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 finite group 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 ${\displaystyle 2n}$ intervals per disk and one ${\displaystyle \star }$-marked interval per disk.

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