# Connected sums and involutive knot Floer homology

## TL;DR

This paper derives a formula for the conjugation action on the knot Floer complex of connected sums, enabling new computations of involutive invariants and linking knot concordance to chain complex automorphisms.

## Contribution

It introduces a formula for the conjugation action on the knot Floer complex of connected sums and constructs a homomorphism from the smooth concordance group to a chain complex automorphism group.

## Key findings

- Derived a formula for conjugation action on connected sum knot Floer complexes
- Constructed a homomorphism from the concordance group to chain complex automorphisms
- Performed example computations of involutive invariants on large surgeries of connected sums

## Abstract

We prove a formula for the conjugation action on the knot Floer complex of the connected sum of two knots. Using the formula we construct a homomorphism from the smooth concordance group to an abelian group consisting of chain complexes with homotopy automorphisms, modulo an equivalence relation. Using our connected sum formula, we perform some example computations of Hendricks and Manolescu's involutive invariants on large surgeries of connected sums of knots.