The flip symmetry on Khovanov-Rozansky homology

The flip symmetry on link diagrams induces an involution on Khovanov-Rozansky ${\mathfrak{g}\mathfrak{l}}_N$ homology. We prove that this involution is diagonalizable with eigenvalues $\pm 1$. On the one hand, it is the identity over ${\mathbb{F}}_2$, generalizing a previous result of Chen and the author. On the other hand, it is expected to be nontrivial over $\mathbb{Z}$ in general. The key ingredients of the proof are (1) a homotopy perturbation argument via a detailed study of the fork twist, which allows us to reduce the computation to planar ${\mathfrak{g}\mathfrak{l}}_N$ webs, and (2) the computation of the flip map for planar ${\mathfrak{g}\mathfrak{l}}_N$ webs via diagrammatics of Soergel bimodules. The latter computation can also be interpreted as a naturality result for the half twist action on type $A$ Soergel bimodules, which might be of independent interest.