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Blanchfield pairings and twisted Blanchfield pairings of torus knots

Koki Yanagida

TL;DR

This work provides explicit, computable matrix presentations for the Blanchfield pairing of the $(m,n)$-torus knot and its metabelian twisted variants. By exploiting a taut identity realizing a genus-2 Heegaard splitting of $X_{T(m,n)}$ (the $0$-surgery manifold), the authors construct a compact chain complex that yields closed formulas: the classical Blanchfield pairing is given by a form on $\mathbb{Z}[t^{\pm1}]/(\Delta_{T(m,n)})$ with matrix $t^{mn} B(m,n)$, where $\Delta_{T(m,n)}$ is the Alexander polynomial factor and $B(m,n)$ is an explicit correction term. For Casson–Gordon type metabelian representations, partial results describe the $(t-\xi)$-primary parts of twisted Alexander modules via $\Theta_{\boldsymbol{b}}(a)$ and the twisted Blanchfield form via $\Psi_{\boldsymbol{b}}(a)$ and $\delta_m(t)$, providing explicit, tractable matrices. The methods avoid Seifert matrices or Wirtinger presentations and scale with torus knot complexity, enabling concrete computations for knots with large genus or crossing number and offering new tools for concordance and four-dimensional knot properties.

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 $ξ$ depending on the representation, we describe the $(t-ξ)$-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.

Blanchfield pairings and twisted Blanchfield pairings of torus knots

TL;DR

This work provides explicit, computable matrix presentations for the Blanchfield pairing of the -torus knot and its metabelian twisted variants. By exploiting a taut identity realizing a genus-2 Heegaard splitting of (the -surgery manifold), the authors construct a compact chain complex that yields closed formulas: the classical Blanchfield pairing is given by a form on with matrix , where is the Alexander polynomial factor and is an explicit correction term. For Casson–Gordon type metabelian representations, partial results describe the -primary parts of twisted Alexander modules via and the twisted Blanchfield form via and , providing explicit, tractable matrices. The methods avoid Seifert matrices or Wirtinger presentations and scale with torus knot complexity, enabling concrete computations for knots with large genus or crossing number and offering new tools for concordance and four-dimensional knot properties.

Abstract

We give explicit matrix presentations of the Blanchfield pairing and certain twisted Blanchfield pairings of the -torus knot . Our method uses a taut identity realizing a genus-two Heegaard splitting of the manifold obtained from by -surgery along . The taut identity allows us to construct a chain complex of with few generators. As a result, we obtain explicit matrix presentations of the Blanchfield pairing of . Moreover, for each Casson-Gordon type metabelian representation and for suitable roots of unity depending on the representation, we describe the -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.
Paper Structure (13 sections, 18 theorems, 89 equations, 1 figure)

This paper contains 13 sections, 18 theorems, 89 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Twisted homology and cohomology groups
  4. Review of Blanchfield pairings
  5. Review of twisted Blanchfield pairings
  6. Recovering chain complexes of $X_K$ from taut identities
  7. Taut identities
  8. A taut identity corresponding to $X_{T(m,n)}$
  9. Proof of Theorem \ref{['thm:preBl']}
  10. Proof of Theorem \ref{['thm:pretAl']} and \ref{['thm:SptBL']}
  11. Preparations for the proofs
  12. Proof of Theorem \ref{['thm:pretAl']}
  13. Proof of Theorem \ref{['thm:SptBL']}

Key Result

Theorem 1.1

Let $\Delta_{T(m,n)} \coloneqq t^{-(m-1)(n-1)/2}\, \frac{(1-t)(1-t^{mn})}{(1-t^{m})(1-t^{n})}$ and Then the Blanchfield pairing associated with $T(m,n)$ is isometric to the following sesquilinear form:

Figures (1)

  • Figure 1: A net of the polyhedron corresponding to the taut identity \ref{['eq:tiden']}. $*$ is the basepoint of the polyhedron, and $\star$ denotes the basepoint on each face. The two faces on the left are oriented clockwise, while the two faces on the right are oriented counterclockwise. From left to right, the faces correspond to $\gamma_1^{-1}, \gamma_1, \gamma_2^{-1}$, and $\gamma_{2}$. For $i \in \{1,2\}$, in the polyhedron represented by this diagram, we obtain a CW–complex structure on $X_{T(m,n)}$ by identifying the face corresponding to $\gamma_i$ with the face corresponding to $\gamma_i^{-1}$ so that the basepoints $\star$ and the labels $x$ and $y$ match

Theorems & Definitions (42)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 3.1
  • Theorem 3.2
  • proof
  • Remark 3.3
  • ...and 32 more