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

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