Homological lifts of Arnold strangeness

Abstract

We construct homological lifts of Arnold's strangeness $\operatorname{St}$ for long and closed plane curves. For long curves, we introduce a new $q$-variable quantization polynomial $\operatorname{St}_q$, an edge state sum, designed for categorification and construct a triply graded homology that recovers $\operatorname{St}_q$. We prove strictness by exhibiting long curves with the same number of crossings and the same $\operatorname{St}_q$ (in particular, the same rotation number and $\operatorname{St}$) that are distinguished by the homology. For closed curves, we introduce a corresponding new polynomial $\operatorname{St}_q$ and construct an edge homology. In this case $\operatorname{St}_q$ is recovered from a signed $q$-character $\operatorname{ch}_q$ defined on the canonical cycles of the homology rather than directly from a graded rank. Again, we exhibit closed curves with the same number of crossings and the same $\operatorname{St}_q$ (in particular, the same rotation number and $\operatorname{St}$) that are distinguished by the homology.