Artículo Original
Publicado 2006-10-24
Palabras clave
- nudos,
- entrelazados,
- suavizaciones de cruces,
- cohomología de nudos,
- característica de Euler
- invariantes de nudos ...Más
Cómo citar
Rodríguez Cárdenas, C. W. (2006). El funtor TQFT y la cohomología de Khovanov. Revista Integración, Temas De matemáticas, 24(2), 51–67. Recuperado a partir de https://revistas.uis.edu.co/index.php/revistaintegracion/article/view/252
Resumen
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.
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Referencias
[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 on Geometric and Topological Methods for The Quamtum Fiel Theory. Villa de Leyva, julio de 2001.
[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 and Zuckerman Functors”. Selecta Math., New Ser., 5:199-241, 1999. ArXiv:math.QA/0002087.
[5] Dror Bar-Natan, David Kazhdan & Dilan Thurston. Categorification for the Busy Mathematician.
http:/www.ma.huji.ac.il/~drorbn/paper/Categorification/ (07/09/2001).
[6] Dror Bar-Natan. On Khovanov’s Categorification of the Jones Polynomial.
http:/www.ma.huji.ac.il/~drorbn/paper/Categorification/ (28/02/2002).
[7] I.B. Frenkel, N. Jing & W. Wang. “Vertex Representations Via Finite Groups and the Mckay Correspondence.” 1999, ArXiv:math.QA/9907166.
[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, 2001.
[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. Press. Princeton, 1987.
[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. 1, 203-271.
[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.
[2] C. Blanchet. “Introduction to Quantum Invariants of 3-Manifols, Topological Quantum Fiel Theories and Modular Categories”. Summer School on Geometric and Topological Methods for The Quamtum Fiel Theory. Villa de Leyva, julio de 2001.
[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 and Zuckerman Functors”. Selecta Math., New Ser., 5:199-241, 1999. ArXiv:math.QA/0002087.
[5] Dror Bar-Natan, David Kazhdan & Dilan Thurston. Categorification for the Busy Mathematician.
http:/www.ma.huji.ac.il/~drorbn/paper/Categorification/ (07/09/2001).
[6] Dror Bar-Natan. On Khovanov’s Categorification of the Jones Polynomial.
http:/www.ma.huji.ac.il/~drorbn/paper/Categorification/ (28/02/2002).
[7] I.B. Frenkel, N. Jing & W. Wang. “Vertex Representations Via Finite Groups and the Mckay Correspondence.” 1999, ArXiv:math.QA/9907166.
[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, 2001.
[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. Press. Princeton, 1987.
[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. 1, 203-271.
[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.