Blanchfield pairings and twisted Blanchfield pairings of torus knots
Abstract
We give explicit matrix presentations of the Blanchfield pairing and certain twisted Blanchfield pairings of the (m,n)-torus knot T(m,n). Our method uses a taut identity realizing a genus-two Heegaard splitting of the manifold XT(m,n) obtained from S3 by 0-surgery along T(m,n). The taut identity allows us to construct a chain complex of XT(m,n) with few generators. As a result, we obtain explicit matrix presentations of the Blanchfield pairing of T(m,n). Moreover, for each Casson-Gordon type metabelian representation and for suitable roots of unity depending on the representation, we describe the (t-)-primary part of the associated twisted Alexander module and give an explicit description of the restriction of the twisted Blanchfield pairing to this primary summand.