### 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 |
---|---|

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 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

**A factorization of the conway polynomial and covering linkage invariants.** / Tsukamoto, Tatsuya; Yasuhara, Akira.

Research output: Contribution to journal › Article

*Journal of Knot Theory and its Ramifications*, vol. 16, no. 5, pp. 631-640. https://doi.org/10.1142/S0218216507005403

}

TY - JOUR

T1 - A factorization of the conway polynomial and covering linkage invariants

AU - Tsukamoto, Tatsuya

AU - Yasuhara, Akira

PY - 2007/5/1

Y1 - 2007/5/1

N2 - 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.

AB - 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.

KW - Conway polynomial

KW - Covering linkage invariants

KW - Deravation on links

KW - Knots

KW - Linking numbers

KW - Links

UR - http://www.scopus.com/inward/record.url?scp=34249896444&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34249896444&partnerID=8YFLogxK

U2 - 10.1142/S0218216507005403

DO - 10.1142/S0218216507005403

M3 - Article

VL - 16

SP - 631

EP - 640

JO - Journal of Knot Theory and its Ramifications

JF - Journal of Knot Theory and its Ramifications

SN - 0218-2165

IS - 5

ER -