Knot at Lunch, July 5, 2007 - Revision history

Drorbn at 01:36, 21 July 2007

2007-07-21T01:36:30Z

← Older revision		Revision as of 21:36, 20 July 2007
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	===Added July 20, 2007===		===Added July 20, 2007===

	Well, following an email from Jana Comstock and Scott Morrison it is clear to me that the answer to the first question above is '''NO''', and hence the other questions above become ~~a bit~~ moot. It remains true that virtual braids are the free product of braids with their skeletons, though this statement has to be properly interpreted - '''not in the category of groups!''' I hope to get back to this at some point.		Well, following an email from Jana Comstock and Scott Morrison it is clear to me that the answer to the first question above is '''NO''', and hence the other questions above become moot.

Drorbn at 18:41, 20 July 2007

2007-07-20T18:41:43Z

← Older revision		Revision as of 14:41, 20 July 2007
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	* Are there other examples in which this definition is interesting?		* Are there other examples in which this definition is interesting?
	* Do we gain any new insight by using this definition?		* Do we gain any new insight by using this definition?

			===Added July 20, 2007===

			Well, following an email from Jana Comstock and Scott Morrison it is clear to me that the answer to the first question above is '''NO''', and hence the other questions above become a bit moot. It remains true that virtual braids are the free product of braids with their skeletons, though this statement has to be properly interpreted - '''not in the category of groups!''' I hope to get back to this at some point.

Drorbn: /* Some Content */

2007-07-06T21:03:05Z

Some Content

← Older revision		Revision as of 17:03, 6 July 2007
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	==Some Content==		==Some Content==

	{{In Preparation}}

	'''Definition.''' Let <math>\varphi:B\to S</math> be a group homomorphism; denote its action by <math>b\mapsto\bar b</math>; i.e., let <math>\bar b:=\varphi(b)</math> for every <math>b\in B</math>. Let "the virtualization <math>\operatorname{VB}</math> of <math>B</math>", or more precisely, "the virtualization <math>\operatorname{VB}_\varphi</math> of <math>B</math> with respect to <math>\varphi</math>", be the following quotient of the free product <math>B\star S</math> of <math>B</math> and <math>S</math>:		'''Definition.''' Let <math>\varphi:B\to S</math> be a group homomorphism; denote its action by <math>b\mapsto\bar b</math>; i.e., let <math>\bar b:=\varphi(b)</math> for every <math>b\in B</math>. Let "the virtualization <math>\operatorname{VB}</math> of <math>B</math>", or more precisely, "the virtualization <math>\operatorname{VB}_\varphi</math> of <math>B</math> with respect to <math>\varphi</math>", be the following quotient of the free product <math>B\star S</math> of <math>B</math> and <math>S</math>:

	{{Equation*\|<math>\operatorname{VB}:=B\star S\left/s^{-1}~~b_1s~~=b_2\right.</math> whenever <math>s\~~in S</math>~~, ~~<math>~~b_{1,2}\in B</math> and <math>s^{-1}~~\bar b_1s~~=~~\bar~~ b_2</math> in <math>S</math>.}}		{{Equation*\|<math>\operatorname{VB}:=B\star S\left/\bar b^{-1}b_1\bar b=b_2\right.</math> whenever <math>b,\,b_{1,2}\in B</math> and <math>b^{-1}b_1b=b_2</math> in <math>B</math>.}}

	In words, this is "if ~~the shadows of~~ two ~~elements~~ of <math>B</math> are conjugate in <math>S</math>, ~~the~~ ~~two~~ are ~~made~~ conjugate, ~~with~~ the ~~same~~ ~~conjugator, in~~ <math>~~\operatorname{VB}~~</math>".		In words, this is "if two element <math>b_{1,2}</math> of <math>B</math> are conjugate with conjugator <math>b</math>, in <math>\operatorname{VB}</math> they are conjugate also using the shadow of <math>b</math>".

