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The Knot Polynomials

  • Richard H. Crowell
  • Ralph H. Fox
Chapter
  • 1.8k Downloads
Part of the Graduate Texts in Mathematics book series (GTM, volume 57)

Abstract

The underlying knot-theoretic structure developed in this book is a chain of successively weaker invariants of knot type. The sequence of knot polynomials, to which this chapter is devoted, is the last in the chain
$$ \begin{gathered} \quad \quad \quad \quad knot\,type\,of\,K \hfill \\ \quad \quad \quad \quad \quad \quad \downarrow \hfill \\ presentation\,type\;of\,\pi \left( {{R^3} - K} \right) \hfill \\ \quad \quad \quad \quad \quad \quad \downarrow \hfill \\ sequence\,of\,elementary\,ideals \hfill \\ \quad \quad \quad \quad \quad \;\;\; \downarrow \hfill \\ sequence\,of\,knot\,polynomials \hfill \\ \end{gathered} $$
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Keywords

Abelianized Group Commutative Ring Elementary Ideal Integral Domain Group Ring 
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References

  1. 1.
    See N. Jacobson, Lectures in Abstract Algebra, Vol. 1 (D. van Nostrand Company, Inc.; Princeton, N.J., 1951), Chap. 3, Sects. 4, 5, 6.Google Scholar
  2. 2.
    See N. Jacobson, Lectures in Abstract Algebra, Vol. 1 (D. van Nostrand Company, Inc.; Princeton, N.J., 1951), Chap. 4, Sect. 6.Google Scholar
  3. 3.
    By the linking invariant of the second cyclic branched covering; cf. H. Seifert, “Die Verschlingungsinvarianten der zyklischen Knotenüberlagerungen,” Hamb. Abh. 11 (1935) pp. 84–101.zbMATHCrossRefGoogle Scholar
  4. 4.
    R. H. Fox, “On the Complementary Domains of a Certain Pair of Inequivalent Knots,” Ned. Akademie Wetensch., Indag. Math. Vol. 14 (1952), pp. 37–40;Google Scholar
  5. 4a.
    H. Seifert, “Verschlingungsinvarianten,” S. B. Preuss. Akad. Wiss. Berlin Vol. 26 (1933), pp. 811–823.Google 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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