The combinatorial and gauge-theoretic foam evaluation functors are not the same

Abstract

Kronheimer and Mrowka used gauge theory to define a functor $J^{♯}$ 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 $J^{♭}$ for $J^{♯}$, 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 $J^{♯}$ to the subcategory of planar webs is not the same as $J^{♭}$.

6 pages, 2 figures