Knot at Lunch on October 23 2008 - Revision history

Drorbn at 19:05, 23 October 2008

2008-10-23T19:05:08Z

← Older revision		Revision as of 15:05, 23 October 2008
Line 35:		Line 35:
	\|		\|
	*Related to metrized Lie algebras!		*Related to metrized Lie algebras!
			*This is the Kontsevich integral!
	\|		\|
	*Related to Chern-Simons theory and configuration space integrals!		*Related to Chern-Simons theory and configuration space integrals!
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	\|- valign=top		\|- valign=top
	\|		\|
	\|colspan=3\|The modifier v means "virtual". Chords become arrows, planar algebras become circuit algebras and <math>4T</math> becomes <math>6T</math>.		\|colspan=3\|The modifier v means "virtual". Chords become arrows, planar algebras become circuit algebras and <math>4T</math> becomes <math>6T</math>. "Algebraic Knot Theory" concepts still apply!
	\|- valign=top		\|- valign=top
	\|		\|
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	\|		\|
	*Related to the Kashiwara-Vergne conjecture and the orbit method!		*Related to the Kashiwara-Vergne conjecture and the orbit method!
	*Via ~~Alkseev~~-Torrosian, explicit tree-level formulas for associators!		*Via Alekseev-Torrosian, explicit tree-level formulas for associators!
	\|		\|
	\|- valign=top		\|- valign=top

Drorbn at 19:03, 23 October 2008

2008-10-23T19:03:30Z

← Older revision		Revision as of 15:03, 23 October 2008
Line 16:		Line 16:

	{{Dror}}.		{{Dror}}.

			==Blackboard Shots==

			See {{BBS Link\|KAL-081023-135536.jpg}}.

	==The Gospel in <math>v</math>, <math>w</math> and Seven <math>Z</math>'s==		==The Gospel in <math>v</math>, <math>w</math> and Seven <math>Z</math>'s==

Drorbn at 18:58, 23 October 2008

2008-10-23T18:58:03Z

← Older revision		Revision as of 14:58, 23 October 2008
Line 17:		Line 17:
	{{Dror}}.		{{Dror}}.

			==The Gospel in <math>v</math>, <math>w</math> and Seven <math>Z</math>'s==
	{{The Gospel}}

			{\|border=1 cellpadding=0 cellspacing=0
			\|- valign=top
			\|
			\|<math>Z:B\to{\mathcal A}^h</math>
			\|colspan=2\|
			*Classifies braids!
			*Detected by Lie algebras!
			\|- valign=top
			\| '''1'''
			\|<math>Z:{\mathcal K}(\bigcirc)\to{\mathcal A}(\bigcirc)</math>
			\|
			*Related to metrized Lie algebras!
			\|
			*Related to Chern-Simons theory and configuration space integrals!
			*Has GPV formulas!
			\|- valign=top
			\| '''2'''
			\|<math>Z:{\mathcal K}^{TG}\to{\mathcal A}^{TG}</math>
			\|
			*Enables "Algebraic Knot Theory"!
			*Finitely presented!
			*Related to Drinfel'd associators!
			\|
			*No-go theorems in bounded degrees.
			*Extremely challenging to compute.
			*Leads to "internal quotients"!
			*Still challenging to compute.
			\|- valign=top
			\|
			\|<math>Z:pT\to{\mathcal A}^{pT}</math>
			\|colspan=2\|
			*Does not exist.
			\|- valign=top
			\|
			\|colspan=3\|The modifier v means "virtual". Chords become arrows, planar algebras become circuit algebras and <math>4T</math> becomes <math>6T</math>.
			\|- valign=top
			\|
			\|colspan=3\|The modifier w means "<u>w</u>elded" or "<u>w</u>eakly virtual". Overcrossings commute, tails commute.
			\|- valign=top
			\| '''3'''
			\|<math>Z:wB\to{\mathcal A}^{wh}</math>
			\|
			*Ribbon tori in <math>{\mathbb R}^4</math>!
			*Flying rings in <math>{\mathbb R}^3</math>!
			*Easy formulas for <math>Z</math>!
			\|
			*Relates to <math>\operatorname{Aut}(F_n)</math>!
			*Classifies w-braids!
			\|- valign=top
			\| '''4'''
			\|<math>Z:w{\mathcal K}(\bigcirc)\to{\mathcal A}^w(\bigcirc)</math>
			\|
			*Related to the Alexander polynomial!
			*Related to general Lie algebras!
			*Related to co-commutative Lie bi-algebras!
			\|
			*Related to BF theory!
			\|- valign=top
			\| '''5'''
			\|<math>Z:w{\mathcal T}\to{\mathcal A}^{wT}</math>
			\|
			*Still easy!
			*Related (by other means) to free Lie algebras, derivations, traces!
			\|
			\|- valign=top
			\| '''6'''
			\|<math>Z:w{\mathcal TT}\to{\mathcal A}^{wTT}</math>
			\|
			*Related to the Kashiwara-Vergne conjecture and the orbit method!
			*Via Alkseev-Torrosian, explicit tree-level formulas for associators!
			\|
			\|- valign=top
			\| '''7'''
			\|<math>Z:v{\mathcal TT}\to{\mathcal A}^{vTT}</math>
			\|
			*Related to general Lie bialgebras, Etingof-Kazhdan and quantum groups in general!
			\|
			\|}

Drorbn: /* The Gospel in $v$ , $w$ and Seven $Z$ 's */

2008-10-23T15:51:00Z

The Gospel in <math>v</math>, <math>w</math> and Seven <math>Z</math>'s

← Older revision		Revision as of 11:51, 23 October 2008
Line 17:		Line 17:
	{{Dror}}.		{{Dror}}.

