In mathematics, planar algebras first appeared in the work of Vaughan Jones on the standard invariant of a II1 subfactor. They also provide an appropriate algebraic framework for many knot invariants (in particular the Jones polynomial), and have been used in describing the properties of Khovanov homology with respect to tangle composition. Any subfactor planar algebra provides a family of unitary representations of Thompson groups. Any finite group (and quantum generalization) can be encoded as a planar algebra.
Definition The idea of the planar algebra is to be a diagrammatic axiomatization of the standard invariant.
Planar tangle A (shaded) planar tangle is the data of finitely many input disks, one output disk, non-intersecting strings giving an even number, say 2 n {\displaystyle 2n} , intervals per disk and one ⋆ {\displaystyle \star } -marked interval per disk.
Here, the mark is shown as a ⋆ {\displaystyle \star } -shape. On each input disk it is placed between two adjacent outgoing strings, and on the output disk it is placed between two adjacent incoming strings. A planar tangle is defined up to isotopy.
Composition To compose two planar tangles, put the output disk of one into an input of the other, having as many intervals, same shading of marked intervals and such that the ⋆ {\displaystyle \star } -marked intervals coincide. Finally we remove the coinciding circles. Note that two planar tangles can have zero, one or several possible compositions.
Planar operad The planar operad is the set of all the planar tangles (up to isomorphism) with such compositions.
Planar algebra A planar algebra is a representation of the planar operad; more precisely, it is a family of vector spaces ( P n , ± ) n ∈ N {\displaystyle ({\mathcal {P}}_{n,\pm })_{n\in \mathbb {N} }} , called n {\displaystyle n} -box spaces, on which acts the planar operad, i.e. for any tangle T {\displaystyle T} (with one output disk and r {\displaystyle r} input disks with 2 n 0 {\displaystyle 2n_{0}} and 2 n 1 , … , 2 n r {\displaystyle 2n_{1},\dots ,2n_{r}} intervals respectively) there is a multilinear map
Z T : P n 1 , ϵ 1 ⊗ ⋯ ⊗ P n r , ϵ r → P n 0 , ϵ 0 {\displaystyle Z_{T}:{\mathcal {P}}_{n_{1},\epsilon _{1}}\otimes \cdots \otimes {\mathcal {P}}_{n_{r},\epsilon _{r}}\to {\mathcal {P}}_{n_{0},\epsilon _{0}}}
with ϵ i ∈ { + , − } {\displaystyle \epsilon _{i}\in \{+,-\}} according to the shading of the ⋆ {\displaystyle \star } -marked intervals, and these maps (also called partition functions) respect the composition of tangle in such a way that all the diagrams as below commute.
Examples
Planar tangles The family of vector spaces ( T n , ± ) n ∈ N {\displaystyle ({\mathcal {T}}_{n,\pm })_{n\in \mathbb {N} }} generated by the planar tangles having 2 n {\displaystyle 2n} intervals on their output disk and a white (or black) ⋆ {\displaystyle \star } -marked interval, admits a planar algebra structure.
Temperley–Lieb The Temperley-Lieb planar algebra T L ( δ ) {\displaystyle {\mathcal {TL}}(\delta )} is generated by the planar tangles without input disk; its 3 {\displaystyle 3} -box space T L 3 , + ( δ ) {\displaystyle {\mathcal {TL}}_{3,+}(\delta )} is generated by
Moreover, a closed string is replaced by a multiplication by δ {\displaystyle \delta } .
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