Concordance maps in $HFK^{-}$

TL;DR

This paper constructs a new type of knot concordance map in knot Floer homology that preserves key gradings, extending previous work and utilizing sutured Floer homology techniques.

Contribution

It introduces a generalized concordance map on $HFK^{-}$ that preserves gradings, building on and extending prior $ ext{hat}$-flavored concordance maps.

Findings

Defines a $ ext{F}[U]$-module homomorphism induced by knot concordance.

Preserves Alexander and $ ext{Z}_2$-Maslov gradings.

Generalizes previous $ ext{hat}$-flavored concordance maps.

Abstract

We show that a decorated knot concordance $\mathcal{C}$ from $K_0$ to $K_1$ induces an $\mathbb{F}[U]$-module homomorphism \[G_{\mathcal{C}}: HFK^{-}(-S^3,K_0) \to HFK^{-}(-S^3,K_1)\] which preserves the Alexander and absolute $\mathbb{Z}_2$-Maslov gradings. Our construction generalizes the concordance maps induced on $\widehat{HFK}$ studied by Juh\'asz and Marengon, but uses the description of $HFK^{-}$ as a direct limit of maps between sutured Floer homology groups discovered by Etnyre, Vela-Vick, and Zarev.