07-1352/Class Notes for March 20

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Today's Agenda. The up-to-vertex-operations uniqueness of an {\mathcal A}-valued algebraic knot theory.

  • Uniqueness up to vertex operations, vaguely.
  • The group {\mathcal A}^V\subset {\mathcal A}(\theta) and its action on the set {\mathcal Z} of {\mathcal A}-valued algebraic knot theories.
  • An aside about trinions (also see 06-1350/Class Notes for Tuesday October 10).
  • The group {\mathcal A}^F\subset {\mathcal A}(\uparrow_2) and its action on the set of all associators.
  • An aside about braided \theta-graphs:
07-1352 A Braided Theta Graph.png
(Also see Dror Bar-Natan's Image Gallery: Knotted Objects: Braided Thetas.)
  • A degree-by-degree construction of a twistor F and the reduction to homology.
  • Computing the homology using unitrivalent graphs and black boxes.
  • Return to the PBW theorem.