-10-10T12:34:46Z

@@ Line 1: / Line 1: @@
 {{06-1350/Navigation}}
 [[Image:PlasticTrinions.jpg|thumb|left|150px|Plastic trinions]]
@@ Line 13: / Line 15: @@
 '''Dror's Speculation.''' I'm clueless.
 '''Question 4.''' A trinion <math>\gamma</math> is embedded in <math>{\mathbb R}^3</math> so that its "strapped boundary" is equivalent to the strapped boundary of the trivially embedded trinion. Does it follow that <math>\gamma</math> is trivial?
@@ Line 18: / Line 22: @@
 '''Dror's Speculation.''' If yes, it will have terrific consequences. If no, it will explain some of the misery we encounter when we deal with "associators". I would really like to understand this one.

Drorbn at 23:09, 9 October 2006

2006-10-09T23:09:41Z

← Older revision		Revision as of 19:09, 9 October 2006
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	'''Question 1.''' Can you embed a trinion (a.k.a. a sphere with three holes, a pair of pants, or a band theta graph) in <math>{\mathbb R}^3</math> so that each boundary component would be unknotted yet each pair of boundary components would be knotted? How about, so that at least one pair of boundary components would be knotted?		'''Question 1.''' Can you embed a trinion (a.k.a. a sphere with three holes, a pair of pants, or a band theta graph) in <math>{\mathbb R}^3</math> so that each boundary component would be unknotted yet each pair of boundary components would be knotted? How about, so that at least one pair of boundary components would be knotted?

			'''Dror's Speculation.''' Yes and yes.
⚫	'''Question 2.''' Suppose two trinions <math>\gamma_1</math> and <math>\gamma_2</math> are knotted so that the pushforwards <math>\gamma_{1\star}L</math> and <math>\gamma_{2\star}L</math> are equal for any link <math>L</math> which is "drawn" on the parameter space <math>\Gamma</math> of <math>\gamma_1</math> and <math>\gamma_2</math>. Does it follow that <math>\gamma_1</math> and <math>\gamma_2</math> are equivalent?

	'''Question 3.''' A trinion <math>\gamma</math> is embedded in <math>{\mathbb R}^3</math> so that its boundary is the trivial 3-component link. Does it follow that <math>\gamma</math> is trivial?		'''Question 2.''' A trinion <math>\gamma</math> is embedded in <math>{\mathbb R}^3</math> so that its boundary is the trivial 3-component link. Does it follow that <math>\gamma</math> is trivial?

			'''Dror's Speculation.''' No.

		⚫	'''Question 3.''' Suppose two trinions <math>\gamma_1</math> and <math>\gamma_2</math> are knotted so that the pushforwards <math>\gamma_{1\star}L</math> and <math>\gamma_{2\star}L</math> are equal for any link <math>L</math> which is "drawn" on the parameter space <math>\Gamma</math> of <math>\gamma_1</math> and <math>\gamma_2</math>. Does it follow that <math>\gamma_1</math> and <math>\gamma_2</math> are equivalent?

			'''Dror's Speculation.''' I'm clueless.

	'''Question 4.''' A trinion <math>\gamma</math> is embedded in <math>{\mathbb R}^3</math> so that its "strapped boundary" is equivalent to the strapped boundary of the trivially embedded trinion. Does it follow that <math>\gamma</math> is trivial?		'''Question 4.''' A trinion <math>\gamma</math> is embedded in <math>{\mathbb R}^3</math> so that its "strapped boundary" is equivalent to the strapped boundary of the trivially embedded trinion. Does it follow that <math>\gamma</math> is trivial?

			'''Dror's Speculation.''' If yes, it will have terrific consequences. If no, it will explain some of the misery we encounter when we deal with "associators". I would really like to understand this one.

	[[Image:StrappedTrinion.svg\|thumb\|center\|200px\|The standardly embedded strapped trinion]]		[[Image:StrappedTrinion.svg\|thumb\|center\|200px\|The standardly embedded strapped trinion]]

Drorbn at 22:52, 9 October 2006

2006-10-09T22:52:37Z

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	{{06-1350/Navigation}}		{{06-1350/Navigation}}

		⚫	[[Image:PlasticTrinions.jpg\|thumb\|left\|150px\|Plastic trinions]]
	'''Question.''' Can you embed a trinion (a.k.a. a sphere with three holes, a pair of pants, or a band theta graph) in <math>{\mathbb R}^3</math> so that each boundary component would be unknotted yet each pair of boundary components would be knotted?		'''Question 1.''' Can you embed a trinion (a.k.a. a sphere with three holes, a pair of pants, or a band theta graph) in <math>{\mathbb R}^3</math> so that each boundary component would be unknotted yet each pair of boundary components would be knotted? How about, so that at least one pair of boundary components would be knotted?

