Rank Bounds in Link Floer Homology and Detection Results

Abstract: Viewing the BRAID invariant as a generator of link Floer homology we generalise work of Baldwin-Vela-Vick to obtain rank bounds on the next to top grading of knot Floer homology. These allow us to classify links with knot Floer homology of rank at most eight, and prove a variant of a classification of links with Khovanov homology of low rank due to Xie-Zhang. In another direction we use a variant of Ozsv\'ath-Szab\'o classification of $E_2$ collapsed $\mathbb{Z} \oplus\mathbb{Z}$ filtered chain complexes to show that knot Floer homology detects $T(2,8)$ and $T(2,10)$. Combining these techniques with the spectral sequences of Batson-Seed, Dowlin, and Lee we can show that Khovanov homology likewise detects $T(2,8)$ and $T(2,10)$.