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.

We withdraw the paper because the operator Δin Section 3 needs to be corrected to adjoint derivation, and the overall theoretical framework requires a more thorough check