Knot theory

Please e-mail suggestions of additional material to a.ranicki@ed.ac.uk

An online archive of source material on knots.

A small personal selection from the 10,000+ items in the Mathematical Reviews which include "knot" in the review.
Geared to high-dimensional knots and knot cobordism.

BIOGRAPHIES OF EARLY KNOT THEORISTS

Links to biographical entries in the MacTutor St. Andrews Mathematics History Archive.
  • M.Dehn
  • P.Heegaard. Heegaard's thesis (1898). English translation by Hans Munkholm. See also the Heegaard home page.
  • T.P.Kirkman
  • J.B.Listing
  • F.Meyer
  • K.Reidemeister
  • O.Schreier
  • P.G.Tait
  • W.Thomson (Lord Kelvin)
  • W.Wirtinger
  • Richard Charles Blanchfield (1928-1955)
    Photo from the Princeton University Library, Princeton University Archives. Department of Rare Books and Special Collection. Reproduced by permission.

    • SCIENTIFIC PAPERS, BIOGRAPHY AND BIBLIOGRAPHY OF P.G.TAIT

  • Scientific papers Volume 1 (1898) and Volume 2 (1900), Cambridge University Press.
  • Life and Scientific Work of Peter Guthrie Tait, by Cargill G. Knott, Cambridge University Press (1911)
  • Bibliography prepared by Chris Pritchard and David Forfar.
  • Wikipedia entry on the Tait Conjectures
  • P.G.Tait, Some elementary properties of closed plane curves, Messenger of Mathematics, New Series, No.69, 1877, (communicated at the 1876 Meeting of the British Association).
  • P.G.Tait, On knots, Proc. Royal Soc. Edinburgh, Vol. 9, 97 (1876-7), 306-317
  • P.G.Tait, On links, Proc. Royal Soc. Edinburgh, Vol. 9, 98 (1876-7), 321-332
  • P.G.Tait, Sevenfold knottiness, Proc. Royal Soc. Edinburgh, Vol. 9, 98 (1876-7), 363-366
  • P.G.Tait Applications of the theorem that two closed plane curves intersect an even number of times Proc. Royal Soc. Edinburgh, Vol. 9, 97 (1876-7), 237-246
  • P.G.Tait Note on the measure of beknottedness Proc. Royal Soc. Edinburgh, Vol. 9, 97 (1876-7), 289-298
  • P.G.Tait On amphicheiral forms and their relations Proc. Royal Soc. Edinburgh, Vol. 9, 98 (1876-7), 391-392
  • P.G.Tait Preliminary note on a new method of investigating the properties of knots Proc. Royal Soc. Edinburgh, Vol. 9, 98 (1876-7), 403
  • P.G.Tait Additional remarks on knots Proc. Royal Soc. Edinburgh, Vol. 9, 98 (1876-7), 405
  • P.G.Tait On knots I. Trans. Roy. Soc. Edinburgh 28 (1876-7), 145-190
  • P.G.Tait On knots II. Trans. Roy. Soc. Edinburgh 32 (1883-4), 327-342
  • P.G.Tait On knots III. Trans. Roy. Soc. Edinburgh 32 (1884-5),493-506
  • P.G.Tait, Johann Benedict Listing, Nature 27 (1882-83), 316.
  • P.G.Tait, Listing's Topologie (Introductory address to the Edinburgh Mathematical Society, November 9, 1883), Philosophical Magazine, January, 1884.
  • Sealed envelope deposited by Tait with the RSE on 16th October 1876, the original statement of the Tait conjecture that the crossing number of a reduced alternating projection of an alternating knot is a topological invariant of the knot. Opened by the RSE Archivist Jean Jones on 15th Octeober 1987, the year in which the conjecture was solved by Kauffman, Murasugi and Thistlethwaite using the (Vaughan) Jones polynomial -- see papers below. Reproduced by permission of the Royal Society of Edinburgh and NLS Trustees from the Papers of the Royal Society of Edinburgh Acc 1000, held on deposit at the National Library of Scotland.
  • State models and the Jones polynomial by L.Kauffman, Topology 26, 395--407 (1987)
  • Jones polynomials and classical conjectures in knot theory by K.Murasugi, Topology 26, 187--194 (1987)
  • A spanning tree expansion of the Jones polynomial by M.Thistlethwaite, Topology 26, 297-309 (1987)
  • New invariants in the Theory of Knots by L.Kauffman, American Math. Monthly 95, 195--242 (1988)
  • Topology, Matter and Space, I: Topological Notions in 19th-Century Natural Philosophy by Moritz Epple, Arch. Hist. Exact Sci. 52, 297--392 (1998). The story of the sealed envelope is on pages 355-356.

