https://github.com/pythonnut/linear-presentation
Program to compute the "linear presentation" of a knot
https://github.com/pythonnut/linear-presentation
Last synced: 18 days ago
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Program to compute the "linear presentation" of a knot
- Host: GitHub
- URL: https://github.com/pythonnut/linear-presentation
- Owner: PythonNut
- Created: 2019-07-14T02:39:01.000Z (almost 6 years ago)
- Default Branch: master
- Last Pushed: 2024-10-30T06:39:16.000Z (6 months ago)
- Last Synced: 2025-04-15T15:09:20.018Z (18 days ago)
- Language: Python
- Homepage:
- Size: 960 KB
- Stars: 2
- Watchers: 2
- Forks: 0
- Open Issues: 2
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Metadata Files:
- Readme: README.md
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README
# A (sort-of) Linear-Time Algorithm for Drawing Linear Knot Diagrams
## Motivation
When doing work in [Knot
Theory](https://en.wikipedia.org/wiki/Knot_theory), it is often
helpful to use _combinatorial encodings_ such as the [_Gauss
code_](http://katlas.math.toronto.edu/wiki/Gauss_Codes) to represent
our knots (another common encoding is the [_DT
code_](http://katlas.org/wiki/DT_%28Dowker-Thistlethwaite%29_Codes),
but we won't talk about that today). Such encodings offer computers an
elegant way of representing / manipulating knots --- indeed, many
invariants can be computed directly from combinatorial
representations.
Basically, the Gauss code is a really elegant tool for applying
algebraic/algorithmic techniques to the study of knots. However, it
can be challenging to build geometric intuition for how manipulations
of the Gauss code translate to manipulations of a knot and vice versa.
The problem is not the translation process is inherently challenging,
but rather that it is tedious and time-consuming.
`linear-presentation` is designed to streamline this process. Given a
Gauss code, it creates a diagram for the associated knot such that
(1) all of the crossings are collinear,
(2) the over/understrands of each crossing are perpendicular at the
crossing point, and
(2) the first time a crossing is encountered, the strand is oriented
pointing to the right.
Since Gauss codes are presented as linear strings, there is a sense in
which this feels like the "natural" diagram to associate to the code.
And in the other direction: In our personal experience, this
presentation makes writing down a Gauss code associated to a diagram
much faster than it would otherwise be.
## A Brief Description of the Algorithm