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

TL;DR

This paper demonstrates that the gauge-theoretic functor $J^ atural$ and the proposed combinatorial functor $J^lat$ differ on planar webs, providing a counterexample to their equivalence.

Contribution

It provides a counterexample showing that the gauge-theoretic and combinatorial foam evaluation functors are not equivalent on planar webs.

Findings

Counterexample showing $J^ atural$ and $J^lat$ differ on planar webs.

$J^lat$ is well-defined on a subcategory of planar webs.

$J^ atural$ and $J^lat$ are not the same functor on this subcategory.

Abstract

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 $J^\flat$.