English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Goldman and Turaev constructed a Lie bialgebra structure on the free$\mathbb{Z}$-module generated by free homotopy classes of loops on a surface.Turaev conjectured that his cobracket $\Delta(\alpha)$ is zero if and only if$\alpha$ is a power of a simple class. Chas constructed examples that showTuraev's conjecture is, unfortunately, false. We define an operation $\mu$ inthe spirit of the Andersen-Mattes-Reshetikhin algebra of chord diagrams. TheTuraev cobracket factors through $\mu$, so we can view $\mu$ as ageneralization of $\Delta$. We show that Turaev's conjecture holds when$\Delta$ is replaced with $\mu$. We also show that $\mu(\alpha)$ gives anexplicit formula for the minimum number of self-intersection points of a loopin $\alpha$. The operation $\mu$ also satisfies identities similar to theco-Jacobi and coskew symmetry identities, so while $\mu$ is not a cobracket,$\mu$ behaves like a Lie cobracket for the Andersen-Mattes-Reshetikhin Poissonalgebra.

A Generalization of the Turaev Cobracket and the Minimal Self-Intersection Number of a Curve on a Surface