An Intrinsic $L_{\infty}$-Algebra on the Khovanov-Sano Complex

Abstract

This paper reinterprets the symmetries of equivariant Khovanov homology, discovered by Khovanov and Sano, within the Batalin-Vilkovisky (BV) formalism. We identify the Shumakovitch operator $\hat{\nu}$ as a BV Laplacian whose nilpotency, a consequence of the algebra's defining relations, induces an $L_{\infty}$-algebra on homology. We prove this structure is non-trivial through explicit computations of higher brackets. Furthermore, we construct a dual $L_{\infty}$-structure, suggesting a unifying homotopy $\mathfrak{sl}_2$ symmetry. The main result of this paper is to lift this structure from homology to the chain level. Applying the Homotopy Transfer Theorem, we construct an intrinsic $L_{\infty}$-algebra on the Khovanov-Sano complex, whose $\infty$-quasi-isomorphism class is a canonical link invariant. This provides a new algebraic framework in which we conjecture the origin of Steenrod operations in knot homology.