Planar algebra

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. 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. The idea of the planar algebra is to be a diagrammatic axiomatization of the standard invariant. 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. 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. The planar operad is the set of all the planar tangles (up to isomorphism) with such compositions. 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 })_{nin 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