A factorization of the conway polynomial and covering linkage invariants

Tatsuya Tsukamoto, Akira Yasuhara

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

Levine showed that the Conway polynomial of a link is a product of two factors: one is the Conway polynomial of a knot which is obtained from the link by banding together the components; and the other is determined by the μ̄-invariants of a string link with the link as its closure. We give another description of the latter factor: the determinant of a matrix whose entries are linking pairings in the infinite cyclic covering space of the knot complement, which take values in the quotient field of ℤ[t, t -1]. In addition, we give a relation between the Taylor expansion of a linking pairing around t = 1 and derivation on links which is invented by Cochran. In fact, the coefficients of the powers of t - 1 will be the linking numbers of certain derived links in S3. Therefore, the first non-vanishing coefficient of the Conway polynomial is determined by the linking numbers in S3. This generalizes a result of Hoste.

Original languageEnglish
Pages (from-to)631-640
Number of pages10
JournalJournal of Knot Theory and its Ramifications
Volume16
Issue number5
DOIs
Publication statusPublished - 2007 May 1
Externally publishedYes

Keywords

  • Conway polynomial
  • Covering linkage invariants
  • Deravation on links
  • Knots
  • Linking numbers
  • Links

ASJC Scopus subject areas

  • Algebra and Number Theory

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