Twisted Blanchfield pairings and decompositions of 3-manifolds

TL;DR

This paper establishes a decomposition formula for twisted Blanchfield pairings of 3-manifolds, showing how they split into components related to the original manifold and an infected knot, with implications for understanding their algebraic structures.

Contribution

It introduces a new decomposition formula for twisted Blanchfield pairings, linking the pairing of a 3-manifold infected by a knot to the pairings of the original manifold and the knot.

Findings

Twisted Blanchfield pairing of infected 3-manifolds decomposes orthogonally.

The pairing splits into the original manifold's pairing and the knot's pairing.

The knot's pairing is tensored from Z[t,t^{-1}] to R.

Abstract

We prove a decomposition formula for twisted Blanchfield pairings of 3-manifolds. As an application we show that the twisted Blanchfield pairing of a 3-manifold obtained from a 3-manifold Y with a representation $\phi: Z[\pi_1(Y)] \to R$, infected by a knot J along a curve $\eta$ with $\phi(\eta) \neq 1$, splits orthogonally as the sum of the twisted Blanchfield pairing of Y and the ordinary Blanchfield pairing of the knot J, with the latter tensored up from $Z[t,t^{-1}]$ to R.