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u, v, and w-Knots: Topology, Combinatorics and Low and High Algebra

Courant Lecture Series
Goettingen April 27,28,29, 2010

Overall Abstract. I will discuss three types of knotted objects - the "u" type, for "usual", the "v" type, for "virtual", and the "w" type, for "welded", or "weakly virtual", or "warm up". I will then discuss an abstract and general yet rather simple machine that in a uniform manner associates to each such class of knotted objects a "combinatorics", and a "low algebra", and a "high algebra". The latter is high indeed - it is the theory of Drinfel'd associators in the u case, most likely it is the Etingof-Kazhdan theory of quantization of Lie bi-algebras in the v case, and it is the Kashiwara-Vergne theory of convolutions on Lie groups and algebras in the w case. Thus these three pieces of high algebra have a simple topological origin. And as on the level of topology u, v, and w are tied together, their respective high algebra theories are closely related, with some of these relationships clearly understood, and some that are yet to be explored.

Day 1
u, v, w: topology and philosophy

  • Dreams and plans.
  • Knots, planar diagrams, Reidemeister moves, virtual knots are to knots as manifolds are to Euclidean spaces, flying rings and knotted tubes in 4D and w-knots.
  • Planar algebras and circuit algebras.
  • The abstract machine - filtered and graded spaces, expansions and homomorphic expansions, equations in graded spaces.

Day 2
u, v, w: combinatorics and low algebra

  • Finite type invariants, weight systems, chord diagrams, arrow diagrams, 4T relations
  • The "bracket-rise" theorem, STU and IHX relations
  • Maps into various kinds of universal enveloping algebras.
  • High algebra in the w case - from R4 to Kashiwara-Vergne - convolutions on Lie groups and algebras. (More in Talks: Bonn-0908).

Day 3
v: 18 Conjectures

  • A w-Map for General Orientation (G3.html, G.pdf).
  • I will state 18=3x3x2 "fundamental" conjectures on finite type invariants of various classes of virtual knots. This done, I will state a few further conjectures about these conjectures and ask a few questions about how these 18 conjectures may or may not interact.

Handout - G1.html, G.pdf
Talk Video.

Handout - G2.html, G.pdf
Talk Video.

Handout - 18C.html, 18C.pdf
Talk Video.
To some extent, the first two lectures follow my in-preparation paper Finite Type Invariants of W-Knotted Objects: From Alexander to Kashiwara and Vergne. The third lecture follows the paper Some Dimensions of Spaces of Finite Type Invariants of Virtual Knots, with Halacheva, Leung, and Roukema.

All Goettingen sources are in G.zip. The Luminy sources are at Talks: Luminy-1004.