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wClips-120201

Speaker: Peter Lee

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Section 2.3 - finite type invariants of v- and w-braids, arrow diagrams, 6T, TC and 4T relations, expansions / universal finite type invariants.


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Resources. How to use this site, Dror's notebook, blackboard shots.

The wClips

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Date Links
Jan 11, 2012 dbnvp 120111-1: Introduction.
dbnvp 120111-2: Section 2.1 - v-Braids.
Jan 18, 2012 dbnvp 120118-1: An introduction to this web site.
dbnvp 120118-2: Section 2.2 - w-Braids by generators and relations and as flying rings.
dbnvp 120118-3: Section 2.2 - w-Braids - other drawing conventions, "wens".
Jan 25, 2012 dbnvp 120125-1: Section 2.2.3 - basis conjugating automorphisms of F_n.
dbnvp 120125-2: A very quick introduction to finite type invariants in the "u" case.
Feb 1, 2012 dbnvp 120201: Section 2.3 - finite type invariants of v- and w-braids, arrow diagrams, 6T, TC and 4T relations, expansions / universal finite type invariants.
Feb 8, 2012 dbnvp 120208: Review of u,v, and w braids and of Section 2.3.
Feb 15, 2012 dbnvp 120215: Section 2.5 - mostly compatibilities of Z^w, also injectivity and uniqueness of Z^w.
Feb 22, 2012 dbnvp 120222: Section 2.5.5, \alpha:{\mathcal A}^u\to{\mathcal A}^v, and Section 3.1 (partially), the definition of v- and w-knots.
Feb 29, 2012 dbnvp 120229: Sections 3.1-3.4: v-Knots and w-Knots: Definitions, framings, finite type invariants, dimensions, and the expansion in the w case.
Mar 7, 2012 dbnvp 120307: Section 3.5: Jacobi diagrams and the bracket-rise theorem.
Mar 14, 2012 dbnvp 120314: Section 3.6 - the relation with Lie algebras.
Mar 21, 2012 dbnvp 120321: Section 4 - Algebraic Structures.
Mar 28, 2012 Out-of-sequence not-on-tape we watched the video of Talks: GWU-1203.
Apr 4, 2012 dbnvp 120404: Section 3.7 - The Alexander Theorem (statement).
Apr 18, 2012 dbnvp 120418: Aside on the Euler trick, the differential of \exp, and the BCH formula.
Apr 25, 2012 dbnvp 120425: Section 3.8, a disorganized lecture towards the proof of the Alexander theorem.
May 2, 2012 dbnvp 120502: Section 4: Algebraic structures (review), circuit algebras, v- and w-tangles.
May 10, 2012 dbnvp 120510: Sections 5.1 and 5.2: tangles, their projectivization and its relationship with Alekseev-Torossian spaces.
May 23, 2012 dbnvp 120523: Section 5.2: Proof of the relationship with A-T spaces.
May 30, 2012 dbnvp 120530: Interpreting {\mathcal A}^w(\uparrow_n) as a universal space of invariant tangential differential operators.
wClips Seminar Group Photo
Group photo on January 11, 2012: DBN, ZD, Stephen Morgan, Lucy Zhang, Iva Halacheva, David Li-Bland, Sam Selmani, Oleg Chterental, Peter Lee.
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0:04:37 [edit] The "rational homotopy" route is often taken in braid theory; however we don't take it because it does not generalize to more complex algebraic structures (such as planar algebras and circuit algebras), whereas the "augmentation ideal" approach that we take generalizes to practically anything. --Drorbn 18:38, 1 February 2012 (EST)
0:06:54 [add] The augmentation ideal and its powers.
0:10:09 [add] Type p invariants.
0:12:05 [edit] Q: What does generation mean? Where is the constraint \sum_i{q_i}=0 captured in the generation equation? -- Lucy
0:14:50 [edit] Question/Claim (not stated in the video): The set of \bar{\simga_{ij}} generate I.
0:14:50 [add] The PvBn case.
0:16:31 [edit] "Also consider \mbox{gr }{\mathbb Q}G = \bigoplus_{p\geq 0}I^p/I^{p+1}"
0:17:20 [edit] Clarification: The questions on the board refer to "What are the generators and relations of gr(QG)", where G=PvBn here.
0:18:09 [edit] Needless to say, much more and much more general appears in Peter's own paper, The Pure Virtual Braid Group Is Quadratic, arXiv:1110.2356. --Drorbn 18:59, 1 February 2012 (EST)
0:21:14 [add] What are the generators? What are the relations?
0:24:41 [add] Arrow diagrams.
0:28:13 [add] Arrow diagrams (2).
0:31:18 [add] Quadratic relations.
0:36:55 [add] 6T / CYB.
0:37:10 [edit] In arrow-diagram notation, the six-term (6T) relation is:

100%
0:42:48 [add] A^v.
0:43:15 [edit] Peter is failing to tell us that P\!v\!B_n is indeed quadratic, and that this is his theorem - see his paper, The Pure Virtual Braid Group Is Quadratic, arXiv:1110.2356. However our lecture series goes in a different direction... --Drorbn 19:35, 1 February 2012 (EST)
0:47:19 [add] A^w.
0:52:41 [edit] Comment: The target space has only quadratic relations here because we are talking about Quadratic UFTI.
0:52:48 [add] Quadratic Universal Finite Type Invariant(s) (QUFTI).
0:56:48 [add] gr is a functor.
0:58:46 [add] QUFTI as filtered maps.
1:02:11 [add] QUFTI as filtered maps (2).
1:07:50 [add] The "central" question of FTI.
1:09:41 [add] The fundamental theorem and QUFTI.
1:09:42 [edit] But of course, the bigger problem is not to show that these three statements are equivalent, but to show that at least one of them (and hence all) is true. --Drorbn 20:00, 1 February 2012 (EST)
1:16:25 [add] The fundamental theorem and QUFTI (2).
1:20:35 [add] The fundamental theorem and QUFTI (3).
1:20:36 [edit] Define Z(\_):=\sum_p\sum_{i\in J_p}x_i(v_i\circ \alpha_p)(\_)
1:22:27 [add] Homomorphic QUFTI.
1:24:17 [edit] In fact, throughout the discussion of Universal Finite we had to talk about completions... --Drorbn 20:20, 1 February 2012 (EST)
1:26:43 [add] Z^w.
1:27:23 [edit] I disagree! Indeed Reidemeister 3 goes through by relatively simple-minded algebra, but it is the hardest of the relations to check, and the fact that it works is a little miracle - in the u and v cases, the same thing does not work. So we really must see how Reidemeister 3 goes through, and this will be one of the first things to do next time. --Drorbn 20:25, 1 February 2012 (EST)