			{{The Gospel}}
	==The Gospel in <math>v</math>, <math>w</math> and Seven <math>Z</math>'s==

	{\|border=1 cellpadding=0 cellspacing=0
	\|- valign=top
	\|
	\|<math>Z:B\to{\mathcal A}^h</math>
	\|colspan=2\|
	*Classifies braids!
	*Detected by Lie algebras!
	\|- valign=top
	\| '''1'''
	\|<math>Z:{\mathcal K}(\bigcirc)\to{\mathcal A}(\bigcirc)</math>
	\|
	*Related to metrized Lie algebras!
	\|
	*Related to Chern-Simons theory and configuration space integrals!
	*Has GPV formulas!
	\|- valign=top
	\| '''2'''
	\|<math>Z:{\mathcal K}^{TG}\to{\mathcal A}^{TG}</math>
	\|
	*Enables "Algebraic Knot Theory"!
	*Finitely presented!
	*Related to Drinfel'd associators!
	\|
	*No-go theorems in bounded degrees.
	*Extremely challenging to compute.
	*Leads to "internal quotients"!
	*Still challenging to compute.
	\|- valign=top
	\|
	\|<math>Z:pT\to{\mathcal A}^{pT}</math>
	\|colspan=2\|
	*Does not exist.
	\|- valign=top
	\|
	\|colspan=3\|The modifier v means "virtual". Chords become arrows, planar algebras become circuit algebras and <math>4T</math> becomes <math>6T</math>.
	\|- valign=top
	\|
	\|colspan=3\|The modifier w means "<u>w</u>elded" or "<u>w</u>eakly virtual". Overcrossings commute, tails commute.
	\|- valign=top
	\| '''3'''
	\|<math>Z:wB\to{\mathcal A}^{wh}</math>
	\|
	*Ribbon tori in <math>{\mathbb R}^4</math>!
	*Flying rings in <math>{\mathbb R}^3</math>!
	*Easy formulas for <math>Z</math>!
	\|
	*Relates to <math>\operatorname{Aut}(F_n)</math>!
	*Classifies w-braids!
	\|- valign=top
	\| '''4'''
	\|<math>Z:w{\mathcal K}(\bigcirc)\to{\mathcal A}^w(\bigcirc)</math>
	\|
	*Related to the Alexander polynomial!
	*Related to general Lie algebras!
	*Related to co-commutative Lie bi-algebras!
	\|
	*Related to BF theory!
	\|- valign=top
	\| '''5'''
	\|<math>Z:w{\mathcal T}\to{\mathcal A}^{wT}</math>
	\|
	*Still easy!
	*Related (by other means) to free Lie algebras, derivations, traces!
	\|
	\|- valign=top
	\| '''6'''
	\|<math>Z:w{\mathcal TT}\to{\mathcal A}^{wTT}</math>
	\|
	*Related to the Kashiwara-Vergne conjecture and the orbit method!
	*Via Alkseev-Torrosian, explicit tree-level formulas for associators!
	\|
	\|- valign=top
	\| '''7'''
	\|<math>Z:v{\mathcal TT}\to{\mathcal A}^{vTT}</math>
	\|
	*Related to general Lie bialgebras, Etingof-Kazhdan and quantum groups in general!
	\|
	\|}

Drorbn: /* The Gospel in $v$ , $w$ and Seven $Z$ 's */

2008-10-23T15:46:35Z

The Gospel in <math>v</math>, <math>w</math> and Seven <math>Z</math>'s

← Older revision		Revision as of 11:46, 23 October 2008
Line 97:		Line 97:
	\|		\|
	\|}		\|}

	(Add the no for ordinary tangles, add a radical statement?)

