Geodesic continued fractions

Geodesic Rosen Continued Fractions

“…Here we describe how to represent Rosen continued fractions by paths in certain graphs of infinite valency that arise naturally in hyperbolic geometry. This perspective sheds light on Rosen's work, and allows us to tackle problems about the length of Rosen continued fractions, in a similar manner to the approach for integer continued fractions found in [2].…”

“…The graph F 3 is the skeleton of a tessellation of the hyperbolic plane by ideal triangles, the vertices of which are the rational numbers and ∞. It has been used already to study continued fractions, in works such as [2,11,21,30] and [27,Chapter 19]. The Farey graphs (for all values of q) also arise in subjects involving hyperbolic geometry that are not directly related to continued fractions; for example, they form a class of universal objects in the theory of maps on surfaces (see [9,10,31]).…”

Circuit complexity of knot states in Chern-Simons theory

“…Proposition 2. The minimal word length (3.6) is also achieved for the geodesic continued fraction constructed in [39] based on the ancestral path between n m and ∞ in the Farey graph.…”

“…Continued fractions with the least number of terms (so-called geodesic continued fractions) have been discussed in [39], where a prescription is given to construct a particular geodesic CF using the so-called ancestral path from n m to ∞ on the Farey graph. The problem here is that in general there are multiple geodesic CFs 9 and it is not clear how to carry out the minimization over coefficients b i within this set of geodesic CFs (the exception here is when |b i | ≥ 3 for all i ≥ 2, in which case the ancestral path CF constructed in [39] is the unique geodesic one and therefore minimizing over b i is trivial). Even though we have compelling numerical evidence that this ancestral path CF indeed minimizes the length (3.6), here we adopt a different (simpler) strategy and prove the following: Proposition 1.…”

“…The construction implements the Euclidean algorithm for finding the greatest common divisor of two integers n and m. It is clear that its coefficients are all positive (except perhaps the first one, which vanishes when 0 ≤ x < 1 or is negative when x < 0). 9 Namely, there are at most F r (the r-th Fibonacci number) geodesic CFs with value x, where r is the minimal number of terms needed to expand the rational x [39]. 10 For an arbitrary (positive or negative) rational x this does not work.…”