From Goeritz Matrices to Quasi-alternating Links

  • Józef H. PrzytyckiEmail author
Part of the Contributions in Mathematical and Computational Sciences book series (CMCS, volume 1)


Knot Theory is currently a very broad field. Even a long survey can only cover a narrow area. Here we concentrate on the path from Goeritz matrices to quasi-alternating links. On the way, we often stray from the main road and tell related stories, especially if they allow us to place the main topic in a historical context. For example, we mention that the Goeritz matrix was preceded by the Kirchhoff matrix of an electrical network. The network complexity extracted from the matrix corresponds to the determinant of a link. We assume basic knowledge of knot theory and graph theory, however, we offer a short introduction under the guise of a historical perspective.


Jones Polynomial Regular Neighborhood Reidemeister Move Link Diagram Alexander Polynomial 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [A28]
    Alexander, J.W.: Topological invariants of knots and links. Trans. Am. Math. Soc. 30, 275–306 (1928) zbMATHGoogle Scholar
  2. [AB27]
    Alexander, J.W., Briggs, G.B.: On types of knotted curves. Ann. Math. 28(2), 562–586 (1926/27) CrossRefMathSciNetGoogle Scholar
  3. [AP01]
    Asaeda, M.M., Przytycki, J.H.: Khovanov homology: torsion and thickness. In: Bryden, J. (ed.) Proceedings of the Workshop, “New Techniques in Topological Quantum Field Theory” Calgary/Kananaskis, Canada, August 2001; October, 2004; arXiv:math.GT/0402402
  4. [BL]
    Baldridge, S., Lowrance, A.: Cube diagrams and a homology theory for knots. arXiv:0811.0225v1 [math.GT]
  5. [BNFK98]
    Bar-Natan, D., Fulman, J., Kauffman, L.H.: An elementary proof that all spanning surfaces of a link are tube-equivalent. J. Knot Theory Ramif. 7, 873–879 (1998) zbMATHCrossRefMathSciNetGoogle Scholar
  6. [Big74]
    Biggs, N.L.: Algebraic Graph Theory, 2nd edn. Cambridge University Press, Cambridge (1993) (first edition, 1974) Google Scholar
  7. [BLW86]
    Biggs, N.L., Lloyd, E.K., Wilson, R.J.: Graph Theory 1736–1936. Clarendon, Oxford (1986) zbMATHGoogle Scholar
  8. [BE82]
    Boltyanskii, V.G., Efremovich, V.A.: Intuitive Combinatorial Topology. Springer, Berlin (1982). English translation 2001 edition Google Scholar
  9. [Bon85]
    Bonner, A. (ed.): Doctor Illuminatus. A Ramon Llull Reader. Princeton University Press, Princeton (1985) Google Scholar
  10. [BSAT40]
    Brooks, R.I., Smith, C.A.B., Stone, A.H., TutteW.T.: The dissection of rectangles into squares. Duke Math. J. 7, 312–340 (1940) CrossRefMathSciNetGoogle Scholar
  11. [BZ85]
    Burde, G., Zieschang, H.: Knots, 2nd edn. de Gruyter, Berlin (2003) (First edition 1985) zbMATHGoogle Scholar
  12. [CL]
    Cha, J.C., Livingston, C.: Knotinfo: Table of knot invariants.
