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

## TL;DR

This paper demonstrates that secondary Upsilon invariants can distinguish knot complexes that are not stably equivalent, extending the understanding of invariants beyond known concordance invariants.

## Contribution

It introduces the use of secondary Upsilon invariants to detect differences in stable equivalence of knot complexes, showing they can distinguish complexes beyond traditional invariants.

## Key findings

- Secondary Upsilon invariants can rule out stable equivalence.
- Relations between Upsilon invariants of torus knots do not extend to stable equivalence.
- Secondary invariants provide new tools for knot complex classification.

## 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 $\tau$, $\varepsilon$, and $\Upsilon$. 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.