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The mathematics and physics of knots

REVIEW ARTICLE

Louis H Kauffman

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This paper is an introduction to relationships between knot theory and theoretical physics. We give an exposition of the theory of polynomial invariants of knots and links, the Witten functional integral formulation of knot and link invariants, and the beginnings of topological quantum field theory, and show how the theory of knots is related to a number of key issues in mathematical physics, including loop quantum gravity and quantum information theory. Along with the references cited in the text below, we also recommend the following as sources of background information [1–13].


PACS

02.10.Kn Knot theory

03.67.-a Quantum information

11.10.-z Field theory

04.60.Pp Loop quantum gravity, quantum geometry, spin foams

02.10.De Algebraic structures and number theory

MSC

81T45 Topological field theories (See also 57R56, 58Dxx)

57M27 Invariants of knots and 3-manifolds

94Axx Communication, information

Subjects

Mathematical physics

Computational physics

Gravitation and cosmology

Particle physics and field theory

Quantum information and quantum mechanics

Dates

Issue 12 (December 2005)

Received 27 July 2005

Published 5 October 2005



  1. The mathematics and physics of knots

    Louis H Kauffman 2005 Rep. Prog. Phys. 68 2829

  2. J-shaped stress/strain curves and crack resistance of biological materials

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  6. Glassy behaviour in the ferromagnetic Ising model on a Cayley tree

    R Mélin et al 1996 J. Phys. A: Math. Gen. 29 5773

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    J Dai et al 2000 J. Phys. D: Appl. Phys. 33 L65

  8. Use of the discriminant Fourier-derived cepstrum with feature-level post-processing for surface electromyographic signal classification

    Xinpu Chen et al 2009 Physiol. Meas. 30 1399

  9. Mass-producible replication of highly hydrophobic surfaces from plant leaves

    Seung-Mo Lee and Tai Hun Kwon 2006 Nanotechnology 17 3189

  10. Geometric properties of two-dimensional O(n) loop configurations

    Chengxiang Ding et al 2007 J. Phys. A: Math. Theor. 40 3305

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