  13. [CK]
    Champanerkar, A., Kofman, I.: Twisting quasi-alternating links. arXiv:0712.2590
  14. [Con69]
    Conway, J.H.: An enumeration of knots and links. In: Leech, J. (ed.) Computational Problems in Abstract Algebra, pp. 329–358. Pergamon, Elmsford (1969) Google Scholar
  15. [Cro89]
    Cromwell, P.R.: Homogeneous links. J. Lond. Math. Soc. 39(2), 535–552 (1989) zbMATHCrossRefMathSciNetGoogle Scholar
  16. [Cz06]
    Czerniakowska, M.: Leonard Euler (1707–1783) and Poland (in Polish). Gdańsk (2006). ISBN 83-924379-2-6 Google Scholar
  17. [DFK+08]
    Dasbach, O., Futer, D., Kalfagianni, E., Lin, X.-S., Stoltzfus, N.: The Jones polynomial and graphs on surfaces. J. Comb. Theory, Ser. B 98/2, 384–399 (2008) CrossRefMathSciNetGoogle Scholar
  18. [Deh07]
    Dehn, M.: In: Jahresbericht der deutschen Mathematiker–Vereinigung, vol. 16, p. 573 (1907) Google Scholar
  19. [DH07]
    Dehn, M., Heegaard, P.: Analysis situs. Encykl. Math. Wiss. III, 153–220 (1907) Google Scholar
  20. [EH]
    Edelsbrunner, H., Harer, J.: Persistent homology—a survey.
  21. [Eul36]
    Euler, L.: Solutio problematis ad geometriam situs pertinentis. Commen. Acad. Sci. Imp. Petropol. 8, 128–140 (1736) Google Scholar
  22. [FP30]
    Frankl, P., Pontrjagin, L.: Ein knotensatz mit Anwendung auf die Dimensionstheorie. Math. Ann. 102, 785–789 (1930) zbMATHCrossRefMathSciNetGoogle Scholar
  23. [FYHLMO85]
    Freyd, P., Yetter, D., Hoste, J., Lickorish, W.B.R., Millett, K., Ocneanu, A.: A new polynomial invariant of knots and links. Bull. Am. Math. Soc. 12, 239–249 (1985) zbMATHCrossRefMathSciNetGoogle Scholar
  24. [Gil82]
    Giller, C.A.: A family of links and the Conway calculus. Trans. Am. Math. Soc. 270(1), 75–109 (1982) zbMATHCrossRefMathSciNetGoogle Scholar
  25. [Goe33]
    Goeritz, L.: Knoten und quadratische Formen. Math. Z. 36, 647–654 (1933) CrossRefMathSciNetGoogle Scholar
  26. [Gor78]
    Gordon, C.M.: Some aspects of classical knot theory. In: Knot Theory. Lecture Notes in Mathematics, vol. 685, pp. 1–60. Springer, Berlin (1978) CrossRefGoogle Scholar
  27. [GL78]
    Gordon, C.M., Litherland, R.A.: On the signature of a link. Invent. Math. 47, 53–69 (1978) zbMATHCrossRefMathSciNetGoogle Scholar
  28. [Gre]
    Greene, J.: Homologically thin, non-quasi-alternating links. arXiv:0906.2222v1 [math.GT]
  29. [Has58]
    Hashizume, Y.: On the uniqueness of the decomposition of a link. Osaka Math. J. 10, 283–300 (1958) MathSciNetGoogle Scholar
  30. [Hem76]
    Hempel, J.: 3-manifolds. Annals of Mathematics Studies, vol. 86. Princeton University Press, Princeton (1976) zbMATHGoogle Scholar
  31. [HW04]
    Hopkins, B., Wilson, R.: The truth about Königsberg. Coll. Math. J. 35, 198–207 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  32. [JS]
    Jablan, S., Sazdanovic, R.: Quasi-alternating links and odd homology: computations and conjectures. arXiv:0901.0075
  33. [JP87]
    Jakobsche, W., Przytycki, J.H.: Topologia 3-wymiarowych Rozmaitości. Warsaw University Press, Warsaw (1987). In Polish (Topology of 3-dimensional manifolds), Second edition in preparation, Script, Warsaw, 2011 Google Scholar
