On grid homology for diagonal knots

TL;DR

This paper advances the understanding of grid homology for diagonal knots, revealing how certain homological terms relate to prime factorization and tangle decompositions, and compares these knots to other well-studied classes.

Contribution

It provides partial computations of grid homology for diagonal knots and links homological features to knot decompositions and prime factors, expanding the combinatorial understanding of these knots.

Findings

Next-to-top homology term detects prime factors

Top minus two term relates to tangle decompositions

Diagonal knots share properties with positive braid and L-space 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.