On grid homology for diagonal knots

Abstract

We partially determine grid homology (combinatorial knot Floer homology) of diagonal knots, which are conjectured to be equivalent to positive braid knots, by exploiting nice grid diagrams. Its next-to-top term detects the number of prime factors, and the $\text{top}-2$ term corresponds to the decompositions of the knot into two non-integer tangles. We compare diagonal knots to various classes of knots, such as positive braids, fibered positive knots, and $L$-space knots.