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References for Knot Theory

Seifert surface of Whitehead link
We list some good references for readers who are interested in learning more about knot theory.
  • Colin Adams: The Knot Book , AMS, 2004. A nice introduction to knot theory which also mentions some applications and some knotty games.
  • Joan Birman: Braids: A Survey , Handbook of Knot Theory, Elsevier, 2005. A good comprehensive introduction to the theory of braids, and also contains some open problems in the field.
  • Burde, Zieschang: Knots , Walter de Gruyter, 2003 (Revised edition). A book for the more serious student, it covers both knots and braids and gives comprehensive proofs to most theorems.
  • Peter Cromwell: Knots and Links , Cambridge University Press, 2004. Another book which is good for beginners and covers a wide range of material.
  • W.B.Raymond Lickorish: An Introduction to Knot Theory , Springer, 1997. Designed for beginning graduate students and covers some more modern aspects of the subject, such as quantum invariants and generalisations of the Jones polynomial.
  • Charles Livingston: Knot Theory (Carus Mathematical Monographs) , The Mathematical Association of America, 1996. Nice introduction to knot theory, and small enough to keep in your pocket to read whilst waiting for the bus.
  • Dale Rolfsen: Knots and Links , Publish or Perish, 1976 (reprinted in 2003). A more topologically oriented book, great for beginners as well as more advanced students.
Here is a list of some important papers where the theorems in these notes were proved for the first time.
  • J.W. Alexander: A lemma on systems of knotted curves , Proc. Nat. Acad. Sci. USA 9 (1923) 93-95.
  • E. Artin: Theory of braids , Annals of Math. (2), 48 (1947) 101-126. His original paper on braids is [Abh. Math. Sem. Hamburgischen Univ. 4 (1926) 47-72].
  • W. Burau: Ueber Zopfgruppen und gleichsinnig verdrilte Verkettungen , Abh. Math. Sem. Hanischen Univ. 11 (1936) 171-178.
  • R. Crowell: Genus of alternating link types , Annals of Math. (2), 69 (1959) 258-275.
  • Joel Hass: Algorithms for recognizing knots and 3-manifolds This paper surveys old and potential algorithms for the key problems in knot theory: to decide whether two given knots are isotopic, and to decide whether a given knot is isotopic to the unknot.
  • G. Hemion: The classifications of knots and 3-dimensional spaces . Oxford University Press, 1992. The first paper with an algorithm deciding whether two given knots are isotopic. The solution, to the best of my knowledge is not readily implementible and has not been implemented on a computer.
  • W. Burau: Hecke Algebra Representation of Braid Groups and Link Polynomials , Annals of Math. 126 (1987) 335-388.

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