			'''Question 2.''' Suppose two trinions <math>\gamma_1</math> and <math>\gamma_2</math> are knotted so that the pushforwards <math>\gamma_{1\star}L</math> and <math>\gamma_{2\star}L</math> are equal for any link <math>L</math> which is "drawn" on the parameter space <math>\Gamma</math> of <math>\gamma_1</math> and <math>\gamma_2</math>. Does it follow that <math>\gamma_1</math> and <math>\gamma_2</math> are equivalent?
⚫	[[Image:PlasticTrinions.jpg\|thumb\|~~center~~\|~~250px~~\|Plastic trinions]]

			'''Question 3.''' A trinion <math>\gamma</math> is embedded in <math>{\mathbb R}^3</math> so that its boundary is the trivial 3-component link. Does it follow that <math>\gamma</math> is trivial?

			'''Question 4.''' A trinion <math>\gamma</math> is embedded in <math>{\mathbb R}^3</math> so that its "strapped boundary" is equivalent to the strapped boundary of the trivially embedded trinion. Does it follow that <math>\gamma</math> is trivial?

			[[Image:StrappedTrinion.svg\|thumb\|center\|200px\|The standardly embedded strapped trinion]]

Drorbn at 21:51, 6 October 2006

2006-10-06T21:51:27Z

New page

{{06-1350/Navigation}}

'''Question.''' Can you embed a trinion (a.k.a. a sphere with three holes, a pair of pants, or a band theta graph) in <math>{\mathbb R}^3</math> so that each boundary component would be unknotted yet each pair of boundary components would be knotted?

[[Image:PlasticTrinions.jpg|thumb|center|250px|Plastic trinions]]

← Older revision		Revision as of 08:40, 10 October 2006
Line 98:		Line 98:
	in {{ref\|Broadhurst_97}} but no progress has been made in the direction of proving or disproving it. The best asymptotic bounds available are:		in {{ref\|Broadhurst_97}} but no progress has been made in the direction of proving or disproving it. The best asymptotic bounds available are:

	'''Theorem.''' For large <math>m</math>, <math>\dim{\mathcal P}_m>e^{c\sqrt m}</math> (for any fixed <math>c<\pi\sqrt{\frac23}</math>, see {{ref\|Dasbach_00}}, {{ref\|~~Kontsevich_93~~}}) and <math>\dim{\mathcal A}_m<6^mm!\sqrt{m}/\pi^{2m}</math> ({{ref\|Stoimenow_98}}, {{ref\|Zagier_01}}).		'''Theorem.''' For large <math>m</math>, <math>\dim{\mathcal P}_m>e^{c\sqrt m}</math> (for any fixed <math>c<\pi\sqrt{\frac23}</math>, see {{ref\|Dasbach_00}}, {{ref\|Kontsevich}}) and <math>\dim{\mathcal A}_m<6^mm!\sqrt{m}/\pi^{2m}</math> ({{ref\|Stoimenow_98}}, {{ref\|Zagier_01}}).

	==Some References==		==Some References==
Line 112:		Line 112:
	{{note\|Kneissler_97}} J. A. Kneissler, ''The number of primitive Vassiliev invariants up to degree twelve,'' preprint, June 1997, {{arXiv\|q-alg/9706022}}.		{{note\|Kneissler_97}} J. A. Kneissler, ''The number of primitive Vassiliev invariants up to degree twelve,'' preprint, June 1997, {{arXiv\|q-alg/9706022}}.

	{{note\|~~Kontsevich_93~~}} M. Kontsevich, ~~''Vassiliev's knot invariants,'' Adv. in Sov. Math., '''16(2)''' (1993) 137-150~~.		{{note\|Kontsevich}} M. Kontsevich, unpublished.

	{{note\|Stoimenow_98}} A. Stoimenow, ''Enumeration of chord diagrams and an upper bound for Vassiliev invariants,'' Jour. of Knot Theory and its Ramifications '''7(1)''' (1998) 94-114.		{{note\|Stoimenow_98}} A. Stoimenow, ''Enumeration of chord diagrams and an upper bound for Vassiliev invariants,'' Jour. of Knot Theory and its Ramifications '''7(1)''' (1998) 94-114.