Presentation of a Knot Group

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


In this chapter we return to knot theory. The major objective here is the description and verification of a procedure for deriving from any polygonal knot K in regular position two presentations of the group of K, which are called respectively the over and under presentations. The classical Wirtinger presentation is obtained as a special case of the over presentation. In a later section we calculate over presentations of the groups of four separate knots explicitly, and the final section contains a proof of the existence of nontrivial knots, in that it is shown that the clover-leaf knot can not be untied. The fact that our basic description in this chapter is concerned with a pair of group presentations represents a concession to later theory. It is of no significance at this stage. One presentation is plenty, and, for this reason, Section 4 is limited to examples of over presentations. The existence of a pair of over and under presentations is the basis for a duality theory which will be exploited in Chapter IX to prove one of the important theorems.


Equivalence Class Fundamental Group Free Basis Homotopy Type Simple Path 
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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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