	Though note that we do not mod out by <math>b^{-1}~~s_1b=s_2</math> when <math>s_{1,2}\in S</math>, <math>b\in B</math> and <math>~~\bar ~~b^{-1}s_1~~\bar ~~b=s_2~~</math>.		Though note that under the same circumstances we do not mod out by <math>b^{-1}\bar b_1b=\bar b_2</math>.

	It is clear that <math>\varphi</math> extends to a homomorphism <math>\hat\varphi:\operatorname{VB}\to S</math>. Let "the pure virtualization <math>\operatorname{PVB}</math> of <math>B</math>" be the kernel of that homomorphism:		It is clear that <math>\varphi</math> extends to a homomorphism <math>\hat\varphi:\operatorname{VB}\to S</math>. Let "the pure virtualization <math>\operatorname{PVB}</math> of <math>B</math>" be the kernel of that homomorphism:

Drorbn: /* Some Content */

2007-07-06T18:16:45Z

Some Content

← Older revision		Revision as of 14:16, 6 July 2007
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	==Some Content==		==Some Content==

			{{In Preparation}}

	'''Definition.''' Let <math>\varphi:B\to S</math> be a group homomorphism; denote its action by <math>b\mapsto\bar b</math>; i.e., let <math>\bar b:=\varphi(b)</math> for every <math>b\in B</math>. Let "the virtualization <math>\operatorname{VB}</math> of <math>B</math>", or more precisely, "the virtualization <math>\operatorname{VB}_\varphi</math> of <math>B</math> with respect to <math>\varphi</math>", be the following quotient of the free product <math>B\star S</math> of <math>B</math> and <math>S</math>:		'''Definition.''' Let <math>\varphi:B\to S</math> be a group homomorphism; denote its action by <math>b\mapsto\bar b</math>; i.e., let <math>\bar b:=\varphi(b)</math> for every <math>b\in B</math>. Let "the virtualization <math>\operatorname{VB}</math> of <math>B</math>", or more precisely, "the virtualization <math>\operatorname{VB}_\varphi</math> of <math>B</math> with respect to <math>\varphi</math>", be the following quotient of the free product <math>B\star S</math> of <math>B</math> and <math>S</math>:

Drorbn: /* Some Content */

2007-07-06T14:47:13Z

Some Content

← Older revision		Revision as of 10:47, 6 July 2007
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	==Some Content==		==Some Content==

	'''Definition.''' Let <math>\varphi:B\to S</math> be a group homomorphism; denote its action by <math>b\mapsto\bar b</math>; i.e., let <math>\bar b:=\varphi(b)</math> for every <math>b\in B</math>. Let "the virtualization <math>VB</math> of <math>B</math>", or more precisely, "the virtualization <math>~~VB_~~\varphi</math> of <math>B</math> with respect to <math>\varphi</math>", be the following quotient of the free product <math>B\star S</math> of <math>B</math> and <math>S</math>:		'''Definition.''' Let <math>\varphi:B\to S</math> be a group homomorphism; denote its action by <math>b\mapsto\bar b</math>; i.e., let <math>\bar b:=\varphi(b)</math> for every <math>b\in B</math>. Let "the virtualization <math>\operatorname{VB}</math> of <math>B</math>", or more precisely, "the virtualization <math>\operatorname{VB}_\varphi</math> of <math>B</math> with respect to <math>\varphi</math>", be the following quotient of the free product <math>B\star S</math> of <math>B</math> and <math>S</math>:

	{{Equation*\|<math>VB:=B\star S\left/s^{-1}b_1s=b_2\right.</math> whenever <math>s\in S</math>, <math>b_{1,2}\in B</math> and <math>s^{-1}\bar b_1s=\bar b_2</math> in <math>S</math>.}}		{{Equation*\|<math>\operatorname{VB}:=B\star S\left/s^{-1}b_1s=b_2\right.</math> whenever <math>s\in S</math>, <math>b_{1,2}\in B</math> and <math>s^{-1}\bar b_1s=\bar b_2</math> in <math>S</math>.}}

			In words, this is "if the shadows of two elements of <math>B</math> are conjugate in <math>S</math>, the two are made conjugate, with the same conjugator, in <math>\operatorname{VB}</math>".

