Knot Logic and Arborescent Links
Louis H Kauffman
TL;DR
The paper develops the crossing algebra, a diagram-based, algebraic tool for counting components in arborescent knots and links and for analysing state expansions of knot invariants. It connects this algebra to continued-fraction encodings of rational tangles, the bracket polynomial, and the beginnings of Khovanov homology, while drawing deep links to logic (boolean and multi-valued), graph theory (medial and checkerboard graphs), and switching circuits. Core contributions include a parity-based method to determine component counts for rational knots/links, a generalization to arborescent links, and a framework that supports algebraic construction of the Khovanov complex via tensor representations. The work offers a foundational algebraic lens on classical knot invariants, with practical implications for algorithmic computation and potential quantum-computational perspectives on homology theories.
Abstract
This paper introduces a new algebra, the crossing algebra, that is applied to count the number of components for arborescent knots, links, tangles or states (of a state polynomial expansion such as the Kauffman bracket). This algebra is foundational, and it is related to generalisations of boolean logic and to aspects of foundations based in diagrams and networks. Applications are given to rational knots, links and tangles and to the structure of the bracket polynomial and the beginnings of Khovanov homology.