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 1 |
| Externally published | Yes |
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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.
In: Journal of Knot Theory and its Ramifications, Vol. 16, No. 5, 01.05.2007, p. 631-640.Research output: Contribution to journal › Article
}
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 -