Drorbn: /* The Gospel in $v$ , $w$ and Seven $Z$ 's */

2008-10-23T15:46:16Z

The Gospel in <math>v</math>, <math>w</math> and Seven <math>Z</math>'s

← Older revision		Revision as of 11:46, 23 October 2008
Line 80:		Line 80:
	\|<math>Z:w{\mathcal T}\to{\mathcal A}^{wT}</math>		\|<math>Z:w{\mathcal T}\to{\mathcal A}^{wT}</math>
	\|		\|
			*Still easy!
	*A "circuit algebra" invariant!
	*Related (by other means) to free Lie algebras, derivations, traces!		*Related (by other means) to free Lie algebras, derivations, traces!
	\|		\|

Drorbn: /* The Gospel in $v$ , $w$ and Seven $Z$ 's */

2008-10-23T15:45:32Z

The Gospel in <math>v</math>, <math>w</math> and Seven <math>Z</math>'s

← Older revision		Revision as of 11:45, 23 October 2008
Line 46:		Line 46:
	*Leads to "internal quotients"!		*Leads to "internal quotients"!
	*Still challenging to compute.		*Still challenging to compute.
			\|- valign=top
			\|
			\|<math>Z:pT\to{\mathcal A}^{pT}</math>
			\|colspan=2\|
			*Does not exist.
			\|- valign=top
			\|
			\|colspan=3\|The modifier v means "virtual". Chords become arrows, planar algebras become circuit algebras and <math>4T</math> becomes <math>6T</math>.
			\|- valign=top
			\|
			\|colspan=3\|The modifier w means "<u>w</u>elded" or "<u>w</u>eakly virtual". Overcrossings commute, tails commute.
	\|- valign=top		\|- valign=top
	\| '''3'''		\| '''3'''
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	\|- valign=top		\|- valign=top
	\| '''7'''		\| '''7'''
	\|<math>Z:v{\mathcal K}~~(\bigcirc)~~\to{\mathcal A}^~~v(\bigcirc)~~</math>		\|<math>Z:v{\mathcal TT}\to{\mathcal A}^{vTT}</math>
	\|		\|
	*Related to general Lie bialgebras, Etingof-Kazhdan and quantum groups in general!		*Related to general Lie bialgebras, Etingof-Kazhdan and quantum groups in general!
	\|		\|
	*Fully extends algebraic knot theory!
	\|}		\|}

Drorbn: /* The Gospel in Seven $Z$ 's */

2008-10-23T11:57:04Z

The Gospel in Seven <math>Z</math>'s

← Older revision		Revision as of 07:57, 23 October 2008
Line 17:		Line 17:
	{{Dror}}.		{{Dror}}.

	==The Gospel in Seven <math>Z</math>'s==		==The Gospel in <math>v</math>, <math>w</math> and Seven <math>Z</math>'s==

	{\|border=1 cellpadding=0 cellspacing=0		{\|border=1 cellpadding=0 cellspacing=0
Line 87:		Line 87:
	*Fully extends algebraic knot theory!		*Fully extends algebraic knot theory!
	\|}		\|}

			(Add the no for ordinary tangles, add a radical statement?)

Drorbn: /* The Gospel in Seven $Z$ 's */

2008-10-23T09:04:51Z

The Gospel in Seven <math>Z</math>'s

@@ Line 46: / Line 46: @@
 *Leads to "internal quotients"!
 *Still challenging to compute.
 |}

Drorbn at 08:39, 23 October 2008

2008-10-23T08:39:02Z

← Older revision		Revision as of 04:39, 23 October 2008
Line 16:		Line 16:

	{{Dror}}.		{{Dror}}.

			==The Gospel in Seven <math>Z</math>'s==

			{\|border=1 cellpadding=0 cellspacing=0
			\|- valign=top
			\|
			\|<math>Z:B\to{\mathcal A}^h</math>
			\|colspan=2\|
			*Classifies braids!
			*Detected by Lie algebras!
			\|- valign=top
			\| '''1'''
			\|<math>Z:{\mathcal K}(\bigcirc)\to{\mathcal A}(\bigcirc)</math>
			\|
			*Related to metrized Lie algebras!
			\|
			*Related to Chern-Simons theory and configuration space integrals!
			*Has GPV formulas!
			\|- valign=top
			\| '''2'''
			\|<math>Z:{\mathcal K}^{TG}\to{\mathcal A}^{TG}</math>
			\|
			*Enables "Algebraic Knot Theory"!
			*Finitely presented!
			*Related to Drinfel'd associators!
			\|
			*No-go theorems in bounded degrees.
			*Extremely challenging to compute.
			*Leads to "internal quotients"!
			*Still challenging to compute.
			\|}

Knot at Lunch on October 23 2008 - Revision history

Drorbn at 19:05, 23 October 2008

Drorbn at 19:03, 23 October 2008

Drorbn at 18:58, 23 October 2008

Drorbn: /* The Gospel in v, w and Seven Z's */

Drorbn: /* The Gospel in v, w and Seven Z's */

Drorbn: /* The Gospel in v, w and Seven Z's */

Drorbn: /* The Gospel in v, w and Seven Z's */

Drorbn: /* The Gospel in Seven Z's */

Drorbn: /* The Gospel in Seven Z's */

Drorbn at 08:39, 23 October 2008

Drorbn: /* The Gospel in $v$ , $w$ and Seven $Z$ 's */

Drorbn: /* The Gospel in $v$ , $w$ and Seven $Z$ 's */

Drorbn: /* The Gospel in $v$ , $w$ and Seven $Z$ 's */

Drorbn: /* The Gospel in $v$ , $w$ and Seven $Z$ 's */

Drorbn: /* The Gospel in Seven $Z$ 's */

Drorbn: /* The Gospel in Seven $Z$ 's */