Skein modules and the noncommutative torus
HTML articles powered by AMS MathViewer
- by Charles Frohman and Răzvan Gelca PDF
- Trans. Amer. Math. Soc. 352 (2000), 4877-4888 Request permission
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 cylinder over a torus, and give a straightforward computation of the $n$-th colored Kauffman bracket of a torus knot, evaluated in the plane or in an annulus.References
- C. Blanchet, N. Habegger, G. Masbaum, and P. Vogel, Three-manifold invariants derived from the Kauffman bracket, Topology 31 (1992), no. 4, 685–699. MR 1191373, DOI 10.1016/0040-9383(92)90002-Y
- Doug Bullock, Rings of $\textrm {SL}_2(\textbf {C})$-characters and the Kauffman bracket skein module, Comment. Math. Helv. 72 (1997), no. 4, 521–542. MR 1600138, DOI 10.1007/s000140050032
- D. Bullock, C. Frohman and J. Kania-Bartoszyńska, Topological Interpretations of Lattice Gauge Field Theory, Commun. Math. Phys. 198 (1998), 47–81.
- Doug Bullock, Charles Frohman, and Joanna Kania-Bartoszyńska, Skein homology, Canad. Math. Bull. 41 (1998), no. 2, 140–144. MR 1624157, DOI 10.4153/CMB-1998-022-1
- D. Bullock, J. Przytycki, Multiplicative structure of Kauffman bracket skein module quantizations, to appear in Proc. Amer. Math. Soc.
- Alain Connes, Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994. MR 1303779
- Răzvan Gelca, Topological quantum field theory with corners based on the Kauffman bracket, Comment. Math. Helv. 72 (1997), no. 2, 216–243. MR 1470089, DOI 10.1007/s000140050013
- Jim Hoste and Józef H. Przytycki, The $(2,\infty )$-skein module of lens spaces; a generalization of the Jones polynomial, J. Knot Theory Ramifications 2 (1993), no. 3, 321–333. MR 1238877, DOI 10.1142/S0218216593000180
- Louis H. Kauffman, State models and the Jones polynomial, Topology 26 (1987), no. 3, 395–407. MR 899057, DOI 10.1016/0040-9383(87)90009-7
- Louis H. Kauffman and Sóstenes L. Lins, Temperley-Lieb recoupling theory and invariants of $3$-manifolds, Annals of Mathematics Studies, vol. 134, Princeton University Press, Princeton, NJ, 1994. MR 1280463, DOI 10.1515/9781400882533
- W. B. R. Lickorish, The skein method for three-manifold invariants, J. Knot Theory Ramifications 2 (1993), no. 2, 171–194. MR 1227009, DOI 10.1142/S0218216593000118
- Józef H. Przytycki, Skein modules of $3$-manifolds, Bull. Polish Acad. Sci. Math. 39 (1991), no. 1-2, 91–100. MR 1194712
- J. H. Przytycki, Symmetric knots and billiard knots, Ideal Knots (A. Stasiak et al., eds.), Ser. Knots Everything, vol. 19, World Sci. Publ., Singapore, 1998, pp. 374–414.
- Józef H. Przytycki and Adam S. Sikora, Skein algebra of a group, Knot theory (Warsaw, 1995) Banach Center Publ., vol. 42, Polish Acad. Sci. Inst. Math., Warsaw, 1998, pp. 297–306. MR 1634463
- Marc A. Rieffel, Deformation quantization of Heisenberg manifolds, Comm. Math. Phys. 122 (1989), no. 4, 531–562. MR 1002830, DOI 10.1007/BF01256492
- Justin Roberts, Skeins and mapping class groups, Math. Proc. Cambridge Philos. Soc. 115 (1994), no. 1, 53–77. MR 1253282, DOI 10.1017/S0305004100071917
- P. Sallenave, Structure of the Kauffman bracket skein algebra of $T^2\times I$, J. Knot Theory Ramif. 8 (1999), 367–372.
- A. Sikora, A New Geometric Method in the theory of $SL_n$-representations of groups, preprint (math.RT/9806016).
- C. N. Yang and M. L. Ge (eds.), Braid group, knot theory and statistical mechanics, Advanced Series in Mathematical Physics, vol. 9, World Scientific Publishing Co., Inc., Teaneck, NJ, 1989. MR 1062420
- Alan Weinstein, Symplectic groupoids, geometric quantization, and irrational rotation algebras, Symplectic geometry, groupoids, and integrable systems (Berkeley, CA, 1989) Math. Sci. Res. Inst. Publ., vol. 20, Springer, New York, 1991, pp. 281–290. MR 1104934, DOI 10.1007/978-1-4613-9719-9_{1}9
Additional Information
- Charles Frohman
- Affiliation: Department of Mathematics, University of Iowa, Iowa City, Iowa 52242
- MR Author ID: 234056
- ORCID: 0000-0003-0202-5351
- Email: frohman@math.uiowa.edu
- Răzvan Gelca
- Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109, and Institute of Mathematics of the Romanian Academy, Bucharest, Romania
- Email: rgelca@math.lsa.umich.edu
- Received by editor(s): June 15, 1998
- Received by editor(s) in revised form: January 20, 1999
- Published electronically: June 12, 2000
- © Copyright 2000 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 352 (2000), 4877-4888
- MSC (1991): Primary 57M27, 58B32, 47L87
- DOI: https://doi.org/10.1090/S0002-9947-00-02512-5
- MathSciNet review: 1675190