Knot at Lunch, June 28, 2007 - Revision history

Drorbn at 15:38, 28 June 2007

2007-06-28T15:38:22Z

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	{{KAL/Navigation}}		{{KAL/Navigation}}
	{{In Preparation}}

	==Invitation==		==Invitation==

Drorbn: /* Some Content */

2007-06-28T15:38:10Z

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← Older revision		Revision as of 11:38, 28 June 2007
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			{{Begin Side Note\|15%}}
⚫	'''Definition.''' The virtual braid group <math>~~VB_n~~</math> on <math>n</math> strands is the group generated by symbols <math>\lambda_{ij}</math> for <math>1\leq i\neq j\leq n</math> (standing for "strand <math>i</math> goes over strand <math>j</math> from λeft to right" ) and <math>\rho_{ij}</math> (standing for "strand <math>i</math> goes over strand <math>j</math> from ρight to left") modulo the relations
			When will it finally be easy to draw on a computer?
⚫	* <math>\lambda_{ij}\rho_{ji}=1</math>,
			{{End Side Note}}
⚫	* <math>\lambda_{ij}\lambda_{kl}=\lambda_{kl}\lambda_{ij}</math> (where <math>i,j,k,l</math> are distinct; variants: <math>\lambda_{ij}\rho_{kl}=\rho_{kl}\lambda_{ij}</math> and <math>\rho_{ij}\rho_{kl}=\rho_{kl}\rho_{ij}</math>)~~, and~~
		⚫	'''Definition.''' The pure virtual braid group <math>\operatorname{PVB}_n</math> on <math>n</math> strands is the group generated by symbols <math>\lambda_{ij}</math> for <math>1\leq i\neq j\leq n</math> (standing for "strand <math>i</math> goes over strand <math>j</math> from λeft to right and returning virtually") and <math>\rho_{ij}</math> (standing for "strand <math>i</math> goes over strand <math>j</math> from ρight to left, virtual return") modulo the relations:
⚫	* <math>\lambda_{ij}\lambda_{jk}\lambda_{ij}=\lambda_{jk}\lambda_{ij}\lambda_{jk}</math> (where <math>i,j,k</math> are distinct; variants?).
		⚫	* <math>\lambda_{ij}\rho_{ij}=1</math>.
		⚫	* <math>\lambda_{ij}\lambda_{kl}=\lambda_{kl}\lambda_{ij}</math> (where <math>i,j,k,l</math> are distinct; variants: <math>\lambda_{ij}\rho_{kl}=\rho_{kl}\lambda_{ij}</math> and <math>\rho_{ij}\rho_{kl}=\rho_{kl}\rho_{ij}</math>).
		⚫	* <math>\lambda_{ij}\lambda_{ik}\lambda_{jk}=\lambda_{jk}\lambda_{ik}\lambda_{ij}</math> (where <math>i,j,k</math> are distinct; variants?).

			{{Begin Side Note\|15%}}
			When will it finally be easy to draw on a computer?
			{{End Side Note}}
			At least at a naive level, the corresponding algebra of chord diagrams is generated by degree 1 symbols <math>a_{ij}</math> (corresponding to <math>\lambda_{ij}-1</math>) and <math>b_{ij}</math> (corresponding to <math>\rho_{ij}-1</math>) subject to the relations:
			* <math>a_{ij}+b_{ij}=0</math> (and hence the <math>b</math>'s can henceforth be ignored).
			* <math>[a_{ij},a_{kl}]=0</math>.
			* <math>a_{ij}a_{ik}+a_{ij}a_{jk}+a_{ik}a_{jk} = a_{jk}a_{ik}+a_{jk}a_{ij}+a_{ik}a_{ij}</math>.

