# Colored knot Floer homology: structures and examples

## TL;DR

This paper introduces a new colored version of knot Floer homology by constructing it as a colimit over infinite full twists, providing explicit descriptions for L-space knots and crossing change maps, inspired by colored Khovanov homology.

## Contribution

It defines a colored knot Floer homology via colimits of link Floer homology with infinite twists, extending the structure and understanding of knot invariants.

## Key findings

- Colimit of infinite full twists forms a module over the unknot's colored knot Floer homology.
- Explicit description of colored Heegaard Floer homology for L-space knots.
- Construction of maps for crossing changes in colored knot Floer homology.

## Abstract

Inspired by the $S^n$ colored version of Khovanov and Khovanov-Rozansky homology, we define a colored version of knot Floer homology by studying the colimit of a directed system of link Floer homology with infinite full twists. Specifically, our $n$-colored knot Floer homology of a knot $K$ is then defined as the colimit of the link Floer homology of $(n, mn)$-cables of $K$ by fixing $n$ and letting $m$ goes to infinity. We show that the colimit of the infinite full twists is a module over the colored knot Floer homology of the unknot. In addition, we give an explicit description of colored Heegaard Floer homology for L-space knots, and maps for colored knot Floer homology of crossing changes.