Kronheimer and Mrowka used gauge theory to define a functor \(J^\sharp \) from a category of webs in \(\mathbb {R}^3\) to the category of finite-dimensional vector spaces over the field of two elements. They also suggested a possible combinatorial replacement \(J^\flat \) for \(J^\sharp \) , which Khovanov and Robert proved is well defined on a subcategory of planar webs. We exhibit a counterexample that shows the restriction of the functor \(J^\sharp \) to the subcategory of planar webs is not the same as \(J^\flat \) .