	Though note that we do not mod out by <math>b^{-1}s_1b=s_2</math> when <math>s_{1,2}\in S</math>, <math>b\in B</math> and <math>\bar b^{-1}s_1\bar b=s_2</math>.		Though note that we do not mod out by <math>b^{-1}s_1b=s_2</math> when <math>s_{1,2}\in S</math>, <math>b\in B</math> and <math>\bar b^{-1}s_1\bar b=s_2</math>.

	It is clear that <math>\varphi</math> extends to a homomorphism <math>\hat\varphi:VB\to S</math>. Let "the pure virtualization <math>PVB</math> of <math>B</math>" be the kernel of that homomorphism:		It is clear that <math>\varphi</math> extends to a homomorphism <math>\hat\varphi:\operatorname{VB}\to S</math>. Let "the pure virtualization <math>\operatorname{PVB}</math> of <math>B</math>" be the kernel of that homomorphism:

	{{Equation*\|<math>PVB:=\ker\hat\varphi\subset VB</math>.}}		{{Equation*\|<math>\operatorname{PVB}:=\ker\hat\varphi\subset\operatorname{VB}</math>.}}

	'''Question.''' Is this definition at all interesting? More precisely:		'''Question.''' Is this definition at all interesting? More precisely:
	* If <math>B</math> is a braid group and <math>S</math> is the corresponding symmetric group, can <math>VB</math> be reasonably identified with "virtual braids"?		* If <math>B</math> is a braid group and <math>S</math> is the corresponding symmetric group, can <math>\operatorname{VB}</math> be reasonably identified with "virtual braids"?
	* Does the <math>PVB</math> that we get here agree with <math>~~PVB_n~~</math> of [[Knot at Lunch, June 28, 2007\|last time]]?		* Does the <math>\operatorname{PVB}</math> that we get here agree with <math>\operatorname{PVB}_n</math> of [[Knot at Lunch, June 28, 2007\|last time]]?
	* Is this definition encountered anywhere else in mathematics?		* Is this definition encountered anywhere else in mathematics?
	* Are there other examples in which this definition is interesting?		* Are there other examples in which this definition is interesting?

Drorbn: /* Some Content */

2007-07-06T14:18:36Z

Some Content

← Older revision		Revision as of 10:18, 6 July 2007
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	==Some Content==		==Some Content==

	'''Definition.''' Let <math>\varphi:B\to S</math> be a group homomorphism; denote its action by <math>~~\beta~~\mapsto\bar~~\beta~~</math>; i.e., let <math>\bar~~\beta~~:=\varphi(~~\beta~~)</math> for every <math>~~\beta~~\in B</math>. Let "the virtualization <math>VB</math> of <math>B</math>", or more precisely, "the virtualization <math>VB_\varphi</math> of <math>B</math> with respect to <math>\varphi</math>", be the following quotient of the free product <math>B\star S</math> of <math>B</math> and <math>S</math>:		'''Definition.''' Let <math>\varphi:B\to S</math> be a group homomorphism; denote its action by <math>b\mapsto\bar b</math>; i.e., let <math>\bar b:=\varphi(b)</math> for every <math>b\in B</math>. Let "the virtualization <math>VB</math> of <math>B</math>", or more precisely, "the virtualization <math>VB_\varphi</math> of <math>B</math> with respect to <math>\varphi</math>", be the following quotient of the free product <math>B\star S</math> of <math>B</math> and <math>S</math>:

	{{Equation*\|<math>VB:=B\star S\left/~~\bar\beta\beta_1~~=~~\beta_2\bar\beta~~\right.</math> whenever <math>\~~beta~~\~~beta_1~~=\~~beta_2\beta~~</math> in <math>B</math>.}}		{{Equation*\|<math>VB:=B\star S\left/s^{-1}b_1s=b_2\right.</math> whenever <math>s\in S</math>, <math>b_{1,2}\in B</math> and <math>s^{-1}\bar b_1s=\bar b_2</math> in <math>S</math>.}}

