Our arborescent tangle tabulation project presents a comprehensive study on the tabulation of arborescent tangles[1]. Arborescent tangles are constructed by taking a collection of twisted bands described by a weighted tree and connecting them with successive plumbing. The arborescent tangles are shown to be equivalent to Conway’s algebraic tangles[2].
Key results include the development of a unique representative for arborescent tangles, an encoding strategy for computational purposes, and an algorithm for generating minimal arborescent tangles. The study also introduces the concept of Right Leaning Identity Canonical Abbreviated Weighted Planar Tangle Trees (RLITT), proving their existence, uniqueness, and providing an efficient generation algorithm.
This work contributes to the field by offering a robust framework for the enumeration and classification of arborescent tangles, supported by both theoretical insights and practical computational tools.
References #
- F. Bonahon and L. Siebenmann,
New Geometric Splittings of Classical Knots and the Classification and Symmetries of Arborescent Knots,
https://dornsife.usc.edu/francis-bonahon/wp-content/uploads/sites/205/2023/06/BonSieb-compressed.pdf, 2016. - J. Conway,
An enumeration of knots and links, and some of their algebraic properties,
in Computational Problems in Abstract Algebra, Elsevier, 1970, pp. 329–358.doi:10.1016/B978-0-08-012975-4.50034-5