A family of ${\mathfrak{sl}}_n$-like invariants in knot Floer homology

Abstract

We define and study a family of link invariants ${HFK}_n(L)$. Although these homology theories are defined using holomorphic disc counts, they share many properties with $s{l}_n$ homology. Using these theories, we give a framework that generalizes the conjectured spectral sequence from Khovanov homology to $\delta$-graded knot Floer homology. In particular, we conjecture that for all links $L$ in $S^3$ and all $n\ge 1$, there is a spectral sequence from the $s{l}_n$ homology of $L$ to ${HFK}_n(L)$.

26 pages