The Free Calculus and the Elementary Ideals

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


In the last chapter a method was developed for calculating a presentation of the group of any knot in regular position. Unfortunately, it does not follow, as was pointed out, that it is now a simple matter to distinguish knot groups, and thus knot types. There is no general algorithm for deciding whether or not two presentations define isomorphic groups, and even in particular examples the problem can be difficult. We are therefore concerned with deriving some powerful, yet effectively calculable, invariants of group presentation type. Such are the elementary ideals. In this chapter we shall study the necessary algebraic machinery for defining these invariants. Specialization to presentations of knot groups in Chapter VIII then leads naturally to the knot polynomials. With these invariants we can easily distinguish many knot types.


Commutative Ring Elementary Ideal Group Ring Ring Homomorphism Commutator Subgroup 
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  1. 1.
    E. Artin, “The Theory of Braids,” American Scientist, Vol. 38, No. 1 (1950), pp. 112–119;MathSciNetGoogle Scholar
  2. 1a.
    F. Bohnenblust, “The Algebraical Braid Group,” Annals of Mathematics, Vol. 48 (1947), pp. 127–136.MathSciNetzbMATHCrossRefGoogle 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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