# The Knot Invariant $\Upsilon$ Using Grid Homologies

## TL;DR

This paper introduces a combinatorial approach to defining the knot invariant d using grid homology, providing an accessible alternative to holomorphic methods and establishing its properties and applications.

## Contribution

It develops a new grid homology-based construction of d, proving its invariance and utility without relying on holomorphic theory.

## Key findings

- d is a well-defined knot invariant using grid diagrams
- d provides a lower bound on the unknotting number
- Reproves key properties of d with new techniques

## 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.