Using secondary Upsilon invariants to rule out stable equivalence of knot complexes

Abstract

Two Heegaard Floer knot complexes are called stably equivalent if an acyclic complex can be added to each complex to make them filtered chain homotopy equivalent. Hom showed that if two knots are concordant, then their knot complexes are stably equivalent. Invariants of stable equivalence include the concordance invariants τ , ε, and Υ. Feller and Krcatovich gave a relationship between the Upsilon invariants of torus knots. We use secondary Upsilon invariants defined by Kim and Livingston to show that these relations do not extend to stable equivalence.

Truncated Heegaard Floer homology and knot concordance invariants

“…Proof. In [1], Allen shows that the staircase model chain complex for CFK ∞ (T p,p+1 ) takes the form…”

A further note on the concordance invariants epsilon and upsilon

“…A very simple alternative proof of the main theorem and a conjecture in [All20] will also be provided in the course of the above computation of the ϕ-invariant. (The conjecture was first proved by Xu [Xu18].)…”

On the secondary Upsilon invariant

“…To see why this is true, simply note that 1 1−t p = ∞ i=0 t pi and use Theorem 3.9. And consequently the steps of the staircase are…”

“…Later, in [1] Allen used Υ 2 to construct pairs of knots where each pair had identical Υ but were not smoothly concordant because of differing Υ 2 . More concretely, she proved that CFK ∞ (T (p, p + 2)) and CFK ∞ (T (2, p)#T (p, p + 1)) are Date: May 25, 2018. not stably equivalent.…”