### 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 S^{3}. Therefore, the first non-vanishing coefficient of the Conway polynomial is determined by the linking numbers in S^{3}. This generalizes a result of Hoste.

Original language | English |
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Pages (from-to) | 631-640 |

Number of pages | 10 |

Journal | Journal of Knot Theory and its Ramifications |

Volume | 16 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2007 May 1 |

Externally published | Yes |

### Keywords

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

### ASJC Scopus subject areas

- Algebra and Number Theory