symmetric monoidal (∞,1)-category of spectra
A planar algebra $\mathcal{P}$ consists of vector spaces $P_n$ together with multilinear operations between them indexed by planar tangles – large discs with internal (“input”) discs all connected up by non-intersecting curves called strings. Thus a planar algebra may be thought of as made up from generators $R_i \in P_n$ to which linear combinations of planar tangles may be applied to obtain all elements of $\mathcal{P}$.
Details are given in
where the author remarks
The definition of a planar algebra has evolved a bit since the original one in Planar algebras I so we give a detailed definition which is, we hope, the ultimate one, at least for shaded planar algebras.
Relations will appear with knot theory, statistical mechanics, combinatorial group theory, quantum groups and most of all, subfactors. They lead to many different algebraic structures according to the particular planar tangles chosen. Of particular interest recently have been graded algebra structures which give connections with the large $N$ limit of random matrices and free probability.
The relation to subfactors and fusion categories is discussed in
Plenty of discussion of planar algebras is at