In this project we implement the Kauffman polynomial in SageMath (Python).
In this project we write implement from scratch the Kauffman polynomial in Python. First we introduce problems in computational knot theory and describe various representations of knots and links and find a good representation to use for the algorithm. We then describe in-depth the algorithm for computing the Kauffman polynomial and how to implement it in Python. Finally we try the algorithm on various knots and links and compare the results with the ones from the KnotInfo Database.
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= Introduction
Actually we don't like Python so we will be using Rust and then write bindings for Python that can be used in SageMath.
@ -181,7 +177,7 @@ We will later be seeing that the Kauffman polynomial can be defined in a more ex
== Computational Knot Theory
The first problem in computational knot theory is to find a good representation for knots and links. There are various common representations, such as:
The first problem in computational knot theory is to find a good representation for knots and links. There are various common representations in the literature, such as:
- *Gauss codes*:
@ -290,7 +286,7 @@ Every directed crossing appears only once as an over-crossing so this algorithm
`[(6,1,7,2), (8,3,5,4), (2,5,3,6), (4,7,1,8)]`
}
=== Signed Gauss Codes
=== SG Codes
Gauss originally developed a notation called *Gauss codes* based on labelling each crossing of a knot with a number and keeping track of when we walk an over-crossing or an under-crossing using a sign. This produces a list of numbers where each number appears exactly twice with different signs. This has a few problems like the fact that this doesn't distinguish a knot vs its mirror.
@ -718,7 +714,7 @@ Let now $K$ be an oriented link with $n$ components so $K = K_1 union dotss unio
The approach has been a mix of bottom-up and top-down. First we defined a couple of classed `SignedGaussCode` and `PDCode` to work with these codes and easily convert between each other.
=== Signed Gauss Codes
=== SG Codes
We are now going to walk thorough the class that lets use work nicely with *SG codes*. The the classes we are going to use are all _frozen data-classes_ to ensure immutability and enforce a more functional programming style.
