Continued Fraction and Twist Vector Converter

2 min read
Resources
Table of Contents

Rational Tangles

A rational tangle is given by alternating NE,SE and SE, SW twisting of the 00 tangle (Conway, 1970). Discussion of canonically of this construction of twist vector can be found in (Goldman & Kauffman, 1997). A twist vector encodes these alternating twists as a list of integers. This induces a unique map from the rational tangles onto the rational numbers (@conwayEnumerationKnotsLinks1970 @goldmanRationalTangles1997 Kauffman & Lambropoulou, 2002). We accomplish this by interpreting a twist vector as a sequence for a continued fraction as:

\left[ a\ b\ c\right]=c+\frac{1}{b+\frac{1}{a}}

Instructions

Twist vectors here are space separated lists of integers.

A rational number here is ”/” separating two integers.




References

Conway, J. H. (1970). An enumeration of knots and links, and some of their algebraic properties. In Computational Problems in Abstract Algebra (pp. 329–358). Elsevier. https://doi.org/10.1016/B978-0-08-012975-4.50034-5
Goldman, J. R., & Kauffman, L. H. (1997). Rational Tangles. Advances in Applied Mathematics, 18(3), 300–332. https://doi.org/10.1006/aama.1996.0511
Kauffman, L. H., & Lambropoulou, S. (2002). On the Classification of Rational Knots. arXiv: Geometric Topology. https://doi.org/10.48550/ARXIV.MATH/0212011