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Zachary Bryhtan, Nicholas Connolly, Isabel Darcy, Ethan Rooke, Joseph Starr*
Mathematics Department at The University of Iowa
“A knot is a smooth embedding of a circle $S^1$ into Euclidean 3-dimensional space $\R^3$ (or the 3-dimensional sphere $S^3$ ).”
Jablan, S., & Sazdanović, R. (2007). Linknot. In Series on Knots and Everything. WORLD SCIENTIFIC. https://doi.org/10.1142/6623
“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
To play with twist vectors and continued fractions visit