El funtor TQFT y la cohomología de Khovanov

• Rodríguez Cárdenas, Carlos Wilson ^[1]
  1. [1] Universidad Industrial de Santander

     Universidad Industrial de Santander

     Colombia

     
• Localización: Integración: Temas de matemáticas, ISSN 0120-419X, Vol. 24, Nº. 2, 2006 (Ejemplar dedicado a: Revista Integración, temas de matemáticas), págs. 51-67
• Idioma: español
• Enlaces
  • Texto completo (pdf)
• Resumen
  • español

       El funtor TQFT (Topological Quantum Field Theory) relaciona la categoría de uno-variedades suaves cerradas con la categoría de módulos sobre un anillo R. Los objetos de la primera categoría son clases de isotopía de curvas suaves cerradas en el plano, y los morfismos son cobordismos entre ellas (superficies suaves confrontera en esas curvas suaves cerradas). En la segunda categoría los morfismos son productos y coproductos definidos sobre un R-módulo A. A través de esta relación se obtienen la cohomología de Khovanov y el polinomio de Kovanov, los cuales son invariantes topológicos de nudos.   

  • English

    A Topological Quantum Field Theory (TQFT) is a funtor between the category of smooth, closed, one dimensional manifolds and the category of R-modules on a ring R. The objects in the first category are classes of isotopy of closed smooth curves and the morphisms are cobordisms (smooth surfaces whose borders are closed soft curves) among them. In the second category the morphisms are homomorphisms of R-modules. Through this functor it is obtained the Khovanov cohomology and Khovanov polinomial, which are topological invariants of knots.

              

• Referencias bibliográficas
  • Citas [1] M.F. Atiyah. Topological Quantum Field Theories. I.H.I.S. Publ., 68:175-186, 1998.
  • [2] C. Blanchet. “Introduction to Quantum Invariants of 3-Manifols, Topological Quantum Fiel Theories and Modular Categories”. Summer School...
  • [3] B. Bakalov & A. Kirillov Jr. Lectures on Tensor Categories and Modular Functors. The American Mathematical Society. USA, 2001.
  • [4] J. Bernstein, I.B. Frenkel & M. Khovanov. “A Categorification of The Temperley-Lieb Algebra and Schur Quotients of U(sl2) Via Projective...
  • [5] Dror Bar-Natan, David Kazhdan & Dilan Thurston. Categorification for the Busy Mathematician. http:/www.ma.huji.ac.il/~drorbn/paper/Categorification/...
  • [6] Dror Bar-Natan. On Khovanov’s Categorification of the Jones Polynomial. http:/www.ma.huji.ac.il/~drorbn/paper/Categorification/ (28/02/2002)....
  • [8] I.B. Frenkel & M. Khovanov. “Canonical Basis in Tensor Products and Graphical Calculus for U(sl2).” Duke Math J., 87(3):409-480, 1997.
  • [9] R.S. Huerfano & M. Khovanov. “A Category for the Adjoint Representation.” Journal of Algebra, 246, 514-542 (2002). Elsevier Science,...
  • [10] V. Jones. “A Polynomial Invariant of Knots Via Von Newmann Algebra.” Bull. AMS 12, 103-111, 1987.
  • [11] V. Jones. “Commuting Transfer Matrices and Link Polinomials.” Internatinal Journal of Mathematics, Vol.3, no.2, 205-212, 1992.
  • [12] L.H. Kauffman. “State Models and the Jones Polinomial.” Topology, 26(3):395-407, 1987.
  • [13] L.H. Kauffman. On Knots. Princeton Univ. Press. Princeton, 1987.
  • [14] L.H. Kauffman. “Temperley-Lieb Recoupling Theory and Invariants of 3- Manifolds.” Annals of Mathematics Studies, no.134. Princeton Univ....
  • [15] M. Khovanov. “A Categorification of The Jones Polynomial.” Duke Math Journal, vol.3, no.101, 359-426, 2000, arXiv:math.QA/9908171, 1999.
  • [16] M. Khovanov. “A Funtor-Valued Invarian of Tangles.” Algebraica and Geometric Topology. vol. 2, 665-741, 2002.
  • [17] M. Khovanov & Sediel. “Quivers, Floer Homology and Braid Group Action.” Journal of the American Mathematical Society. vol. 15, no....
  • [18] D. Rolfsen. “Knots and Links.” Mathematics Lecture Serie. Publish or Perish, 1976.
  • [19] V. Turaev. “Quantum Invariants on Knots and 3-Manifolds.” De Gruiter Studies in Mathematics. 18. 1994.
  • [20] E. Witten. “Quantum Field Theory and The Jones Polynomial.” Comm. Math. Phys., 121(3):351-399, 1989.