The Knot Invariant ${\rm Y}$ Using Grid Homologies

Abstract

According to the idea of Ozsváth, Stipsicz and Szabó, we define the knot invariant ${\rm Y}$ without the holomorphic theory, using constructions from grid homology. We develop a homology theory using grid diagrams, and show that ${\rm Y}$, as introduced this way, is a well-defined knot invariant. We reprove some important propositions using the new techniques, and show that ${\rm Y}$ provides a lower bound on the unknotting number.