GridPyM: a Python module to handle grid diagrams

TL;DR

Motivated by questions from (bio)-physical knot theory, GridPyM, a Sage compatible Python module that handles grid diagrams is introduced, which focuses on generating and simplifying grids, and on modelling local transformations between them.

Abstract

Grid diagrams are a combinatorial version of classical link diagrams, widely used in theoretical, computational and applied knot theory. Motivated by questions from (bio)-physical knot theory, we introduce GridPyM, a Sage compatible Python module that handles grid diagrams. GridPyM focuses on generating and simplifying grids, and on modelling local transformations between them.

Figures (8)

• Figure 1: From left to right: an example of a grid diagram, and how to obtain an oriented link diagram from it by resolving each double point as a vertical overpass.
• Figure 2: From left to right: a grid diagram representing the (right) trefoil, the effect of applying a stabilisation, the effect of an interleaving commutation on the stabilised diagram and a shift along the vertical axis.
• Figure 3: The effect of the destabilize_all function on a scrambled grid representing the $5_2$ knot. This function performs all possible "generalised" destabilisations on the grid.
• Figure 4: Some examples of grid diagrams obtained with GridPyM's drawing function draw_grid (from top to bottom): the $8$ crossings negatively clasped twist knot, a random knot in grid number $20$, the $(11,9)$ torus knot and its flat $(3,1)$ cable.
• Figure 5: A schematic representation showing one way to associate a braid diagram to a grid diagram.