Lecture notes are available here. Videos of the lectures:
  1. Lecture 1

Exercises

First lecture

  1. Show that in an unoriented trivalent skein theory $\mathcal{P}$ (as described in the lectures), with $\dim \mathcal{P}_4 \leq 2$, if the Temperley-Lieb-Jones diagrams are linearly dependent, "something goes wrong".
    • Specifically, show that the value of the theta graph must be zero, and hence that the trivalent vertex itself must be negligible.
    • To do this, it may be helpful to calculate the pairwise "inner products" of the four simplest diagams in $\mathcal{P}_4$, and to think about the rank of the resulting matrix. (See p. 11 of Categories generated by a trivalent vertex for further hints.)
  2.  
    • Come up with a definition of a morphism of planar algebras
    • Explain why there is a morphism $\mathsf{TLJ} \to \mathcal{P}$ for any planar algebras $\mathcal{P}$ with $\dim \mathcal{P}_0 = 1$.
    • Discover and prove a formula for $\dim \mathsf{TLJ}_n$. (Hint: Eric told you the answer.)
    • Invent some generalisations of the definition of a planar algebra from the lectures:
      • What if we want to allow oriented spaghetti?
      • What is a Colombian planar algebra?
    • Check that in any monoidal category. How many of the axioms did you need to use?
    • Check that in any pivotal category. How many of the axioms did you need to use?
  4. In our construction of a monoidal category from a planar algebra, check that it is rigid, and write down the "obvious" pivotal structure.
  5. In our construction of a planar algebra from a (symmetrically self-dual) object in a pivotal category, explain why the given data satisfies the axioms.
    • Write down the equivalence of categories $\mathsf{Kar}(\mathcal{C}) \cong \mathcal{C}$.
    • If $\mathcal{C}$ already has direct sums, show $\mathsf{Mat}(\mathcal{C}) \cong \mathcal{C}$.
  7. In our diagrammatic definition of $\mathsf{Fib}$,
    • Explain why there are no maps between $1$ and $X$.
    • Prove that $X^{\otimes n} \cong F_{n-1} 1 \oplus F_n X$, where the $F_n$ are Fibonacci numbers.
    • Argue that every object in the idempotent completion is a subobject of $X^{otimes n}$ for some $n$, and hence show that the idempotent completion is semisimple.
    • Find all the braidings on our diagrammatic presentation of $\mathsf{Fib}$. (Hints in the lecture notes.)