Przytycki–Silvero wedge-of-spheres conjecture for circle graphs

Prove that for every circle graph, the independence simplicial complex is homotopy equivalent to a wedge of spheres; in particular, show that for every link diagram the extreme Khovanov homology is torsion-free.

Background

The paper reviews the result of González-Meneses, Manchón, and Silvero that the extreme Khovanov complex of a link diagram can be expressed via the independence complex of its Lando graph. Building on this connection, Przytycki and Silvero formulated a conjecture predicting a uniform topological structure for independence complexes associated with circle graphs.

If the independence complex of every circle graph is homotopy equivalent to a wedge of spheres, then the corresponding extreme Khovanov homology groups computed from these complexes are torsion-free. The authors’ constructions for pretzel links provide additional evidence supporting this conjecture.