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

Akira Yasuhara

KNOTS in WASHINGTON XV (2nd Japan-USA Workshop in Knot Theory)
January 10-15, 2003
George Washington University and Johns Hopkins University
Washington, DC and Baltimore, MD, USA

Organizers
Kazuaki Kobayashi, Jozef H. Przytycki, Yongwu Rong, Shin-ichi Suzuki, Kouki Taniyama, Tatsuya Tsukamoto, Akira Yasuhara

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Linking numbers in rational homology 3-spheres, cyclic branched covers and infinite cyclic covers
by
Akira Yasuhara
Tokyo Gakugei University
Coauthors: Jozef H. Przytycki (The George Washington University)

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.

Date received: December 18, 2002