Octagon and tropical octagon yield braid invariants

TL;DR

This paper introduces a new braid invariant in the real projective plane derived from solutions to the octagon relation, expanding the understanding of braid invariants through geometric and algebraic methods.

Contribution

It constructs a novel braid invariant in RP^2 based on octagon relations, linking geometric graph actions to algebraic braid properties.

Findings

Constructed a braid invariant in RP^2 from octagon relations.

Demonstrated that solutions to the octagon relation produce braid invariants.

Extended previous work on braid invariants and geometric relations.

Abstract

In the present paper, we construct an invariant of braids in the real projective plane which corresponds to an ``action'' of braids on certain graphs in $\R{}P^{2}$ with labels. This paper is a sequel of papers \cite{M},\cite{KM}. It demonstrates once more that solutions to the octagon relation in various forms give rise to invariants of braids.