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 [math]\displaystyle{ T }[/math], [math]\displaystyle{ R }[/math], [math]\displaystyle{ \Phi }[/math] and [math]\displaystyle{ B^{\pm} }[/math] in verbal form.
• Sweeping clean a tree and [math]\displaystyle{ {\mathcal A}(\Gamma)\equiv{\mathcal A}(\uparrow_{b_1(\Gamma)}) }[/math].

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