Knot logic and arborescent links

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 elementary and foundational, and it is related to generalizations of Boolean logic and to aspects of foundations based in diagrams and distinctions. Applications are given to rational knots, links and tangles and to the structure of the bracket polynomial and the beginnings of Khovanov homology.