We show that the reduced Hochschild homology of a DG open Frobenius algebra has the natural structure of a Batalin-Vilkovisky coalgebra, and the reduced cyclic homology has the natural structure of a gravity coalgebra. This gives an algebraic model for a Batalin-Vilkovisky coalgebra structure on the reduced homology of the free loop space of a simply connected closed oriented manifold, and a gr...

Hess and Shipley defined an invariant of coalgebra spectra called topological coHochschild homology, Bohmann–Gerhardt–Høgenhaven–Shipley–Ziegenhagen developed a coBökstedt spectral sequence to compute the homology $$\mathrm {coTHH}$$ for coalgebras over sphere spectrum. We construct relative study any commutative ring spectrum R. Further, we use algebraic structures in this complete some calcul...

We prove that Khovanov homology detects the trefoils. Our proof incorporates an array of ideas in Floer and contact geometry. It uses open books; invariants we defined instanton setting; a bypass exact triangle sutured homology, proven here; Kronheimer Mrowka's spectral sequence relating with singular knot homology. As byproduct, also strengthen result Mrowka on $SU(2)$ representations group.