Conway classification of alternating knots

TL;DR

This paper introduces a novel classification scheme for alternating knots based on their correspondence with dynamical systems and graph permutations, providing explicit polynomial invariants and a systematic categorization of rational knots.

Contribution

It develops a new classification method linking knot theory with dynamical systems, using matrix decompositions and Chebyshev polynomials, and classifies rational knots up to five ribbons.

Findings

Unique matrix decomposition for knots, not links.

Explicit polynomial invariants for knot families.

Classification of rational knots up to five ribbons.

Abstract

The alternating knots, links and twists projected on the $S_2$ sphere were identified with the phase space of a Hamiltonian dynamic system of one degree of freedom. The saddles of the system correspond to the crossings, the edges correspond to the stable and unstable manifolds connecting the saddles. Each face is then oriented in one of two different senses determined by the direction of these manifolds. This correspondence can be also realized between the knot and the Poincar\'e section of a two degrees of freedom integrable dynamical system. The crossings corresponding to unstable orbits, and the faces to foliated torus, around a stable orbit. The associated matrix to that connected graph was decomposed in two permutations. The separation was shown unique for knots not for links. The characteristic polynomial corresponding to some knot, link or twist families was explicitly computed in terms of Chebyschev polynomials. A classification of rational knots was formulated in terms of the first derivative of the polynomial of a knot computed in $x=2$, equal to the number of crossings of the knot multiplying the same number used previously by Conway for tabulation of knot properties. This leads to a classification of knots exemplified for the families having up to five ribbons. We subdivide the families of $N$ ribbons in subfamilies related to the prime knots of $N$ crossings.