On Farber's invariants for simple $2q$-knots

TL;DR

This paper demonstrates how Farber's invariants determine the homotopy type of the exterior of simple 2q-knots under certain conditions, and explores their algebraic and duality properties.

Contribution

It provides a direct method to recover the homotopy type of the knot exterior from Farber's quintuple and reformulates these invariants as hermitian self-dual objects.

Findings

Farber quintuple determines the homotopy type when torsion subgroup has odd order.

Connection between duality pairings and boundary inclusion via EHP sequence.

Reformulation of Farber invariants as hermitian self-duality in an additive category.

Abstract

Let $K$ be a simple $2q$-knot with exterior $X$. We show directly how the Farber quintuple $(A,\Pi,\alpha,\ell,\psi)$ determines the homotopy type of $X$ if the torsion subgroup of $A=\pi_q(X)$ has odd order. We comment briefly on the possible role of the EHP sequence in recovering the boundary inclusion from the duality pairings $\ell $ and $\psi$. Finally we reformulate the Farber quintuple as an hermitian self-duality of an object in an additive category with involution.