Distribution of periodic orbits in the homology group of a knot complement

The paper proves precise window asymptotics for linking numbers and full homology in a knot complement for transitive Anosov flows via a symbolic suspension, a locally constant Z^{b+N}-valued cocycle, Legendre duality for pressure, and a direct application of Babillot–Ledrappier’s constrained-orbit theorem. The candidate gives a parallel route: build Hölder cocycles by integrating closed 1-forms (ambient homology and meridians), invoke Ruelle–Parry–Pollicott spectral perturbation, and obtain the same T^{−1−d/2}e^{h(ρ)T} law. The main asymptotic shape, analyticity/positivity of the entropy function, and the reductions agree with the paper. Two caveats for the candidate: (i) they do not explicitly verify the key group hypotheses (AF=Z^d and ÃF=R×AF) that the paper establishes; and (ii) they claim the prefactor c_{ρ,·}(T) can be made constant under the non-arithmetic (not locally constant) roof, whereas the paper retains a bounded positive T-dependent factor (arising, e.g., from the e^{⟨u(ρ), Tρ−⌊Tρ⌋⟩} term). These are minor fixable points; the core argument and conclusions match the paper’s results.