Unresolved cases for geometric realizations of P(p,−q,−r) when min{q,r}=p+1 and q≠r

Determine explicit geometric realizations of the real-extreme Khovanov homology of the pretzel links P(p,−q,−r) in the parameter regime min{q,r}=p+1 and q≠r by constructing suitable link diagrams whose Lando graph independence complexes are non-contractible and identifying their homotopy types.

Background

The authors’ method constructs link diagrams for pretzel links P(p,−q,−r) whose Lando graphs yield independence complexes with known homotopy types, enabling explicit geometric realizations of real-extreme Khovanov homology. However, when both vertices z and z′ in the relevant subgraph have attached small trees and the vertex x is present, the subgraph G* contains an independent vertex, making its independence complex contractible and causing the method to fail.

This obstruction arises precisely when min{q,r}=p+1 and q≠r, leaving these cases unresolved by the current techniques. The open task is to develop an approach (possibly via alternative diagrams or graph manipulations) that yields non-contractible independence complexes for these parameter values.