Juliet Cooke

Research

Higher Rank Askey-Wilson Algebras as Skein Algebras

In this paper we give a topological interpretation and diagrammatic calculus for the rank $(n−2)$ Askey-Wilson algebra by proving there is an explicit isomorphism with the Kauffman bracket skein algebra of the $(n+1)$-punctured sphere. To do this we consider the Askey-Wilson algebra in the braided tensor product of n copies of either the quantum group $\mathcal{U}_q(\mathfrak{sl}_2)$ or the reflection equation algebra. We then use the isomorpism of the Kauffman bracket skein algebra of the $(n+1)$-punctured sphere with the $\mathcal{U}_q(\mathfrak{sl}_2)$ invariants of the Aleeksev moduli algebra to complete the correspondence. We also find the graded vector space dimension of the $\mathcal{U}_q(\mathfrak{sl}_2)$ invariants of the Aleeksev moduli algebra and apply this to finding a presentation of the skein algebra of the five-punctured sphere and hence also find a presentation for the rank $2$ Askey-Wilson algebra.

On the genus two skein algebra

with Peter Samuelson in the Journal of the London Mathematical Society

We study the skein algebra of the genus $2$ surface and its action on the skein module of the genus $2$ handlebody. We compute this action explicitly, and we describe how the module decomposes over certain subalgebras in terms of polynomial representations of double affine Hecke algebras. Finally, we show that this algebra is isomorphic to the $t=q$ specialisation of the genus two spherical double affine Hecke algebra recently defined by Arthamonov and Shakirov.

Excision of Skein Categories and Factorisation Homology

We prove that the skein categories of Walker--Johnson-Freyd satisfy excision. This allows us to conclude that skein categories are k-linear factorisation homology and taking the free cocompletion of skein categories recovers locally finitely presentable factorisation homology. An application of this is that the skein algebra of a punctured surface related to any quantum group with generic parameter gives a quantisation of the associated character variety.

Kauffman Skein Algebras and Quantum Teichmüller Spaces via Factorisation Homology

in the Jornal of Knot Theory & Its Ramifications

We compute the factorisation homology of the four-punctured sphere and punctured torus over the quantum group $\mathcal{U}_q(\mathfrak{sl}_2)$ explicitly as categories of equivariant modules using the framework of `Integrating Quantum Groups over Surfaces' by Ben-Zvi, Brochier, and Jordan. We identify the algebra of invariants (quantum global sections) with the spherical double affine Hecke algebra of type $(C^\vee_1,C_1)$, in the four-punctured sphere case, and with the `cyclic deformation' of $U(su_2)$ in the punctured torus case. In both cases, we give an identification with the corresponding quantum Teichmüller space as proposed by Teschner and Vartanov as a quantization of the moduli space of flat connections.

PhD Thesis

Factorisation Homology and Skein Categories of Surfaces
Supervisor David Jordan

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In this thesis we prove skein categories can be computed by the mechanism of factorisation homology. We also use factorisation homology to compute a quantisation of the $\mathop{SL}_2$-character variety of the four--punctured sphere and punctured torus.

In the first part of this thesis, we study in detail the presentable factorisation homology of the four--punctured sphere and punctured torus with coefficients in the integrable representations of the quantum group $\mathcal{U}_q(\mathfrak{sl}_2)$. These factorisation homologies are categories of A-modules for an algebra A, and the algebra of $\mathcal{U}_q(\mathfrak{sl}_2)$-invariant endomorphisms of A gives a quantisation of the $\mathop{SL}_2$-character variety of the surface. We obtain presentations and Poincaré-Birkhoff-Witt bases for this algebra of invariants for both our example surfaces. As an application, we explicitly identify these algebras of invariants with two other quantisations of the $\mathop{SL}_2$-character variety for these surfaces: Teschner and Vartanov's quantisation of the moduli space of flat connections and the Kauffman bracket skein algebra.

In the second part of this thesis, we pursue the relation between factorisation homology and skein theory further. We prove that skein categories satisfy excision and that the skein categories of any ribbon category $\mathscr{V}$ are k-linear factorisation homologies with coefficients in $\mathscr{V}$. A corollary of this is that the free cocompletion of the skein category of the ribbon category of finite--dimensional representations of the quantum group $\mathcal{U}_q(\mathfrak{g})$ is the presentable factorisation homology with coefficients in the integrable representations of the quantum group $\mathcal{U}_q(\mathfrak{g})$. Hence, the free cocompletion of the Kauffman bracket skein category is the factorisation homology which we considered in the first part of the thesis.