### 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 language | English |
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Pages (from-to) | 3669-3685 |

Number of pages | 17 |

Journal | Transactions of the American Mathematical Society |

Volume | 356 |

Issue number | 9 |

DOIs | |

Publication status | Published - 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