	==References==		==References==

Drorbn: /* Some Content */

2007-06-28T15:00:54Z

Some Content

← Older revision		Revision as of 11:00, 28 June 2007
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	'''Definition.''' The virtual braid group <math>VB_n</math> on <math>n</math> strands is the group generated by symbols <math>\lambda_{ij}</math> for <math>1\leq i\neq j\leq n</math> (standing for "strand <math>i</math> goes over strand <math>j</math> from λeft to right" ) and <math>\rho_{ij}</math> (standing for "strand <math>i</math> goes over strand <math>j</math> from ρight to left") modulo the relations		'''Definition.''' The virtual braid group <math>VB_n</math> on <math>n</math> strands is the group generated by symbols <math>\lambda_{ij}</math> for <math>1\leq i\neq j\leq n</math> (standing for "strand <math>i</math> goes over strand <math>j</math> from λeft to right" ) and <math>\rho_{ij}</math> (standing for "strand <math>i</math> goes over strand <math>j</math> from ρight to left") modulo the relations
	* <math>\lambda_{ij}\rho_{ji}=1</math>,		* <math>\lambda_{ij}\rho_{ji}=1</math>,
	* <math>\lambda_{ij}\lambda_{kl}=\lambda_{kl}\lambda_{ij}</math>, <math>\lambda_{ij}\rho_{kl}=\rho_{kl}\lambda_{ij}</math> and <math>\rho_{ij}\rho_{kl}=\rho_{kl}\rho_{ij}</math> ~~(where <math>i,j,k,l</math> are distinct~~), and		* <math>\lambda_{ij}\lambda_{kl}=\lambda_{kl}\lambda_{ij}</math> (where <math>i,j,k,l</math> are distinct; variants: <math>\lambda_{ij}\rho_{kl}=\rho_{kl}\lambda_{ij}</math> and <math>\rho_{ij}\rho_{kl}=\rho_{kl}\rho_{ij}</math>), and
	* <math>\lambda_{ij}\lambda_{jk}\lambda_{ij}=\lambda_{jk}\lambda_{ij}\lambda_{jk}</math> ~~(and variants?)~~ (where <math>i,j,k</math> are distinct).		* <math>\lambda_{ij}\lambda_{jk}\lambda_{ij}=\lambda_{jk}\lambda_{ij}\lambda_{jk}</math> (where <math>i,j,k</math> are distinct; variants?).

	==References==		==References==

Drorbn: /* Some Content */

2007-06-28T14:39:33Z

Some Content

← Older revision		Revision as of 10:39, 28 June 2007
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	==Some Content==		==Some Content==

	'''Definition.''' The virtual braid group <math>VB_n</math> on <math>n</math> strands is the group generated by symbols <math>\~~sigma_~~{ij}</math> for <math>1\leq i\neq j\leq n</math> (standing for "strand <math>i</math> goes over strand <math>j</math>") modulo the relations		'''Definition.''' The virtual braid group <math>VB_n</math> on <math>n</math> strands is the group generated by symbols <math>\lambda_{ij}</math> for <math>1\leq i\neq j\leq n</math> (standing for "strand <math>i</math> goes over strand <math>j</math> from λeft to right" ) and <math>\rho_{ij}</math> (standing for "strand <math>i</math> goes over strand <math>j</math> from ρight to left") modulo the relations
	* <math>\~~sigma_~~{ij}\~~sigma_~~{ji}=1</math>,		* <math>\lambda_{ij}\rho_{ji}=1</math>,
	* <math>\~~sigma_~~{ij}\~~sigma_~~{kl}=\~~sigma_~~{kl}\~~sigma_~~{ij}</math> (where <math>i,j,k,l</math> are distinct), and		* <math>\lambda_{ij}\lambda_{kl}=\lambda_{kl}\lambda_{ij}</math>, <math>\lambda_{ij}\rho_{kl}=\rho_{kl}\lambda_{ij}</math> and <math>\rho_{ij}\rho_{kl}=\rho_{kl}\rho_{ij}</math> (where <math>i,j,k,l</math> are distinct), and
	* <math>\~~sigma_~~{ij}\~~sigma_~~{jk}\~~sigma_~~{ij}=\~~sigma_~~{jk}\~~sigma_~~{ij}\~~sigma_~~{jk}</math> (where <math>i,j,k</math> are distinct).		* <math>\lambda_{ij}\lambda_{jk}\lambda_{ij}=\lambda_{jk}\lambda_{ij}\lambda_{jk}</math> (and variants?) (where <math>i,j,k</math> are distinct).

