Research and Innovation Articles
Published 2006-10-24
Keywords
- nudos,
- entrelazados,
- suavizaciones de cruces,
- cohomología de nudos,
- característica de Euler
- invariantes de nudos ...More
How to Cite
Rodríguez Cárdenas, C. W. (2006). El funtor TQFT y la cohomología de Khovanov. Revista Integración, Temas De matemáticas, 24(2), 51–67. Retrieved from https://revistas.uis.edu.co/index.php/revistaintegracion/article/view/252
Abstract
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.
Downloads
Download data is not yet available.
References
[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.