Knot Logic and Arborescent Links

TL;DR

This paper introduces a new algebraic framework called crossing algebra, which helps analyze arborescent knots and links, with applications to polynomial invariants and homology theories.

Contribution

It develops a foundational algebraic approach to count components of arborescent knots and links, connecting to logic, diagrams, and network structures.

Findings

Applied crossing algebra to rational knots, links, and tangles.

Connected crossing algebra to the structure of the bracket polynomial.

Initiated exploration of Khovanov homology using this algebra.

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.