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

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 |

Externally published | Yes |

### Fingerprint

### 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

### Cite this

**Linking numbers in rational homology 3-spheres, cyclic branched covers and infinite cyclic covers.** / Przytycki, Józef H.; Yasuhara, Akira.

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 356, no. 9, pp. 3669-3685. https://doi.org/10.1090/S0002-9947-04-03423-3

}

TY - JOUR

T1 - Linking numbers in rational homology 3-spheres, cyclic branched covers and infinite cyclic covers

AU - Przytycki, Józef H.

AU - Yasuhara, Akira

PY - 2004/9/1

Y1 - 2004/9/1

N2 - 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.

AB - 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.

KW - Covering space

KW - Framed link

KW - Goeritz matrix

KW - Linking matrix

KW - Linking number

KW - Rational homology 3-sphere

KW - Seifert matrix

UR - http://www.scopus.com/inward/record.url?scp=4344576566&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=4344576566&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-04-03423-3

DO - 10.1090/S0002-9947-04-03423-3

M3 - Article

VL - 356

SP - 3669

EP - 3685

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 9

ER -