  34. [Jan01]
    Januszajtis, A.: Scientists in old Gdansk: 17th and 18th century. Task Q. 5(3), 389–399 (2001). Google Scholar
  35. [Jon85]
    Jones, V.F.R.: A polynomial invariant for knots via Von Neumann algebras. Bull. Am. Math. Soc. 12, 103–111 (1985) zbMATHCrossRefGoogle Scholar
  36. [Kau80]
    Kauffman, L.H.: The Conway polynomial. Topology 20, 101–108 (1980) CrossRefMathSciNetGoogle Scholar
  37. [Kau83]
    Kauffman, L.H.: Formal Knot Theory. Mathematical Notes, vol. 30. Dover, New York (2006). (First edition, Princeton University Press (1983)) Google Scholar
  38. [Kau87a]
    Kauffman, L.H.: On Knots. Annals of Math. Studies, vol. 115. Princeton University Press, Princeton (1987) zbMATHGoogle Scholar
  39. [Kau87a]
    Kauffman, L.H.: State models and the Jones polynomial. Topology 26, 395–407 (1987) zbMATHCrossRefMathSciNetGoogle Scholar
  40. [Kaw96]
    Kawauchi, A.: A Survey of Knot Theory. Birkhäusen, Basel (1996) zbMATHGoogle Scholar
  41. [Kho00]
    Khovanov, M.: A categorification of the Jones polynomial. Duke Math. J. 101(3), 359–426 (2000). arXiv:math.QA/9908171 zbMATHCrossRefMathSciNetGoogle Scholar
  42. [Kir47]
    Kirchhoff, G.R.: Über die Auflosung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer strome geführt wird. Ann. Phys. Chem. 72, 497–508 (1847) CrossRefGoogle Scholar
  43. [KP53]
    Kneser, M., Puppe, D.: Quadratische Formen und Verschlingungsinvarianten von Knoten. Math. Z. 58, 376–384 (1953) zbMATHCrossRefMathSciNetGoogle Scholar
  44. [Kue56]
    Kuehnio, H.: Meditationes de quantitatibus imaginariis construendis et radicibus imaginariis exhibendis. Novi Commen. Acad. Sci. Imp. Petropol. 1750–1751 III, 170–223 (1756). Google Scholar
  45. [Kyl54]
    Kyle, R.H.: Branched covering spaces and the quadratic forms of links. Ann. Math. 59(2), 539–548 (1954) CrossRefMathSciNetGoogle Scholar
  46. [Lei66]
    Leibniz, G.W.: Dissertatio de arte combinatoria. In: Sämtliche Schriften und Briefe, vol. A VI 1, p. 163. Akademie, Berlin (1666) Google Scholar
  47. [Lev69]
    Levine, J.: Knot cobordism groups in codimension two. Comment. Math. Helv. 44, 229–244 (1969) zbMATHCrossRefMathSciNetGoogle Scholar
  48. [Lic97]
    Lickorish, W.B.R.: An Introduction to Knot Theory. Graduate Texts in Mathematics, vol. 175. Springer, Berlin (1997) zbMATHGoogle Scholar
  49. [Lis47]
    Listing, J.B.: Vorstudien zur Topologie. Göttinger Studien (Abtheilung 1), vol. 1, pp. 811–875 (1847). The part we quote was translated by Maxim Sokolov in 1997 Google Scholar
  50. [Liv93]
    Livingston, C.: Knot Theory. Carus Mathematical Monographs, vol. 24. Math. Assoc. Am., Washington, D.C. (1993) zbMATHGoogle Scholar
  51. [Llu05]
    Llull, R.: Ars generalis ultima (Ars magna). 1305. Figure 9.1 is taken from
  52. [Mag78]
    Magnus, W., Dehn, M., Math. Intell. 1, 132–143 (1978). Also in: Wilhelm Magnus Collected Paper, Edited by G. Baumslag and B. Chandler. Springer (1984) zbMATHCrossRefMathSciNetGoogle Scholar
  53. [MO]
    Manolescu, C., Ozsvath, P.: On the Khovanov and knot Floer homologies of quasi-alternating links. arXiv:0708.3249
