Introduction to Knot Theory pp 110-133 | Cite as
The Knot Polynomials
Chapter
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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 .
$$ \begin{gathered} \quad \quad \quad \quad knot\,type\,of\,K \hfill \\ \quad \quad \quad \quad \quad \quad \downarrow \hfill \\ presentation\,type\;of\,\pi \left( {{R^3} - K} \right) \hfill \\ \quad \quad \quad \quad \quad \quad \downarrow \hfill \\ sequence\,of\,elementary\,ideals \hfill \\ \quad \quad \quad \quad \quad \;\;\; \downarrow \hfill \\ sequence\,of\,knot\,polynomials \hfill \\ \end{gathered} $$
Keywords
Abelianized Group Commutative Ring Elementary Ideal Integral Domain Group Ring
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References
- 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.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.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.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
- 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