V.D234 — T² QNM Mode Structure

The T²=(R·S¹)×(r·S¹) torus with r/R=ι_τ carries Laplacian eigenmodes labeled by (n,m)∈ℤ² with eigenvalue λ_{n,m}=n²/R²+m²/r²=n²+(m·ι_τ⁻¹)²/R². The three primitive modes (1,0), (0,1), (1,1) give frequency ratio f(0,1)/f(1,0)=ι_τ⁻¹≈2.9299. T² spectrum is anisotropic vs S² l(l+1); first 15 distinct eigenvalues tabulated in the Lab (see scripts/bh_topology_lab.py).