`$\newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\R}{\mathbb{R}} \newcommand{\LP}{\left(} \newcommand{\RP}{\right)} \newcommand{\LS}{\left\lbrace} \newcommand{\RS}{\right\rbrace} \newcommand{\LA}{\left\langle} \newcommand{\RA}{\right\rangle} \newcommand{\LB}{\left[} \newcommand{\RB}{\right]} \newcommand{\MM}{\ \middle|\ } \newcommand{\exp}{\text{exp}} \newcommand{\abs}[1]{\left\vert#1\right\vert} \newcommand{\msr}[1]{m\left(#1\right)} \newcommand{\inv}[1]{#1^{-1}} \newcommand{\bkt}[1]{\LA \img{#1}\RA} \require{color}$`s
Zachary Bryhtan, Nicholas Connolly, Isabel Darcy, Ethan Rooke, Joseph Starr*
Mathematics Department at The University of Iowa
“We define a tangle as a portion of a knot diagram from which there emerge just 4 arcs pointing in the compass directions NW, NE, SW, SE.”
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
For any $N$ an obvious twist vector is the twist vector of all $1$s $$[1\ 1\ 1\ \cdots\ 1]$$ Noting that when we write this sequence, we have $N-1$ spaces.
If we choose to place a $+$ instead of the left most space we get $$[1+1\ 1\ \cdots\ 1]=[2\ 1\ \cdots\ 1]$$ we’re free to make this choice for each space
this gives $N-1$ choices between ‘$+$’ and space $$[1\square 1\square 1\square\cdots\square1]$$ letting us generate twist vectors by simply counting from $0\to 2^{N-1}$.
We can write a canonical twist vector by taking the odd length vectors (appending $0$ where needed).
The rational number for a twist vector is computed by taking the twist vector as a finite continued fraction that is: $$\LB a\ b\ c\RB=c+\frac{1}{b+\frac{1}{a}}$$
Louis H. Kauffman and Sofia Lambropoulou. Classifying and applying rational knots and rational tangles. In DeTurck, editor, Contemporary Mathematics, volume 304, pages 223-259, 2001
F. Bonahon and L. Siebenmann, New geometric splittings of classical knots, and the classification and symmetries of arborescent knots, http://www-bcf.usc.edu/~fbonahon/Research/Publications.html
The Montesinos tangles of crossing number $N$ have a slightly simpler generation strategy compared to rational tangles. We again generate twist vectors but require that each entry $e$ of the twist vector satisfies $2\leq e < N.$ We call these restricted set of twist vectors stencils.
Now for each entry $e_i$ of the stencil, we generate a list of rational tangles of crossing number equal to $e_i$, with the restriction $0<\frac{p_i}{q_i}<1$. We then take all combinations of elements of these lists.
The construction for the canonical Montesinos tangles includes a trailing $\frac{k}{1}$ tangle. Our generation strategy seems to miss these.
What we’re actually generating with this algorithm is equivalent to allowing the boundary components of the tangle to move. To recover fixed boundary tangles we need to generate the next larger class of tangles.
We can note which stage a tangle was generated at to allow users to choose datasets for fixed or non-fixed boundary tangles.
Moon, Hyeyoung, and Isabel K. Darcy. “Tangle Equations Involving Montesinos Links.” Journal of Knot Theory and Its Ramifications 30, no. 08 (July 2021): 2150060. https://doi.org/10.1142/S0218216521500607.
We just need to take our lists of Montesinos and rational tangles and glue them together with $\circ$.