  54. [Mat77]
    Matumoto, T.: On the signature invariants of a non-singular complex sesquilinear form. J. Math. Soc. Jpn. 29, 67–71 (1977) zbMATHCrossRefMathSciNetGoogle Scholar
  55. [Mil68]
    Milnor, J.W.: Infinite cyclic covers. In: Conference on the Topology of Manifolds, pp. 115–133 (1968) Google Scholar
  56. [MH73]
    Milnor, J.W., Husemoller, D.: Symmetric Bilinear Forms. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 73. Springer, Berlin (1973) zbMATHGoogle Scholar
  57. [Mur65]
    Murasugi, K.: On a certain numerical invariant of link types. Trans. Am. Math. Soc. 117, 387–422 (1965) zbMATHCrossRefMathSciNetGoogle Scholar
  58. [Mur70]
    Murasugi, K.: On the signature of links. Topology 9, 283–298 (1970) CrossRefMathSciNetGoogle Scholar
  59. [Mur96]
    Murasugi, K.: Knot Theory and Its Applications. Birkhauser, Basel (1996). Translated from the 1993 Japanese original by B. Kurpita zbMATHGoogle Scholar
  60. [Mye71]
    Myers, B.R.: Number of spanning trees in a wheel. Trans. Circuit Theory CT-18, 280–282 (1971) CrossRefGoogle Scholar
  61. [New76]
    Newton, I.: Newton’s letter to Robert Hooke, February 5, 1676 (Julian calendar) (1676). See [Wes83] Google Scholar
  62. [OS05]
    Ozsvath, P., Szabo, Z.: On the Heegaard Floer homology of branched double covers. Adv. Math. 194, 1–33 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  63. [PPS09]
    Pabiniak, M.D., Przytycki, J.H., Sazdanovic, R.: On the first group of the chromatic cohomology of graphs. Geom. Dedic. 140(1), 19–48 (2009). arXiv:math.GT/0607326 zbMATHCrossRefMathSciNetGoogle Scholar
  64. [Poi95]
    Poincaré, H.: Analysis situs (&12). J. Ecole Polytech. Norm. 1, 1–121 (1895) Google Scholar
  65. [Prz86]
    Przytycki, J.H.: Survey on recent invariants in classical knot theory. Warsaw University Preprints 6, 8, 9. Warszawa (1986). arXiv:0810.4191
  66. [Prz88]
    Przytycki, J.H.: In: t k-Moves on Links. Contemporary Math., vol. 78, pp. 615–656 (1988). arXiv:math.GT/0606633 Google Scholar
  67. [Prz98]
    Przytycki, J.H.: Classical roots of knot theory. Chaos Solitons Fractals 9(4–5), 531–545 (1998) zbMATHCrossRefMathSciNetGoogle Scholar
  68. [Prz12]
    Przytycki, J.H.: KNOTS: From Combinatorics of Knot Diagrams to the Combinatorial Topology Based on Knots, 600 pages. Cambridge University Press, Cambridge (2012, to appear). Chapter II, arXiv:math/0703096, Chapter V, arXiv:math.GT/0601227, Chapter IX, arXiv:math.GT/0602264, Chapter X, arXiv:math.GT/0512630
  69. [PT]
    Przytycki, J.H., Taniyama, K.: Almost positive links have negative signature. arXiv:0904.4130. Preprint (1991). To be published in February 2010 issue of JKTR
  70. [PT87]
    Przytycki, J.H., Traczyk, P.: Invariants of links of Conway type. Kobe J. Math. 4, 115–139 (1987) zbMATHMathSciNetGoogle Scholar
  71. [PT87’]
    Przytycki, J.H., Traczyk, P.: Conway algebras and skein equivalence of links. Proc. Am. Math. Soc. 100(4), 744–748 (1987) zbMATHMathSciNetGoogle Scholar
  72. [Rei27]
    Reidemeister, K.: Elementare Begrundung der Knotentheorie. Abh. Math. Sem. Univ. Hamb. 5, 24–32 (1927) CrossRefGoogle Scholar
