Narrow Results

This paper studies planar algebras of Jones’ style associated with the Young graph. We first see that, given a positive real valued function on the Young graph, we may obtain a planar algebra whose structure is defined in terms of a state sum over the ways of filling planar tangles with Young diagrams. We delve into the case that the function is harmonic and related to the Plancherel measures on Young diagrams. Along with an element that is depicted as a cross of two strings, we see that the defining relations among morphisms for the Khovanov Heisenberg category are recovered in the planar algebra. We also identify certain elements in the planar algebra with particular functions of Young diagrams that include the moments, Boolean cumulants and normalized characters. This paper thereby bridges diagrammatical categorification and asymptotic representation theory. In fact, the Khovanov Heisenberg category is one of the most fundamental examples of diagrammatical categorification whereas the harmonic functions on the Young graph have been a central object in the asymptotic representation theory of symmetric groups.

Most known examples of subfactors occur in families, coming from algebraic objects such as groups, quantum groups and rational conformal field theories. The Haagerup subfactor is the smallest index finite depth subfactor which does not occur in one of these families. In this paper we construct the planar algebra associated to the Haagerup subfactor, which provides a new proof of the existence of the Haagerup subfactor. Our technique is to find the Haagerup planar algebra as a singly generated subfactor planar algebra, that is contained inside a graph planar algebra.

In recent joint work with Jones and Shlyakhtenko, we have given a diagrammatic description of Popa's symmetric enveloping inclusion for planar algebra subfactors. In this paper, we give a diagrammatic construction of the associated Jones tower, in the case that the planar algebra is finite-depth. We then use this construction to describe the planar algebra of the symmetric enveloping inclusion, which is known to be isomorphic to the planar algebra of Ocneanu's asymptotic inclusion by a result of Popa. As an application we give a planar algebraic computation of the (reduced) fusion algebra of the asymptotic inclusion, recovering some well-known results of Ocneanu and Evans–Kawahigashi.

In this paper, we show in a combinatorial way that the 0-box space of the E₈ subfactor planar algebra is 1-dimensional. In the proof, we improve on Bigelow's relations for the E₈ subfactor planar algebra and give an efficient algorithm to reduce any planar diagram to the empty diagram.

We define generalized notions of biunitary elements in planar algebras and show that objects arising in quantum information theory such as Hadamard matrices, quantum Latin squares and unitary error bases are all given by biunitary elements in the spin planar algebra. We show that there are natural subfactor planar algebras associated with biunitary elements.

We discuss the structure of the Motzkin algebra $M_k(D)$ by introducing a sequence of idempotents and the basic construction. We show that ${\bigcup }_{k\ge 1}{M}_k(D)$ admits a factor trace if and only if $D\in \{2cos(\pi /n)+1|n\ge 3\}\cup [3,\infty )$ and the higher commutants of these factors depend on $D$. Then a family of irreducible bimodules over these factors is constructed. A tensor category with $A_n$ fusion rule is obtained from these bimodules.

Starting with a vertex-weighted pointed graph $(\mathrm{\Gamma },\mu ,v_0)$, we form the free loop algebra $S_0$ defined in Hartglass–Penneys’ article on canonical ${\mathrm{\text{C}}}^{\ast }$-algebras associated to a planar algebra. Under mild conditions, ${{𝒮}}_0$ is a non-nuclear simple ${\mathrm{\text{C}}}^{\ast }$-algebra with unique tracial state. There is a canonical polynomial subalgebra $A\subset {{𝒮}}_0$ together with a Dirac number operator $N$ such that $(A,L^{2}A,N)$ is a spectral triple. We prove the Haagerup-type bound of Ozawa–Rieffel to verify $({{𝒮}}_0,A,N)$ yields a compact quantum metric space in the sense of Rieffel.

We give a weighted analog of Benjamini–Schramm convergence for vertex-weighted pointed graphs. As our ${\mathrm{\text{C}}}^{\ast }$-algebras are non-nuclear, we adjust the Lip-norm coming from $N$ to utilize the finite dimensional filtration of $A$. We then prove that convergence of vertex-weighted pointed graphs leads to quantum Gromov–Hausdorff convergence of the associated adjusted compact quantum metric spaces.

As an application, we apply our construction to the Guionnet–Jones–Shyakhtenko (GJS) ${\mathrm{\text{C}}}^{\ast }$-algebra associated to a planar algebra. We conclude that the compact quantum metric spaces coming from the GJS ${\mathrm{\text{C}}}^{\ast }$-algebras of many infinite families of planar algebras converge in quantum Gromov–Hausdorff distance.

It is a well known result that the Jones polynomial of a non-split alternating link is alternating. We find the right generalization of this result to the case of non-split alternating tangles. More specifically: the Jones polynomial of tangles is valued in a certain skein module; we describe an alternating condition on elements of this skein module, show that it is satisfied by the Jones invariant of the single crossing tangles (⤲) and (⤲), and prove that it is preserved by appropriately "alternating" planar algebra compositions. Hence, this condition is satisfied by the Jones polynomial of all alternating tangles. Finally, in the case of 0-tangles, that is links, our condition is equivalent to simple alternation of the coefficients of the Jones polynomial.

In [The planar algebra of a bipartite graph, Knots in Hellas '98 (World Scientific, 2000), pp. 94–117, MR1865703], Jones found two copies of the cyclic category cΔ in the annular Temperley–Lieb category Atl. We give an abstract presentation of Atl to discuss how these two copies of cΔ generate Atl together with the coupling constants and the coupling relations. We then discuss modules over the annular category and homologies of such modules, the latter of which arises from the cyclic viewpoint.

We give a new construction of the one-variable Alexander polynomial of an oriented knot or link, and show that it generalizes to a vector valued invariant of oriented tangles.

The Jones–Wenzl idempotents are elements of the Temperley–Lieb (TL) planar algebra that are important, but complicated to write down. We will present a new planar algebra, the pop-switch planar algebra (PSPA), which contains the TL planar algebra. It is motivated by Jones' idea of the graph planar algebra (GPA) of type A_n. In the tensor category of idempotents of the PSPA, the nth Jones–Wenzl idempotent is isomorphic to a direct sum of n + 1 diagrams consisting of only vertical strands.

To a planar algebra in the sense of Jones we associate a natural non-commutative ring, which can be viewed as the ring of non-commutative polynomials in several indeterminates, invariant under a symmetry encoded by . We show that this ring carries a natural structure of a non-commutative probability space. Non-commutative laws on this space turn out to describe random matrix ensembles possessing special symmetries. As application, we give a canonical construction of a subfactor and its symmetric enveloping algebra associated to a given planar algebra . This talk is based on joint work with A. Guionnet and V. Jones.