First values of the equivariant invariants τ₀ and τ₁ of strongly invertible knots

Abstract

Framba has recently defined an equivariant grid homology for strongly invertible knots and, from it, two integer invariants τ₀ and τ₁ satisfying τ₀(K) ≤ τ(K) ≤ τ₁(K). That paper computes the two invariants for no knot, and asks for a strongly invertible knot at which an inequality is strict, and for two strong inversions on one knot that τ₀ or τ₁ tells apart. We compute the invariants. For the figure-eight knot with the strong inversion carried by an explicit 6 × 6 symmetric grid, τ₀=-1<0=τ=τ₁, so the first inequality is strict; the same three values come out of a 7 × 7 grid for the same knot, which exhibits the invariance rather than assuming it. We also exhibit two 7 × 7 symmetric grid diagrams that agree on the fully blocked grid homology in every bigrading, on the simply blocked homology — which is widehatmathit(HFK)(5₂) — on mathit(GH)⁻, and on the dimensions of the equivariant homology, and whose τ₀ are 1 and 0. They therefore carry inequivalent strongly invertible knots, and if their underlying knots agree, as the computed widehatmathit(HFK) and their common grid number indicate, then τ₀ separates two strong inversions on 5₂. Every homology computed here is the 𝔽₂-homology of a finite bigraded block, obtained by exhaustive enumeration of the n! grid states, and every numerical statement below is machine-checked in Lean 4.

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