	(Though note that we do not mod out by <math>~~\beta\bar\beta_1~~=\~~bar\beta_2~~\~~beta~~</math> ~~under~~ ~~the~~ ~~same~~ ~~circumstances)~~.		Though note that we do not mod out by <math>b^{-1}s_1b=s_2</math> when <math>s_{1,2}\in S</math>, <math>b\in B</math> and <math>\bar b^{-1}s_1\bar b=s_2</math>.

	It is clear that <math>\varphi</math> extends to a homomorphism <math>\hat\varphi:VB\to S</math>. Let "the pure virtualization <math>PVB</math> of <math>B</math>" be the kernel of that homomorphism:		It is clear that <math>\varphi</math> extends to a homomorphism <math>\hat\varphi:VB\to S</math>. Let "the pure virtualization <math>PVB</math> of <math>B</math>" be the kernel of that homomorphism:

Drorbn at 15:51, 5 July 2007

2007-07-05T15:51:53Z

← Older revision		Revision as of 11:51, 5 July 2007
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	==Some Content==		==Some Content==

	'''Definition.''' Let <math>\varphi:B\to S</math> be a group homomorphism; denote its action by <math>\beta\mapsto\bar\beta</math>; i.e., let <math>\bar\beta:=\varphi(\beta)</math> for every <math>\beta\in B</math>.		'''Definition.''' Let <math>\varphi:B\to S</math> be a group homomorphism; denote its action by <math>\beta\mapsto\bar\beta</math>; i.e., let <math>\bar\beta:=\varphi(\beta)</math> for every <math>\beta\in B</math>. Let "the virtualization <math>VB</math> of <math>B</math>", or more precisely, "the virtualization <math>VB_\varphi</math> of <math>B</math> with respect to <math>\varphi</math>", be the following quotient of the free product <math>B\star S</math> of <math>B</math> and <math>S</math>:

			{{Equation*\|<math>VB:=B\star S\left/\bar\beta\beta_1=\beta_2\bar\beta\right.</math> whenever <math>\beta\beta_1=\beta_2\beta</math> in <math>B</math>.}}

			(Though note that we do not mod out by <math>\beta\bar\beta_1=\bar\beta_2\beta</math> under the same circumstances).

			It is clear that <math>\varphi</math> extends to a homomorphism <math>\hat\varphi:VB\to S</math>. Let "the pure virtualization <math>PVB</math> of <math>B</math>" be the kernel of that homomorphism:

			{{Equation*\|<math>PVB:=\ker\hat\varphi\subset VB</math>.}}

			'''Question.''' Is this definition at all interesting? More precisely:
			* If <math>B</math> is a braid group and <math>S</math> is the corresponding symmetric group, can <math>VB</math> be reasonably identified with "virtual braids"?
			* Does the <math>PVB</math> that we get here agree with <math>PVB_n</math> of [[Knot at Lunch, June 28, 2007\|last time]]?
			* Is this definition encountered anywhere else in mathematics?
			* Are there other examples in which this definition is interesting?
			* Do we gain any new insight by using this definition?

Drorbn at 15:31, 5 July 2007

2007-07-05T15:31:45Z

New page

{{KAL/Navigation}}

==Invitation==

Dear Knot at Lunch People,

We will have our next summer lunch on Thursday July 5, 2007, at the usual place, Bahen 6180, at 12 noon.

As always, please bring brown-bag lunch and fresh ideas. I'm not sure what we will be talking about; perhaps just continue with last week's topics.

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.

==Some Content==

'''Definition.''' Let <math>\varphi:B\to S</math> be a group homomorphism; denote its action by <math>\beta\mapsto\bar\beta</math>; i.e., let <math>\bar\beta:=\varphi(\beta)</math> for every <math>\beta\in B</math>.