Knots and Knot Types

  • Richard H. Crowell
  • Ralph H. Fox
Part of the Graduate Texts in Mathematics book series (GTM, volume 57)


Almost everyone is familiar with at least the simplest of the common knots, e.g., the overhand knot, Figure 1, and the figure-eight knot, Figure 2. A little experimenting with a piece of rope will convince anyone that these two knots are different: one cannot be transformed into the other without passing a loop over one of the ends, i.e., without “tying” or “untying.” Nevertheless, failure to change the figure-eight into the overhand by hours of patient twisting is no proof that it can’t be done. The problem that we shall consider is the problem of showing mathematically that these knots (and many others) are distinct from one another.


Multiple Point Double Point Orientation Preserve Infinite Order Mathematical Definition 
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  1. 1.
    Any knot which lies in a plane is necessarily trivial. This is a well-known and deep theorem of plane topology. See A. H. Newman, Elements of the Topology of Plane Sets of Points, Second edition (Cambridge University Press, Cambridge, 1951), p. 173.zbMATHGoogle Scholar
  2. 2.
    R. H. Fox, “A Remarkable Simple Closed Curve,” Annals of Mathematics, Vol. 50 (1949), pp. 264, 265.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    For an account of the concepts used in this proof, see O. Veblen and J. W. Young, Projective Geometry (Ginn and Company, Boston, Massachusetts, 1910), Vol. 1 pp. 11, 299, 301.zbMATHGoogle Scholar
  4. 6.
    G. M. Fisher, “On the Group of all Homeomorphisms of a Manifold,” Transactions of the American Mathematical Society; Vol. 97 (1960), pp. 193–212.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 7.
    H. F. Trotter, “Noninvertible knots exist.” Topology, vol. 2 (1964), pp. 275–280.MathSciNetCrossRefGoogle Scholar

Copyright information

© R. H. Crowell and C. Fox 1963

Authors and Affiliations

  • Richard H. Crowell
    • 1
  • Ralph H. Fox
    • 2
  1. 1.Department of MathematicsDartmouth CollegeHanoverUSA
  2. 2.Princeton UniversityPrincetonUSA

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