Topology of positive polytopes in the Grassmannian

Determine whether every positive polytope in \(\mathrm{Gr}(2,4)\) is connected and contractible.

Background

Positive polytopes in Gr(2,4)\mathrm{Gr}(2,4) are intersections of the positive Grassmannian with polytopal half-spaces. The examples studied in Section 5 are connected and contractible, but the paper does not establish these properties in general. The open question asks whether this observed topology holds for all positive pentahedra and hexahedra, or more generally for all positive polytopes in the paper’s class.

References

What can one say about the topology of positive polytopes in $\mathrm{Gr}(2,4)$? Are they connected? Are they contractible? This is the case for the examples in Section \ref{sec:5} but does this hold in general?

— Positive Polytopes with Few Facets in the Grassmannian  (2503.01652 - Pavlov et al., 3 Mar 2025) in Question environment, Section 6, “Open questions and future directions”