Ribbon distance bounds from Bar-Natan Homology and $\alpha$-Homology

TL;DR

This paper establishes new lower bounds on ribbon distance using Bar-Natan and alpha-homology, aligning with previous bounds on unknotting number and revealing a consistent pattern across different homology theories.

Contribution

It introduces novel lower bounds on ribbon distance derived from Bar-Natan and alpha-homology, extending known relationships between homology theories and knot invariants.

Findings

Lower bound on ribbon distance via Bar-Natan Homology.

Lower bound on ribbon distance via alpha-homology.

Pattern consistency between ribbon distance and unknotting number bounds.

Abstract

We prove a lower bound on the ribbon distance via Bar-Natan Homology. This lower bound agrees with Alishahi's lower bound on the unknotting number via Bar-Natan Homology, which furthers a pattern first observed by Sarkar: Sarkar's lower bound on the ribbon distance via Lee Homology agreed with Alishahi and Dowlin's lower bound on the unknotting number via Lee Homology. We also prove a lower bound on the ribbon distance via $\alpha$-homology, a variant of Khovanov homology defined recently by Khovanov and Robert.