Professor Sir Tom Davis in a note - Conway's Rational Tangles has said $$\\$$ If the students are a bit advanced, you can point out that the process of reducing the fraction down to zero is almost exactly the same as finding the greatest common divisor (the GCD) of the numerator and denominator. $\\$.
I understand the Euclidean algorithm to find gcd and the rational tangle algorithm but cannot translate the steps of Euclidean algorithms in terms of rational tangles algorithm.
I was trying to understand with an example, with two number $19$ and $11$.
To find the gcd we follow the steps:-
$$19 = 11.1+8\\11 = 8.1 + 3\\8 = 3.2 + 2\\3 = 2.1 + 1\\2 = 1.2+0$$
where gcd is $1$.
Now using rational tangle algorithm we have, $$\frac{19}{11}{\longrightarrow}^{R}\longrightarrow \frac{-11}{19}{\longrightarrow}^{T}\longrightarrow \frac{8}{19}{\longrightarrow}^{R}\longrightarrow \frac{-19}{8}{\longrightarrow}^{T^3}\longrightarrow \frac{5}{8}{\longrightarrow}^{R}\longrightarrow \frac{-8}{5}{\longrightarrow}^{T^2}\longrightarrow \frac{2}{5}{\longrightarrow}^{R}\longrightarrow \frac{-5}{2}{\longrightarrow}^{T^3}\longrightarrow \frac{1}{2}{\longrightarrow}^{R}\longrightarrow -2{\longrightarrow}^{T^2}\longrightarrow0$$
where $T=x+1, \ R=\frac{-1}{x}$ and $x$ is initial stage before any twist or rotation and $T^n$ denotes $n$ times twists.
In both algorithms, we see $0$ at the final step but the gcd in Euclidean algorithm is $1$ and according to the rational tangle algorithm the total number of attempts to make $0$ tangles is $-1+1-1+3-1+2-1+3-1+2=6$ which is greater than $1$.
1.Then how the process of reducing the fraction down to zero is almost exactly the same as finding the greatest common divisor? $$\\$$ 2. Is it correct to assume the multiplication of divisor and quotient as rotation and the remainder as twist?