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wClips-120229

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Sections 3.1-3.4: v-Knots and w-Knots: Definitions, framings, finite type invariants, dimensions, and the expansion in the w case.


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Announcements. small circle, UofT, LDT Blog (also here). Email Dror to join our mailing list!

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

The wClips

http://katlas.math.toronto.edu/drorbn/dbnvp/dbnvp.png
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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0:00:00 [edit] Blackboard shots by Zsuzsanna Dancso.
0:00:14 [add] Handout view 2: 2.5.5: Important Leftover
0:00:22 [add] Handout view 3: 3: Knots are the wrong objects for study
0:01:02 [add] Handout view 4: 3.1: v- and w-Knots
0:02:26 [add] Handout view 5: Warning 3.2 and Warning 3.3
0:02:53 [add] Handout view 4: 3.1: v- and w-Knots
0:03:02 [add] Handout view 5: Warning 3.2 and Warning 3.3
0:05:11 [add] Handout view 6: Remark 3.4: Gauss Diagrams
0:06:38 [add] Handout view 7: Remark 3.5 Framings
0:12:17 [add] Framed and unframed long v-knots, day 2.
0:12:47 [add] Classical linkinf and self-linking numbers.
0:14:14 [add] Framed and unframed long v-knots: $\iota$ and $sl$.
0:20:32 [add] Framed and unframed long v-knots: $h$.
0:27:38 [add] Handout view 6: Remark 3.4: Gauss Diagrams
0:27:42 [add] Handout view 4: 3.1: v- and w-Knots
0:30:11 [add] Framed and unframed long v-knots: the compositions.
0:31:20 [add] Handout view 7: Remark 3.5 Framings
0:31:42 [add] Framed and unframed long v-knots: proofs.
0:33:39 [add] Handout view 9: A bit on the topology of v and w
0:35:49 [edit] Our other handout is Terrific Pictures by Shin Satoh:

http://katlas.math.toronto.edu/drorbn/AcademicPensieve/Classes/12-wClips/Terrific_Pictures_by_Shin_Satoh_400.jpg

0:36:11 [edit] First Satoh picture.
0:39:11 [edit] A picture by Ichimori and Kanenobu.
0:40:57 [edit] Second Satoh picture.
0:42:26 [edit] Third Satoh picture.
0:45:31 [add] Handout view 10: 3.2: Finite Type Invariants
0:47:21 [add] Semi-virtual crossings.
0:51:31 [add] Type $m$ invariants and their weight systems.
0:52:49 [add] ${\mathcal A}^{v,w}$.
0:56:15 [add] 6T.
0:59:13 [add] 4T.
1:01:24 [add] Handout view 2: 2.5.5: Important Leftover
1:05:25 [add] $\alpha:{\mathcal A}^u\to{\mathcal A}^v$.
1:06:29 [add] Handout view 11: 3.9: The bi-algebra structure and primitives
1:08:06 [add] ${\mathcal A}^{v,w}(\uparrow)$ and ${\mathcal A}^{v,w}(\bigcirc)$.
1:13:12 [add] The product and the co-product.
1:16:49 [add] Handout view 12: A table of dimensions
1:17:02 [add] The Milnor-Moore theorem.
1:22:48 [add] Handout view 13: 3.4: The w-Expansion