The Knot Invariant $\Upsilon$ Using Grid Homologies

Abstract

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