The knot invariant Υ using grid homologies

Abstract

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