  73. [Rei32]
    Reidemeister, K.: Knotentheorie, vol. 1. Springer, Berlin (1932). English translation: Knot theory, BSC Associates, Moscow, Idaho, USA, 1983 Google Scholar
  74. [Rol76]
    Rolfsen, D.: Knots and Links. Publish or Perish, Houston (1976). Second edition, 1990; third edition, AMS Chelsea Publishing, 2003 zbMATHGoogle Scholar
  75. [Sch49]
    Schubert, H.: Die eindeutige Zerlegbarkeit eines Knoten in Primknoten. Sitzungsber. Acad. Wiss. Heildernerg, Math.-nat. Kl. 3, 57–104 (1949) MathSciNetGoogle Scholar
  76. [Sch53]
    Schubert, H.: Knotten und Vollringe. Acta Math. 90, 131–286 (1953) zbMATHCrossRefMathSciNetGoogle Scholar
  77. [Sed70]
    Sedlacek, J.: On the skeletons of a graph or digraph. In: Calgary International Conference of Combinatorial Structures and Their Applications, pp. 387–391. Gordon & Breach, New York (1970) Google Scholar
  78. [Sei34]
    Seifert, H.: Über das Geschlecht von Knoten. Math. Ann. 110, 571–592 (1934) CrossRefMathSciNetGoogle Scholar
  79. [Thi85]
    Thistlethwaite, M.B.: Knot Tabulations and Related Topics. LMS Lecture Notes Series, vol. 93, pp. 1–76. Cambridge University Press, Cambridge (1985) Google Scholar
  80. [Tra04]
    Traczyk, P.: A combinatorial formula for the signature of alternating diagrams. Fund. Math. 184, 311–316 (2004). A new version of the unpublished manuscript, 1987 zbMATHCrossRefMathSciNetGoogle Scholar
  81. [Tra85]
    Traldi, L.: On the Goeritz matrix of a link. Math. Z. 188, 203–213 (1985) zbMATHCrossRefMathSciNetGoogle Scholar
  82. [Tri69]
    Tristram, A.G.: Some cobordism invariants for links. Proc. Camb. Philos. Soc. 66, 251–264 (1969) zbMATHCrossRefMathSciNetGoogle Scholar
  83. [Tro62]
    Trotter, H.F.: Homology of group systems with applications to knot theory. Ann. Math. 76, 464–498 (1962) CrossRefMathSciNetGoogle Scholar
  84. [Tur87]
    Turaev, V.G.: A simple proof of the Murasugi and Kauffman theorems on alternating links. Enseign. Math. (2) 33(3–4), 203–225 (1987) zbMATHMathSciNetGoogle Scholar
  85. [Van71]
    Vandermonde, A.T.: Remarques sur les problèmes de situation. Mém. Acad. R. Sci. 566–574 (1771) Google Scholar
  86. [Wes83]
    Westfall, R.S.: Never at Rest: A Biography of Isaac Newton. Cambridge University Press, Cambridge (1983) zbMATHGoogle Scholar
  87. [Wid09]
    Widmer, T.: Quasi-alternating Montesinos links. J. Knot Theory Ramif. 18(10), 1459–1469 (2009). arXiv:0811.0270 [math.GT] zbMATHCrossRefMathSciNetGoogle Scholar
  88. [Wir05]
    Wirtinger, W.: Über die Verzweigungen bei Funktionen von zwei Veränderlichen. In: Jahresbericht d. Deutschen Mathematiker Vereinigung, vol. 14, p. 517 (1905). The title of the talk supposedly given at September 26 1905 at the annual meeting of the German Mathematical Society in Meran Google Scholar

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© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.Dept. of MathematicsThe George Washington UniversityWashingtonUSA
  2. 2.University of Texas at DallasDallasUSA

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