Blanchfield pairings and twisted Blanchfield pairings of torus knots

TL;DR

This paper provides explicit matrix formulas for the Blanchfield and twisted Blanchfield pairings of torus knots using a genus-two Heegaard splitting, enabling detailed analysis of their algebraic invariants.

Contribution

It introduces a new method to explicitly compute Blanchfield pairings of torus knots via a genus-two Heegaard splitting and chain complexes with minimal generators.

Findings

Explicit matrix presentations of Blanchfield pairings for torus knots.

Description of twisted Blanchfield pairings for Casson-Gordon type representations.

Analysis of the primary parts of twisted Alexander modules at roots of unity.

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 $X_{T(m,n)}$ obtained from $S^3$ by $0$-surgery along $T(m,n)$. The taut identity allows us to construct a chain complex of $X_{T(m,n)}$ 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 $\xi$ depending on the representation, we describe the $(t-\xi)$-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.