Knottingham

Alexander Polynomial:

Gauss:
Dowker-Thislethwaite:

Force Smoothness: |

Drag Neighboring Points: | Try reverting Non-Reidemeister moves:

Additional Invariants, calculated once:

Styling:
Gap Width:
Stroke Width:
Show Intersections: Drag Handles Independently:

Exporting:
Importing:

From Knottingham JSON:

Manual

Knottingham is a tool that lets you draw and manipulate knot diagrams, sporting a clean yet somewhat hand-drawn look. To start knotting away, you may want to follow these steps:

Create a knot by:

Clicking on segments to add new nodes
Clicking and dragging nodes to move them (you can use 'Drag Neighbors' to rescue nodes from below crossings)
and finally clicking on the (red) crossings to switch them (with "try reverting Non-Reidemeister moves" turned off)
or importing JSON
or clicking on 'Drawing' to draw a new knot from scratch!
Turning it into a link by clicking on 'Add Component' and drawing further closed curves

Adjust the knot by:

Removing nodes with Shift+Click
Smoothing segments with Control+Click
Moving and rotating the selected component with the WASD+EQ keys (handy for arranging links), mirroring everything with M
Undoing with the Z key
Selecting it through the select button and adjust Bezier handles
Creating a minimal-bend diagram on the integer grid with the button "Orthogonalize". You can follow up with:
Forcing smoothness (twice continuous differentiability, to be precise)
Adjusting the style with the sliders

Show the knot to your friends by:
- Exporting it to SVG!
- Exporting it to JSON, to be imported back into Knottingham!
- Exporting it to TikZ! (No support for crossing information yet, to adjust it in the final TikZ render, use the flipcrossings or the flip crossing key as documented in the knots package. Import tikz and the tikz package knots to use.)

Knottingham can also try detecting non-Reidemeister moves! Check the corresponding box and start thinking through knot equivalences. Discontious operations like smoothing or deleting segments might lead to breaking the equivalence.

Multi-component links can be drawn with 'Add Component' or imported by name (for example L6a4 in the Rolfsen field). For links, a per-component Gauss code and the pairwise linking numbers are shown, and the Jones and HOMFLY polynomials as well as orthogonalization work through the diagram's PD code. The Alexander polynomial and the Dowker-Thistlethwaite code are only computed for knots so far.

How does it work?

You can read all about that in our preprint about Knottingham here or the finished paper in the IEEE Transactions on Visualization and Computer Graphics. If Knottingham helped you with your research or teaching, we are very happy to be cited as

          @article{finke2024,
            author={Finke, Lennart and Weitz, Edmund},
            journal={IEEE Transactions on Visualization and Computer Graphics}, 
            title={A Phenomenological Approach to Interactive Knot Diagrams}, 
            year={2024},
            volume={30},
            number={8},
            pages={5901-5907},
            doi={10.1109/TVCG.2024.3405369}}

The Jones and HOMFLY polynomial are calculated with SageCellMath. To help them cover server costs incurred through websites like this, they accept contributions and donations here.

Any and all feedback is appreciated! You can mail to developer/at/fi-le.net.
Knottingham is open source - you can read and contribute to the code over here.