Blanchfield pairings and twisted Blanchfield pairings of torus knots

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.

💡 Research Summary

The paper by Koki Yanagida presents explicit matrix formulas for both the classical Blanchfield pairing and certain twisted Blanchfield pairings associated to the (m,n)‑torus knot T(m,n). The central innovation is a method that avoids the traditional reliance on Seifert matrices or Wirtinger presentations, which become unwieldy for torus knots because their Seifert genus and crossing number grow like (m‑1)(n‑1)/2. Instead, the author exploits a “taut identity” that encodes a genus‑two Heegaard splitting of the 3‑manifold X_T(m,n) obtained by 0‑surgery on the knot. This identity yields a cellular chain complex for the universal cover of X_T(m,n) together with a diagonal approximation D♯, both of which involve a bounded number of generators independent of m and n.

Using this compact chain complex, the author computes the Blanchfield pairing directly from its definition. Let Δ_T(m,n)=t^{-(m‑1)(n‑1)/2}(1‑t)(1‑t^{mn})(1‑t^m)(1‑t^n) be the Alexander polynomial of the torus knot, and let Λ=ℤ