Grid Movies

TL;DR

This paper introduces a grid diagram analogue of the smooth movie theorem, establishing a correspondence between grid and smooth planar isotopy classes of knots and surfaces, using a new planar grid algorithm.

Contribution

It defines grid movies and isotopies, and proves a correspondence theorem linking grid and smooth isotopy classes, extending to surfaces with boundary.

Findings

Grid planar isotopy classes correspond to smooth planar isotopy classes.

A new planar grid algorithm converts smooth knot diagrams to grid diagrams.

Generalizations of the movie theorem apply to surfaces with boundary.

Abstract

We present a grid diagram analogue of Carter, Rieger and Saito's smooth movie theorem. Specifically, we give definitions for grid movies, grid movie isotopies and present a definition of grid planar isotopy as a particular subset of the grid diagram moves: stabilization, destabilization and commutation. We show that grid planar isotopy classes are in 1-1 correspondence with smooth planar isotopy classes by using a new planar grid algorithm that takes a smooth knot diagram to a grid diagram. We then present generalizations of both the smooth and grid movie theorems that apply to surfaces with boundary.