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 language | English |
---|---|
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 |
Externally published | Yes |
Keywords
- Conway polynomial
- Covering linkage invariants
- Deravation on links
- Knots
- Linking numbers
- Links
ASJC Scopus subject areas
- Algebra and Number Theory