Twisted Blanchfield Pairings and Symmetric Chain Complexes

Abstract

We define the twisted Blanchfield pairing of a symmetric triad of chain complexes over a group ring Z[π], together with a unitary representation of π over an Ore domain with involution. We prove that the pairing is sesquilinear, and we prove that it is hermitian and nonsingular under certain extra conditions. A twisted Blanchfield pairing is then associated to a 3-manifold together with a decomposition of its boundary into two pieces and a unitary representation of its fundamental group.T H 1 (N ; Z) × T H 1 (N ; Z) → Q/Z.

Two-solvable and two-bipolar knots with large four-genera

“…Note that in the above circumstance H φ 1 pM K q is a torsion Qpξ m qrt˘1s-module, by the corollary to [CG86, Lemma 4]; see also [FP12]. In Section 6.1, we will need to know that this form is sesquilinear [Pow16]. That is, letting s denote the involution of Qpξ m qrt˘1s induced by sending t Ñ t´1 and a`bi Þ Ñ a´bi, we have Bl φ ppx, qyq " ps q Bl φ px, yq for every p, q P Qpξ m qrt˘1s and x, y P H φ 1 pM K q.…”

The $\mathbb{Z}$-genus of boundary links

“…Let M be a closed, oriented, connected 3-manifold equipped with a homomorphism π 1 (M ) → Z, giving rise to twisted homology and cohomology with coefficients in the Λ-modules Λ, Q(t), and Q(t)/Λ. The Blanchfield form [Bla57] Bl M is the nonsingular, sesquilinear, Hermitian form [Pow16] defined on the torsion submodule T H 1 (M ; Λ) of H 1 (M ; Λ).…”

The Blanchfield pairing of colored links

“…As the matrix H(t) is hermitian, Theorem 1.2 provides a proof that the Blanchfield pairing Bl(L) is hermitian. To the best of our knowledge, in the case of links, the only other proof of this fact was recently given by Powell [17]. Moreover, we know of no computation of the Blanchfield form for links which are not (homology) boundary links [5,10].…”