    • EARLY PAPERS AND BOOKS ON KNOT THEORY

  • A.T.Vandermonde, Remarques sur les problems de situation. Memoires de l'Academie Royale des Sciences (Paris), 566-574 (1771)
  • J.B.Listing, Vorstudien zur Topologie, Goettinger Studien (Abtheilung 1) 1 (1847), 811-875
  • A.Cayley, On a problem of arrangements, Proc. Royal Soc. Edinburgh, Vol. 9, 98 (1876-7), 338-342
  • Crum Brown, On a case of interlacing surfaces, Proc. Royal Soc. Edinburgh, Vol. 13, 121 (1885-6), 382-386
  • M.G.Haseman On knots, with a census of the amphicheirals with twelve crossings Trans. Roy. Soc. Edinburgh, 52 (1917-8), 235-255. Ph.D thesis, Bryn Mawr College, 1918
    Printed by Neill & Co. Limited, 212 Causewayside, Edinburgh - 1918.
  • M.G.Haseman Amphicheiral knots Trans. Roy. Soc. Edinburgh 52 (1919-20), 597-602.
  • P.Heegaard Forstudier til en topologisk teori for de algebraiske fladers sammehaeng Thesis (in Danish, Nordiske Forlag, Copemhagen, 1898), Sur l'Analysis situs (French translation, Bulletin de la S.M.F. 44, 161-242 (1916))
  • T.P.Kirkman The enumeration, description, and construction of knots of fewer than 10 crossings Trans. Roy. Soc. Edinburgh 32 (1883-4), 281-309.
  • T.P.Kirkman The 364 unifilar knots of ten crossings, enumerated and described Trans. Roy. Soc. Edinburgh 32 (1884-5), 483-491. Two appendices in Proc. Roy. Soc. Edinburgh 13, p. 359.
  • T.P.Kirkman Examples upon the Reading of the Circle or Circles of a Knot Proc. Royal Soc. Edinburgh, Vol. ?? (1885-6), 693-698
  • T.P.Kirkman On the twists of Listing and Tait Proc. Royal Soc. Edinburgh, Vol. 13, 120 (1884-5), 363-367
  • T.P.Kirkman Demonstration of Theorems A,B,C etc. Proc. Royal Soc. Edinburgh, Vol. 13, 120 (1884-5), 359-363
  • T.P.Kirkman, On the linear section PR of a knot M_n, which passes through two crossings P and R, which meets no edge, and which cuts away a (3+r)-gonal mesh of M_n, Proc. Royal Soc. Edinburgh, Vol. 13, 121 (1884-5), 514-522
  • C.N.Little, On knots, with a census for order 10 Trans. Connecticut Academy Sci. 18, Vol. 7 (1885), 27-43 (1-17 ??).
  • C.N.Little, Alternate +/-Knots of order eight and nine Trans. Royal. Soc. Edinburgh 35 (1889-90), 253-255.
  • C.N.Little Alternate +/-Knots of order eleven Trans. Royal. Soc. Edinburgh 36 (1890-1), 253-255.
  • C.N.Little, Non-Alternate +/-Knots, Trans. Royal. Soc. Edinburgh 39 (1898-9), 771-778.
  • F.Meyer Ueber algebraische Knoten Proc. Royal Soc. Edinburgh, Vol. 13, 97 (1885-6), 931-946
  • W.Thomson Vortex statics Proc. Royal Soc. Edinburgh Vol. 9, 94 (1875-6), 59-73
  • H.Brunn Ueber Verkettung Sitzungsber. Bayr. Akad. 22, 77-99 (1892)
  • J.W.Alexander and G.B.Briggs On types of knotted curves Ann. of Maths. 28, 562-586 (1926)
  • K.Reidemeister Knotentheorie Springer (1932)
  • H.Tietze Ein Kapitel Topologie. Zur Einführung in die Lehre von den verknoteten Linien. Teubner (1942)

    • BOOKS AND PAPERS ABOUT THE HISTORY OF KNOT THEORY

  • R.Ricca and B.Nipoti Gauss' linking number revisited, J. Knot Theory Ramifications 20, 1325-1343 (2011).
  • M.Epple, Die Enstehung der Knotentheorie, Vieweg (1999), Mathematical review
  • J.Przytycki Little and Haseman -- early American tabulators of knots
  • J.Przytycki The classical roots of knot theory. Chaos, Solitons and Fractals 9, 531--545 (1998)

    • THE ORIGINAL PAPERS ON THE ALEXANDER POLYNOMIAL, SEIFERT SURFACES AND MATRICES, CONWAY'S ENUMERATION OF KNOTS AND LINKS

  • Topological invariants of knots and links by J.W.Alexander, Transactions of A.M.S. 30, 275--306 (1928)
  • Ein Knotensatz mit Anwendung auf Dimensionstheorie by P.Frankl and L.Pontrjagin, Math. Ann. 102, 785--789 (1930)
  • Ueber das Geschlecht von Knoten by H.Seifert, Math. Ann. 110, 571--592 (1934)
  • An enumeration of knots and links, and some of their algebraic properties by J.H.Conway, Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967), 329--358, Pergamon (1970)

    • SOME KNOT WEBSITES

  • A Bibliography of Literature on Knots and Braids by Joyce Rile (1991).
  • Knot Atlas (wiki)
  • SeifertView Windows programme for the visualisation of knots and links, and their Seifert surfaces. Try the roller coaster ride! (misc, esc for full screen)
  • Knotilus
  • KnotPlot
  • Knots worth knowing
  • The hagfish (Thanks to John Murray for the weblink).
  • Julia Collins.
  • The Beauty, Dynamics and Design of String Patterns in Folk Arts
  • Celtic knot theory Edinburgh senior year undergraduate project by Jessica Connor and Nick Ward (2012)

    • Macedonian translation of the website