Q-anisotropy of positive knots

Establish that every positive knot is Q-anisotropic; specifically, prove that for each positive knot K, the rational cohomology H^1(\bar{X}; Q) of the infinite cyclic cover \bar{X} of the knot exterior contains no nontrivial invariant Q-isotropic subspace under the deck transformation action.

Background

Q-anisotropy is an algebraic condition on the Alexander module that constrains behavior under concordance. By classical work of Kervaire and Gilmer, if concordant knots are Q-anisotropic and admit non-singular Seifert matrices, then their rational Alexander modules are isomorphic.

In this paper, proving Q-anisotropy for a large subclass of positive knots yields strong concordance consequences. Establishing Q-anisotropy for all positive knots would remove technical hypotheses from theorems in the paper and imply algebraic ribbon concordance minimality for positive knots (as observed by Gilmer). Scharlemann proved this property for torus knots.