Linking numbers in rational homology 3-spheres, cyclic branched covers and infinite cyclic covers

Józef H. Przytycki, Akira Yasuhara

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4 Citations (Scopus)

Abstract

We study the linking numbers in a rational homology 3-sphere and in the infinite cyclic cover of the complement of a knot. They take values in ℚ and in Q(ℤ[t,t -1]), respectively, where Q(ℤ[t,t -1]) denotes the quotient field of Z[t, t -1]. It is known that the modulo-Z linking number in the rational homology 3-sphere is determined by the linking matrix of the framed link and that the modulo-Z[t, t -1] linking number in the infinite cyclic cover of the complement of a knot is determined by the Seifert matrix of the knot. We eliminate 'modulo Z' and 'modulo Z[t, t -1]'. When the finite cyclic cover of the 3-sphere branched over a knot is a rational homology 3-sphere, the linking number of a pair in the preimage of a link in the 3-sphere is determined by the Goeritz/Seifert matrix of the knot.

Original languageEnglish
Pages (from-to)3669-3685
Number of pages17
JournalTransactions of the American Mathematical Society
Volume356
Issue number9
DOIs
Publication statusPublished - 2004 Sep 1

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Keywords

  • Covering space
  • Framed link
  • Goeritz matrix
  • Linking matrix
  • Linking number
  • Rational homology 3-sphere
  • Seifert matrix

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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