2000
DOI: 10.1090/s0002-9947-00-02512-5
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Skein modules and the noncommutative torus

Abstract: Abstract. We prove that the Kauffman bracket skein algebra of the cylinder over a torus is a canonical subalgebra of the noncommutative torus. The proof is based on Chebyshev polynomials. As an application, we describe the structure of the Kauffman bracket skein module of a solid torus as a module over the algebra of the cylinder over a torus, and recover a result of Hoste and Przytycki about the skein module of a lens space. We establish simple formulas for Jones-Wenzl idempotents in the skein algebra of a cy… Show more

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Cited by 107 publications
(97 citation statements)
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“…Remark 4.3. In the special case where the cluster algebra A has trivial coefficients, a similar formula can be found in [FG00].…”
Section: Proof Of the Main Resultsmentioning
confidence: 89%
“…Remark 4.3. In the special case where the cluster algebra A has trivial coefficients, a similar formula can be found in [FG00].…”
Section: Proof Of the Main Resultsmentioning
confidence: 89%
“…Remark It is interesting to compare factorization homology with the theory of skein algebras and skein categories (see, for example, ), which provide a convenient graphical calculus for quantizing SL2 (and more generally SLn) character varieties and for constructing associated 3‐manifold invariants. Roughly speaking, to each surface S and each braided tensor category scriptA with a choice of presentation (that is, a collection of objects generating under tensor product, morphisms generating under composition, and a specification of ‘local’ relations on morphisms), there is an associated skein category in which an object is a configuration of disks, each colored by a generating object of scriptA, and in which a morphism is the quotient of the vector space of colored tangles by local relations in scriptA.…”
Section: Introductionmentioning
confidence: 99%
“…(8) we see that the coefficient α 1 is equal to − [2]. Continuing on this way we see that this skein has to be…”
Section: The Yang-mills Measure On a Closed Surfacementioning
confidence: 76%