knot-theory

SageMath and GAP code related to low-dimensional topology

In particular, we study the Casson-Whitney unknotting number via lower bounds from the fundamental group of the complement.

This is applied to Suciu's infinite family of ribbon 2-knots with the trefoil knot group.

Requirements

• SageMath version >= 9.0

Instructions

From the main git directory start sage

• attach('src/suciu_ribbon_knots.sage') to load the file with Suciu's examples
• attach('src/bridge_trisection.sage') to load the file with the functions for triplane diagrams, this will look for the data files in /data

Using SageMath from docker image

Docker repository for SageMath

Start SageMath command line: sudo docker run -it sagemath/sagemath:latest

Start SageMath in Jupyter Notebook: sudo docker run -p8888:8888 sagemath/sagemath:latest sage-jupyter

Using Sage's GAP interface

GAP finitely presented groups through SageMath: (https://doc.sagemath.org/html/en/reference/groups/sage/groups/finitely_presented.html)

Projective Special Linear Group: (https://groupprops.subwiki.org/wiki/Projective_special_linear_group:PSL(2,Z))

Logging Sage output and keeping processes running over ssh

(https://askubuntu.com/questions/8653/how-to-keep-processes-running-after-ending-ssh-session) Start tmux

Start process in the tmux session:

(https://doc.sagemath.org/html/en/tutorial/interactive_shell.html) In SageMath, activate logging with logstart -o -t Trefoil4commutator

Options: -o to also log output, -t for timestamps, can give any other name instead of 'Trefoil4commutator'

Leave/detach the tmux session by typing Ctrl + B and then D

(In tmux, Ctrl + B is the control key, while D is the command key for detaching a session)

You can now safely log off from the remote machine, your process will keep running inside tmux. When you come back again and want to check the status of your process you can use tmux attach to attach to your tmux session.

If you want to have multiple sessions running side-by-side, you should name each session using Ctrl + B and $. Alternative from the terminal tmux rename-session -t 5 brown_band_commutator where 5 is the old name and brown_band_commutator new.

You can get a list of the currently running sessions using tmux list-sessions or tmux ls, now attach to a running session with command tmux attach-session -t <session-name>