Date(s)

Link(s)

2010/02/22

???

2010/01/20

Formal integration

2010/01/13

Combing wB

2010/01/06

Exponentiation in tder

2009/09/22

descending vknots

2009/08/26

red over green vtangles

2009/08/19

Polyak Algebra

2009/07/08

Immanants

2009/07/01

Alexander modules

2009/06/24

Alexander modules

2009/06/10

Alexander, PBW for A^w, class videos

2009/06/03

Low key

2009/05/06

Low key

2009/04/29

Winter on Ribbons

2009/04/22

Misc

2009/04/15

KV

2009/03/25

KV

2009/03/18

Peter Lee

2009/03/04

Kirby calculus

2009/02/25

Karene on ReidemeisterSchreier

2009/02/11

Dror on Trotter, Jana on Alexander

2009/02/04

Bracelets

2009/01/28

gl(N) chickens

2009/01/15

2D Gauss Diagrams, FiC

2009/01/08

S&G update and more

2008/12/11

Chu on Garside, II

2008/12/04

wZ is 11

2008/11/27

The Wen

2008/11/20

The Zoom Space

2008/11/13

Chu on Garside

2008/11/06

Z and GPV

2008/10/30

Peter Lee on EHKR

2008/10/23

Map of the Field

2008/09/25

Hirasawa on Open Books

2008/09/18

Odd Khovanov

2008/09/17

Categorification.m

2008/09/11

More wAlex

2008/09/03

?

2008/08/27

Dexp and BCH

2008/08/06

Z, A, det, tr, log

2008/07/30

Alexander Relations Marathon

2008/07/02

Peter Lee on horizontal Aw

2008/06/25

wAlexander

2008/06 1622

Thomas Fiedler Marathon

2008/06/11

?

2008/06/04

Dylan Thurston

2008/05/28

Welded Tangles

2008/05/21

Bruce, Lucy

2008/04/23

Welded Knots

2008/04/16

Quandles and Lie algebras

2008/04/09

Nodiv AlekseevTorossian

2008/04/02

Knotted Kung Fu Pandas

2008/03/26

Homotopy invariants

2008/03/19

Infinitesimal Artin

2008/03/12

Infinitesimalization of Artin

2008/03/05

Krzysztof Putyra on Odd Khovanov Homology

2008/02/27

Karene Chu on Proof of Artin

2008/02/20

Organizational, Hecke algebras

2008/02/13

Exponential and Magnus expansions

2008/02/06

cancelled

2008/01/30

Artin's theorem

2008/01/16

Hutchings' work, 2

2008/01/09

Hutchings' work, 1

2007/12/12

Bone soup

2007/12/05

Expansions

2007/11/28

Quantum groups

2007/11/21

Surfaces and gl(N)/so(N)

2007/11/07

Expansions for Groups

2007/10/31

Louis Leung on bialgebra weight systems

2007/10/24

Zsuzsi Dancso, continued

2007/10/17

Jana Archibald on the multivariable Alexander

2007/10/10

Zsuzsi Dancso on diagrammatic su(2)

2007/10/03

Hernando Burgos on alternating tangles

2007/09/26

Peter Lee on homology

2007/09/06

Garoufalidis' visit

2007/08/30

Art and enumeration

2007/08/23

My Hanoi talk?

2007/08/16

Lie bialgebra weight systems and more

2007/07/19

Subdiagram formulas

2007/07/12

Playing with Brunnians

2007/07/05

Virtualization

2007/06/28

Virtual braids

2007/06/07

Virtual knots

2007/05/31

Social gathering

2007/05/24

Lee on Frozen Feet


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 brownbag 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 $\varphi :B\to S$ be a group homomorphism; denote its action by $b\mapsto {\bar {b}}$; i.e., let ${\bar {b}}:=\varphi (b)$ for every $b\in B$. Let "the virtualization $\operatorname {VB}$ of $B$", or more precisely, "the virtualization $\operatorname {VB} _{\varphi }$ of $B$ with respect to $\varphi$", be the following quotient of the free product $B\star S$ of $B$ and $S$:
$\operatorname {VB} :=B\star S\left/{\bar {b}}^{1}b_{1}{\bar {b}}=b_{2}\right.$ whenever $b,\,b_{1,2}\in B$ and $b^{1}b_{1}b=b_{2}$ in $B$.
In words, this is "if two element $b_{1,2}$ of $B$ are conjugate with conjugator $b$, in $\operatorname {VB}$ they are conjugate also using the shadow of $b$".
Though note that under the same circumstances we do not mod out by $b^{1}{\bar {b}}_{1}b={\bar {b}}_{2}$.
It is clear that $\varphi$ extends to a homomorphism ${\hat {\varphi }}:\operatorname {VB} \to S$. Let "the pure virtualization $\operatorname {PVB}$ of $B$" be the kernel of that homomorphism:
$\operatorname {PVB} :=\ker {\hat {\varphi }}\subset \operatorname {VB}$.
Question. Is this definition at all interesting? More precisely:
 If $B$ is a braid group and $S$ is the corresponding symmetric group, can $\operatorname {VB}$ be reasonably identified with "virtual braids"?
 Does the $\operatorname {PVB}$ that we get here agree with $\operatorname {PVB} _{n}$ of 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?
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 moot.