A$_\infty$ deformations of extended Khovanov arc algebras and Stroppel's conjecture

Extended Khovanov arc algebras $\mathrm{K}_m^n$ are graded associative algebras which naturally appear in a variety of contexts, from knot and link homology, low-dimensional topology and topological quantum field theory to representation theory and symplectic geometry. C. Stroppel conjectured in her ICM 2010 address that the bigraded Hochschild cohomology groups of $\mathrm{K}_m^n$ vanish in a certain range, implying that the algebras $\mathrm K_m^n$ admit no nontrivial A$_\infty$ deformations, in particular that the algebras are intrinsically formal. Whereas Stroppel's Conjecture is known to hold for the algebras $\mathrm K_m^1$ and $\mathrm K_1^n$ by work of Seidel and Thomas, we show that $\mathrm K_m^n$ does in fact admit nontrivial A$_\infty$ deformations with nonvanishing higher products for all $m, n \geq 2$. We describe both $\mathrm K_m^n$ and its Koszul dual concretely as path algebras of quivers with relations and give an explicit algebraic construction of A$_\infty$ deformations of $\mathrm K_m^n$ by using the correspondence between A$_\infty$ deformations of a Koszul algebra and filtered associative deformations of its Koszul dual. These deformations can also be viewed as A$_\infty$ deformations of Fukaya--Seidel categories associated to Hilbert schemes of surfaces based on recent work of Mak and Smith.