	==References==		==References==

Drorbn at 14:21, 28 June 2007

2007-06-28T14:21:37Z

← Older revision		Revision as of 10:21, 28 June 2007
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	Dror.		Dror.

			==Some Content==

			'''Definition.''' The virtual braid group <math>VB_n</math> on <math>n</math> strands is the group generated by symbols <math>\sigma_{ij}</math> for <math>1\leq i\neq j\leq n</math> (standing for "strand <math>i</math> goes over strand <math>j</math>") modulo the relations
			* <math>\sigma_{ij}\sigma_{ji}=1</math>,
			* <math>\sigma_{ij}\sigma_{kl}=\sigma_{kl}\sigma_{ij}</math> (where <math>i,j,k,l</math> are distinct), and
			* <math>\sigma_{ij}\sigma_{jk}\sigma_{ij}=\sigma_{jk}\sigma_{ij}\sigma_{jk}</math> (where <math>i,j,k</math> are distinct).

	==References==		==References==

Drorbn: /* Invitation */

2007-06-28T12:32:28Z

Invitation

← Older revision		Revision as of 08:32, 28 June 2007
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	2007, at the usual place, Bahen 6180, at 12 noon, though this time it will		2007, at the usual place, Bahen 6180, at 12 noon, though this time it will
	be a short meeting - I'd like to attend a lecture at the Fields Institute		be a short meeting - I'd like to attend a lecture at the Fields Institute
	at 1PM (titled "Proof of Banach-Tarski paradox") so we have to finish		at 1PM (titled "[http://www.utoronto.ca/reg/list_full.pl?20070528-1310.27056 Proof of Banach-Tarski paradox]") so we have to finish
	before that.		before that.

Drorbn at 12:24, 28 June 2007

2007-06-28T12:24:49Z

New page

{{KAL/Navigation}}
{{In Preparation}}

==Invitation==

Dear Knot at Lunch People,

We will have the fourth of our weekly summer lunches on Thursday June 28,
2007, at the usual place, Bahen 6180, at 12 noon, though this time it will
be a short meeting - I'd like to attend a lecture at the Fields Institute
at 1PM (titled "Proof of Banach-Tarski paradox") so we have to finish
before that.

As always, please bring brown-bag lunch and fresh ideas. I have a few
ideas for the agenda: a bit of propaganda about computing, and some words
about an email I sent Dylan a few days ago (attached below). But of
course, we may end up talking about something else.

As always, if you know anyone I should add to this mailing list or if you
wish to be removed from this mailing list please let me know. To prevent
junk accumulation in mailboxes, I will actively remove inactive people
unless they request otherwise.

Best,

Dror.

----

Dylan,

In all things finite type (and elsewhere in knot theory) braids have
always been one of the simplest examples.

So here's a bunch of *specific* questions:

What are virtual braids? Can they be combed? What are finite type
invariants of virtual braids? Is there a structure-preserving universal
finite type invariant of virtual braids? Does it separate virtual braids?
Does every finite type invariant of braids extends to a finite type
invariant of virtual braids, of the same degree? Do "Lie-bialgebraic"
weight systems span all weight systems, for virtual braids? Can you write
honest formulas in terms of arrow diagrams for an R-matrix?

Dror.

==References==

The "computing" subject is around [http://www.math.toronto.edu/~drorbn/Misc/OneParameter/index.html Some One Parameter Knot Theory Computations] by Dror Bar-Natan and Thomas Fiedler.
The defining work on virtual knots is Kauffman's [http://www.math.uic.edu/~kauffman/Papers.html ''Virtual Knot Theory''],
Europ. J. Combinatorics '''20''' (1999) 663–691.
Other highly relevant papers are Goussarov-Polyak-Viro's ''Finite Type Invariants of Classical and Virtual Knots'', {{arXiv|math/9810073}} and Polyak's ''[http://www.springerlink.com/content/p32716130747l815/ On the Algebra of Arrow Diagrams]'', Letters in Mathematical Physics '''51-4''' (2000) 275-291.