Narrow Results

The skein algebra of an oriented three-manifold is a classical limit of the Kauffman bracket skein module and gives the coordinate ring of the ${SL}_2(ℂ)$-character variety. In this paper, we determine the quotient of a polynomial ring which is isomorphic to the skein algebra of a group with three generators and two relators. As an application, we give an explicit formula for the skein algebra of the Borromean rings complement in $S^3$.

The Kauffman bracket skein algebra of a compact oriented surface when the variable $A$ in the Kauffman bracket is set equal to $e^{\pi i/N}$, where $N$ is an odd counting number, is a central extension of the ring of $S{L}_2ℂ$-characters of the fundamental group of the underlying surface. In this paper, we construct symmetric Frobenius algebras from the Kauffman bracket skein algebra of some simple surfaces by two strategies. The first is to localize the skein algebra at the characters so it becomes an algebra over the function field of the character variety of the surface, and the second is to specialize at a place of the character ring.

We give an explicit basis ${ℬ}$ of the quotient of the Kauffman bracket skein algebra ${𝒮}(\mathrm{\Sigma })$ on a surface $\mathrm{\Sigma }$ by the square of an augmentation ideal. Moreover, we construct an embedding of the mapping class group of a compact connected oriented surface of genus $0$ into the Kauffman bracket skein algebra on the surface completed with respect to a filtration coming from the augmentation ideal.

Based on the presentation of the Kauffman bracket skein algebra of the thickened torus given by the third author in previous work [4], Frohman and Gelca established a complete description of the multiplicative operation leading to a famous product-to-sum formula. In this paper, we study the multiplicative structure of the Kauffman bracket skein algebra of the thickened four-holed sphere. We present an algorithm to compute the product of any two elements of the algebra, and give an explicit formula for some families of curves. We surmise that the algorithm has quasi-polynomial growth with respect to the number of crossings of a pair of curves. Further, we conjecture the existence of a positive basis for the algebra.