On the $A_\infty$-extension of Bar-Natan multicurves

Abstract

Kotelskiy, Watson and Zibrowius show that the Bar-Natan invariant $\unicode{x0414}(T)$ associated to a $4$-ended oriented tangle $T$ extends to a twisted complex $\unicode{x0414}^\infty(T)$ over an $A_\infty$-enhancement of the Bar-Natan algebra. We prove that $\unicode{x0414}^\infty(T)$ itself is a tangle invariant, settling a conjecture from their work. We deduce this from an invariance statement for a \emph{matrix multifactorization} $\mathcal M(T)$, generalizing a construction of Ballinger to the coefficient ring $\mathbb Z[G]$. This multifactorization is well defined up to a notion of 1-homotopy equivalence. The proof rests on explicit formulas for special deformation retracts of Koszul matrix factorizations, which realize delooping in this setting and may be of independent interest.