2001/11/19 by Jozef H. Przytycki, Przytycki, Jozef H., Akira Yasuhara +1
Mathematics · #57M 12 (Secondary) #57M25 (Primary) 57M10 #FOS: Mathematics #Geometric Topology (math.GT) #math.GT #msc:12 #msc:57M #msc:57M10 #msc:57M25
paper · pdf · doi:10.48550/arxiv.math/0111203
LaTeX, 24 pages, 6 figures
arxiv created 2001/11/19 · arxiv updated 2009/11/30
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 \Bbb Q and in Q(\Bbb Z[t,t-1]) respectively, where Q(\Bbb Z[t,t-1]) denotes the quotient field of \Bbb Z[t,t-1]. It is known that the modulo-\Bbb Z linking number in the rational homology 3-sphere is determined by the linking matrix of the framed link and that the modulo-\Bbb 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 \Bbb Z' and ` modulo \Bbb 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.