Knot at Lunch, June 28, 2007

Knot at Lunch/Navigation  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 v-knots 2009/08/26 red over green v-tangles 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 Reidemeister-Schreier 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 1-1 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 w-Alexander 2008/06 16-22 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 No-div Alekseev-Torossian 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 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.

Some Content

When will it finally be easy to draw on a computer?

Definition. The pure virtual braid group $\operatorname {PVB} _{n}$ on $n$ strands is the group generated by symbols $\lambda _{ij}$ for $1\leq i\neq j\leq n$ (standing for "strand $i$ goes over strand $j$ from λeft to right and returning virtually") and $\rho _{ij}$ (standing for "strand $i$ goes over strand $j$ from ρight to left, virtual return") modulo the relations:

• $\lambda _{ij}\rho _{ij}=1$.
• $\lambda _{ij}\lambda _{kl}=\lambda _{kl}\lambda _{ij}$ (where $i,j,k,l$ are distinct; variants: $\lambda _{ij}\rho _{kl}=\rho _{kl}\lambda _{ij}$ and $\rho _{ij}\rho _{kl}=\rho _{kl}\rho _{ij}$).
• $\lambda _{ij}\lambda _{ik}\lambda _{jk}=\lambda _{jk}\lambda _{ik}\lambda _{ij}$ (where $i,j,k$ are distinct; variants?).

When will it finally be easy to draw on a computer?

At least at a naive level, the corresponding algebra of chord diagrams is generated by degree 1 symbols $a_{ij}$ (corresponding to $\lambda _{ij}-1$) and $b_{ij}$ (corresponding to $\rho _{ij}-1$) subject to the relations:

• $a_{ij}+b_{ij}=0$ (and hence the $b$'s can henceforth be ignored).
• $[a_{ij},a_{kl}]=0$.
• $a_{ij}a_{ik}+a_{ij}a_{jk}+a_{ik}a_{jk}=a_{jk}a_{ik}+a_{jk}a_{ij}+a_{ik}a_{ij}$.

References

The "computing" subject is around Some One Parameter Knot Theory Computations by Dror Bar-Natan and Thomas Fiedler. The defining work on virtual knots is Kauffman's 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 On the Algebra of Arrow Diagrams, Letters in Mathematical Physics 51-4 (2000) 275-291.