We describe, for a few small examples, the Kauffman bracket skein algebra of a surface crossed with an interval. If the surface is a punctured torus the result is a quantization of the symmetric algebra in three variables (and an algebra closely related to a cyclic quantization of $U(so_3$). For a torus without boundary we obtain a quantization of "the symmetric homologies" of a torus (equivalently, the coordinate ring of the $SL_2(C)$-character variety of $Z \oplus Z$). Presentations are also given for the four punctured sphere and twice punctured torus. We conclude with an investigation of central elements and zero divisors.
We give a presentation of the Kauffman (BMW) skein algebra of the torus, which is the "type BCD" analogue of the Homflypt skein algebra of torus which was computed by the first and third authors. In the appendix we show this presentation is compatible with the Frohman-Gelca description of the Kauffman bracket (Temperley-Lieb) skein algebra of the torus [FG00].
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 correspondi…
In this work, we extend some results from the Kauffman bracket and HOMFLYPT skein theories to the Kauffman (Dubrovnik) skein theory. A definition is given for ``power sum" type elements $\widetilde{P}_k$ in the Dubrovnik skein algebra of the annulus $\mathcal{D}(A)$. These elements generalize the Chebyshev polynomials often used when studying Kauffman bracket skein algebras. Threadings of the $\widetilde{P}_k$ are used as generators in a presentation of the Dubrovnik skein algebra of the torus $\mathcal{D}(T^2)$, where they are shown to satisfy simple relations. This description of $\mathcal{D…
In this paper we introduce the notion of admissible skein modules associated to an ideal in a pivotal category. We explain how these modules are generalizations of the Kauffman skein algebra and how they relate to renormalized quantum invariants coming from non-semisimple categories.
In this paper we compute the Kauffman bracket skein module of the complement of $(2, 2p+1)$-torus knots, $KBSM(T_{(2, 2p+1)}^c)$, via braids. We start by considering geometric mixed braids in $S^3$, the closure of which are mixed links in $S^3$ that represent links in the complement of $(2, 2p+1)$-torus knots, $T_{(2, 2p+1)}^c$. Using the technique of parting and combing, we obtain algebraic mixed braids, that is, mixed braids that belong to the mixed braid group $B_{2, n}$ and that are followed by their ``coset'' part, that represents $T_{(2, 2p+1)}^c$. In that way we show that links in $T_{(…
We relate the Kauffman bracket stated skein modules to two independent constructions of quantum representation spaces of Habiro and Van der Veen with the second author. We deduce from this relation a description of the classical limit of stated skein modules, a quantum Van Kampen theorem and a quantum HNN extension theorem for stated skein modules and obtain a new description of the skein modules of mapping tori and links exteriors.
We obtain a family of skein identities in the Kauffman bracket skein module which relate Frobenius elements to Jones-Wenzl projectors at roots of unity. We view these skein identities as certain incarnations of Steinberg tensor product formulae from the theory of tilting modules of the quantum group $U_q(sl_2)$. We show that the simplest skein identities yield a short new proof of the existence of the Chebyshev-Frobenius homomorphism of Bonahon-Wong.
We describe presentations of the Roger-Yang generalized skein algebras for punctured spheres with an arbitrary number of punctures. This skein algebra is a quantization of the decorated Teichmuller space and generalizes the construction of the Kauffman bracket skein algebra. In this paper, we also obtain a new interpretation of the homogeneous coordinate ring of the Grassmannian of planes in terms of skein theory.
