Abstract
A brief review of the development of Chern-Simons gauge theory since its relation to knot theory was discovered in 1988 is presented. The presentation is done guided by a dictionary which relates knot theory concepts to quantum field theory ones. From the basic objects in both contexts the quantities leading to knot and link invariants are introduced and analysed. The quantum field theory approaches that have been developed to compute these quantities are reviewed. Perturbative approaches lead to Vassiliev or finite type invariants. Non-perturbative ones lead to polynomial or quantum group invariants. In addition, a brief discussion on open problems and future developments is included.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
D. Altschuler and L. Friedel, `Vassiliev knot invariants and Chern-Simons perturbation theory to all orders,“ Commun. Math. Phys. 187 (1997) 261, and `On universal Vassiliev invariants,” 170 (1995) 41.
M. Alvarez and J. M. F. Labastida, “Analysis of observables in Chern-Simons perturbation theory,” Nucl. Phys. B395 (1993) 198, hep-th/9110069, and `Numerical knot invariants of finite type from Chern-Simons gauge theory,“ B433 (1995) 555, hep-th/9407076; Erratum, ibid. B441 (1995) 403.
M. Alvarez and J. M. F. Labastida, “Vassiliev invariants for torus knots,” Journal of Knot Theory and its Ramifications 5 (1996) 779; q-alg/9506009.
M. Alvarez and J. M. F. Labastida, “Primitive Vassiliev invariants and factorization in Chern-Simons gauge theory,” Commun. Math. Phys. 189 (1997) 641, qalg/9604010.
Y. Akutsu and M. Wadati, “Exactly solvable models and knot theory,” Phys. Rep. 180 (1989) 247.
D. Bar-Natan “On the Vassiliev knot invariants,”Topology34 (1995) 423.
D. Bar-Natan “Perturbative aspects of Chern-Simons topological quantum field theory”, Ph.D. Thesis, Princeton University, 1991.
J. S. Birman, “New points of view in knot theory,” Bull. AMS 28 (1993) 253.
J. S. Birman and X. S. Lin, “Knot polynomials and Vassiliev’s invariants,” Invent. Math. 111 (1993) 225.
R. Bott and C. Taubes, “On the self-linking of knots,” Jour. Math. Phys. 35 (1994) 5247.
A. S. Cattaneo, P. Cotta-Ramusino, J. Frohlich and M. Martellini, “Topological BF theories in three-dimensions and four-dimensions,” J. Math. Phys. 36 (1995) 6137.
P. Freyd, D. Yetter, J. Hoste, W. B. R. Lickorish, K. Millet and A. Ocneanu, “A new polynomial invariant of knots and links,” Bull. AMS 12 (1985) 239.
M. Goussarov, M. Polyak and O. Viro, “Finite Type Invariants of Classical and Virtual Knots”, preprint, 1998, math.GT/9810073.
E. Guadagnini, M. Martellini and M. Mintchev, `Perturbative aspects of the ChernSimons field theory,“ Phys. Lett. B227(1989) 111; ”Chern-Simons model and new relations between the HOMFLY coefficients,“ B228 (1989) 489, and ”Wilson lines in Chern-Simons theory and link invariants,“ Nucl. Phys. B330 (1990) 575.
A. C. Hirshfeld and U. Sassenberg “Derivation of the total twist from Chern-Simons theory,”Journal of Knot Theory and its Ramifications5(1996) 489 and “Explicit formulation of a third order finite knot invariant derived from Chern-Simons theory,” 5(1996) 805
V. F. R. Jones, “Hecke algebras representations of braid groups and link polynomials,” Ann. of Math. 126 (1987) 335.
C. Kassel, M. Rosso and V. Turaev “Quantum groups and knot invariants”, Panoramas et syntheses 5, Societe Mathematique de France, 1997.
C. Kassel and V. Turaev, “Chord diagram invariants of tangles and graphs,” Duke Math. J. 92 (1998) 497–552.
L. Kauffman, “Witten’s Integral and Kontsevich Integral”, Particles, Fields and Gravitation, Lodz, Poland 1998, Ed. Jakub Rembieliski; AIP Proceedings 453 (1998), 368.
L. H. Kauffman, “An invariant of regular isotopy,” Trans. Am. Math. Soc. 318 (1990) 417.
M. Kontsevich, “Vassiliev’s knot invariants,” Advances in Soviet Math. 16, Part 2 (1993) 137.
J. M. F. Labastida, “Chern-Simons Gauge Theory: Ten Years After”, Trends in Theoretical Physics II, H. Falomir, R. Gamboa, F. Schaposnik, eds., American Institute of Physics, New York, 1999, CP 484, 1–41, hep-th/9905057.
J. M. F. Labastida and E. Pérez, “Kontsevich integral for Vassiliev invariants from Chern-Simons perturbation theory in the light-cone gauge,” J. Math. Phys. 39 (1998) 5183; hep-th/9710176.
J. M. F. Labastida and E. Pérez, “Gauge-invariant operators for singular knots in Chern-Simons gauge theory,” Nucl. Phys. B527 (1998) 499, hep-th/9712139.
J. M. F. Labastida and E. Pérez, “Combinatorial Formulae for Vassiliev Invariants from Chern-Simons Perturbation Theory”, J. Math. Phys. 41 (2000), 2658–2699.
J. M. F. Labastida and E. Pérez, “Vassiliev Invariants in the Context of ChernSimons Gauge Theory”, Santiago de Compostela preprint, US-FT-18/98; hepth/9812105.
G. Leibbrandt, “Introduction to noncovariant gauges,” Rev. Mod. Phys. 59 (1987) 1067.
M. Polyak and O. Viro, “Gauss diagram formulas for Vassiliev invariants,” Int. Math. Res. Notices 11 (1994) 445.
H. Ooguri and C. Vafa, “Knot Invariants and Topological Strings”, Harvard preprint, HUTP-99/A070, hep-th/9912123.
D. Thurston, “Integral expressions for the Vassiliev knot Invariants”, Harvard University senior thesis, April 1995; math/9901110.
V. A. Vassiliev, “Cohomology of knot spaces”, Theory of singularities and its applications, Advances in Soviet Mathematics, vol. 1,Arvericam Math. Soc., Providence, RI, 1990, 23–69.
J. F. W. H. van de Wetering, “Knot invariants and universal R-matrices from perturbative Chern-Simons theory in the almost axial gauge,” Nucl. Phys. B379 (1992) 172.
S. Willerton, “On Universal Vassiliev Invariants, Cabling, and Torus Knots”, University of Melbourne preprint (1998).
E. Witten, “Quantum field theory and the Jones polynomial,” Commun. Math. Phys. 121 (1989) 351.
S.-W. Yang, “Feynman integral, knot invariant and degree theory of maps”, National Taiwan University preprint, September 1997; q-alg/9709029.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer Basel AG
About this paper
Cite this paper
Labastida, J.M.F. (2001). Knot Invariants and Chern-Simons Theory. In: Casacuberta, C., Miró-Roig, R.M., Verdera, J., Xambó-Descamps, S. (eds) European Congress of Mathematics. Progress in Mathematics, vol 202. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8266-8_40
Download citation
DOI: https://doi.org/10.1007/978-3-0348-8266-8_40
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9496-8
Online ISBN: 978-3-0348-8266-8
eBook Packages: Springer Book Archive