Planar algebra
Encyclopedia
In mathematics
, planar algebras first appeared in the work of Vaughan Jones
on the standard invariant of a II1 subfactor http://www.math.berkeley.edu/~vfr/plnalg1.ps. 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 http://www.math.toronto.edu/~drorbn/papers/Cobordism/ http://front.math.ucdavis.edu/math.GT/0410495.
Given a label set
with an involution, and a fixed set of words 
in the elements of the label set, a planar algebra consists of a collection of modules 
, one for each element 
in 
, together with an action of the operad of tangles labelled by 
.
, and a single word 
, we define a tangle from 
to 
to be a disk D in the plane, with points around its circumference labelled in order by the letters of 
, with 
internal disks removed, indexed 1 through k, with the i-th internal disk having points around its circumference labelled in order by the letters of 
, and finally, with a collection of oriented non-intersecting curves lying in the remaining portion of the disk, with each component being labelled by an element of the label set, such that the set of end points of these curves coincide exactly with the labelled points on the internal and external circumferences, and at the initial points of the curves, the label on the curves coincides with the label on the circumference, while at the final points, the label on the curve coincides with the involute of the label on the circumference.
While this sounds complicated, an illustrated example does wonders!
Such tangles can be composed. With this notion of composition, the collection of tangles with labels in
and boundaries labelled by 
forms an operad.
This operad acts on the modules
as follows. For each tangle 
from 
to 
, we need a module homomorphism 
. Further, for a composition of tangles, we must get the corresponding composition of module homomorphisms.
Fix an element
in the ground ring. Take a one element label set, and allow words of even length. (Thus the words correspond exactly to nonnegative even integers.) For each even integer 
, let 
be the free module generated by (isotopy classes of) diagrams consisting of n non-intersecting arcs drawn in a disk, with the endpoints of the arcs lying on the boundary of the disk. The action of tangles is simply by gluing the appropriate disks into the tangle, removing any closed arcs, replacing each with a factor of 
.
We can generalise this to allow more complicated label and word sets (including for example, the planar algebra version of the Fuss–Catalan algebras). For each label
, fix 
in the ground ring. For a word 
, the module 
is generated by (again, isotopy classes) of diagrams consisting on non-intersecting arcs drawn in a disk, labelled by elements of 
with endpoints on the boundary of disk, such that the induced labels on these points, when read in order, give 
. The action of tangles is defined as before, with closed arcs labelled by 
being replaced by a factor of 
.
, living in V++−−. Knot polynomial
s satisfying skein relations can be succinctly described as quotient maps from this planar algebra, which are rank 1 on
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, planar algebras first appeared in the work of Vaughan Jones
Vaughan Jones
Sir Vaughan Frederick Randal Jones, KNZM, FRS, FRSNZ is a New Zealand mathematician, known for his work on von Neumann algebras, knot polynomials and conformal field theory. He was awarded a Fields Medal in 1990, and famously wore a New Zealand rugby jersey when he accepted the prize...
on the standard invariant of a II1 subfactor http://www.math.berkeley.edu/~vfr/plnalg1.ps. 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
Khovanov homology
In mathematics, Khovanov homology is an invariant of oriented knots and links that arises as the homology of a chain complex. It may be regarded as a categorification of the Jones polynomial....
with respect to tangle composition http://www.math.toronto.edu/~drorbn/papers/Cobordism/ http://front.math.ucdavis.edu/math.GT/0410495.
Given a label set
with an involution, and a fixed set of words in the elements of the label set, a planar algebra consists of a collection of modules , one for each element in , together with an action of the operad of tangles labelled by .Definition
In more detail, given a list of words, and a single word , we define a tangle from to to be a disk D in the plane, with points around its circumference labelled in order by the letters of , with internal disks removed, indexed 1 through k, with the i-th internal disk having points around its circumference labelled in order by the letters of , and finally, with a collection of oriented non-intersecting curves lying in the remaining portion of the disk, with each component being labelled by an element of the label set, such that the set of end points of these curves coincide exactly with the labelled points on the internal and external circumferences, and at the initial points of the curves, the label on the curves coincides with the label on the circumference, while at the final points, the label on the curve coincides with the involute of the label on the circumference.While this sounds complicated, an illustrated example does wonders!
Such tangles can be composed. With this notion of composition, the collection of tangles with labels in
and boundaries labelled by forms an operad.This operad acts on the modules
as follows. For each tangle from to , we need a module homomorphism . Further, for a composition of tangles, we must get the corresponding composition of module homomorphisms.Temperley–Lieb
The Temperley–Lieb algebras can be retrofitted as a planar algebra.Fix an element
in the ground ring. Take a one element label set, and allow words of even length. (Thus the words correspond exactly to nonnegative even integers.) For each even integer , let be the free module generated by (isotopy classes of) diagrams consisting of n non-intersecting arcs drawn in a disk, with the endpoints of the arcs lying on the boundary of the disk. The action of tangles is simply by gluing the appropriate disks into the tangle, removing any closed arcs, replacing each with a factor of .We can generalise this to allow more complicated label and word sets (including for example, the planar algebra version of the Fuss–Catalan algebras). For each label
, fix in the ground ring. For a word , the module is generated by (again, isotopy classes) of diagrams consisting on non-intersecting arcs drawn in a disk, labelled by elements of with endpoints on the boundary of disk, such that the induced labels on these points, when read in order, give . The action of tangles is defined as before, with closed arcs labelled by being replaced by a factor of .Tangles
The (oriented) tangle planar algebra is a planar algebra with a two element label set, the nontrivial involution on it, and balanced even length words. It is generated, as a planar algebra, by the diagrams of the positive and negative crossings in knot theoryKnot theory
In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a...
, living in V++−−. Knot polynomial
Knot polynomial
In the mathematical field of knot theory, a knot polynomial is a knot invariant in the form of a polynomial whose coefficients encode some of the properties of a given knot.-History:The first knot polynomial, the Alexander polynomial, was introduced by J. W...
s satisfying skein relations can be succinctly described as quotient maps from this planar algebra, which are rank 1 on
.