Use Case

classDiagram
    note_tv --|> notation
    note_tv *-- note_tv_t

    class note_tv_t {
        <<struct>>
        List~uint8_t~ twist_vec
    }

    class notation {
        <<interface>>
    }

Language

C

Implements

Notations Interface

Uses

N/A

Libraries

N/A

Functionality

A rational tangle is given by alternating NE,SE and SE,SW twisting of the $0$ tangle${}^{[2]}$${}^{[1]}$. Discussion of canonicality of this construction of twist vector can be found in ${}^{[2]}$. A twist vector encodes these alternating twists as a list of integers.

Example

Starting with the $0$ tangle then twisting NE/SE clockwise 3 times SW/SE clockwise 2 times NE/SE clockwise 2 times we have noted as $\LB3\ 2\ 2\RB$

Data Structure Description

When encoding a twist vector as a string the standard indexing is $\LB x_n\ x_{n-1}\ \cdots\ x_0\RB$ we follow the convention but align the indexing to the array index, that is

$ $ \LB xn\ x\ \cdots\ x0\RB\to \begin{array}{|c|c|c|c|} \hline \text{0x0000}& & &\text{0x0000}+n\cdot \text{size_t}\\hline x_0&\cdots & x $ $} &x_n\ \hline \end{array

Encoding

When encoding the data structure to a string we need to reverse the order we read the twist vector array. This will left align the $0$ index in the string.

Decoding

We need to insert entries into the twist vector data structure, to do this we will traverse the string in reverse.

Cite

Conway, J.H. “An Enumeration of Knots and Links, and Some of Their Algebraic Properties.” In Computational Problems in Abstract Algebra, 329–58. Elsevier, 1970. https://doi.org/10.1016/B978-0-08-012975-4.50034-5.
Kauffman, Louis H., and Sofia Lambropoulou. “On the Classification of Rational Knots,” 2002. https://doi.org/10.48550/ARXIV.MATH/0212011.