06-1350/Class Notes for Tuesday November 14

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In Preparation

The information below is preliminary and cannot be trusted! (v)

Today's Agenda

• Recall planar algebras and the planar algebra of shielded chord diagrams.
• Finish describing the necessary relations between $T$, $R$, $\Phi$ and $B^{\pm}$ in verbal form.
• Sweeping clean a tree and ${\mathcal A}(\Gamma)\equiv{\mathcal A}(\uparrow_{b_1(\Gamma)})$.

• ${\mathcal A}(\uparrow_n)$ is a VS-algebra (see more at VS, TS and TG Algebras).
• In the coordinates above, write the $TR\Phi B$ relations in various algebraic notations.
  • R4: $(1230)^\star B^\pm\cdot(1213)^\star B^\pm\cdot(1023)^\star\Phi=(1123)^\star\Phi\cdot(1233)^\star B^\pm$.
  • $B^{\pm}$ in terms of $\Phi$ and $R$ and $R$ in terms of $T$.
  • R3, R2, R1
  • Symmetry of $\Phi$ and of $B^{\pm}$.
  • $u$, $d$ and $\#$
  • Idempotence for $T$, $R$, $\Phi$ and $B^{\pm}$.