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

Józef H. Przytycki, Akira Yasuhara

Research output: Contribution to journalArticle

4 Citations (Scopus)

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 languageEnglish
Pages (from-to)3669-3685
Number of pages17
JournalTransactions of the American Mathematical Society
Volume356
Issue number9
DOIs
Publication statusPublished - 2004 Sep 1
Externally publishedYes

Fingerprint

Branched Cover
Linking number
Knot
Homology
Cover
Modulo
Complement
Linking
Quotient
Eliminate
Denote

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.

In: Transactions of the American Mathematical Society, Vol. 356, No. 9, 01.09.2004, p. 3669-3685.

Research output: Contribution to journalArticle

@article{7e90171aa19142f8bf8f5715766a2f86,
title = "Linking numbers in rational homology 3-spheres, cyclic branched covers and infinite cyclic covers",
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.",
keywords = "Covering space, Framed link, Goeritz matrix, Linking matrix, Linking number, Rational homology 3-sphere, Seifert matrix",
author = "Przytycki, {J{\'o}zef H.} and Akira Yasuhara",
year = "2004",
month = "9",
day = "1",
doi = "10.1090/S0002-9947-04-03423-3",
language = "English",
volume = "356",
pages = "3669--3685",
journal = "Transactions of the American Mathematical Society",
issn = "0002-9947",
publisher = "American Mathematical Society",
number = "9",

}

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

AN - SCOPUS:4344576566

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 -