Table of Families of Alternating Knots with their Conway's Function

TL;DR

This paper presents a comprehensive table of alternating knot families using Conway's function, employing linear algebra to classify knots based on crossings and handedness, with detailed figures illustrating their structure.

Contribution

It introduces a systematic classification of alternating knots via Conway's function using linear algebra and detailed visualizations, including color-coded tangles and factorizations.

Findings

Conway's function can be expressed as an internal product of vectors.

Color-coded figures illustrate knot dissection and handedness.

Full factorizations of Conway's function are presented, showing non-uniqueness.

Abstract

A table of the families of alternating knots formed by conways is presented. The Conway's function is shown with the use of linear algebra in terms of natural numbers, called conways, that represent the number of crossings along a direction, as it was used by J. Conway for the classification of knots. Colored figures and tangles show the parts of the knots or tangles with a definite handedness: all the colored parts of the knot family are associated to a particular orientation. For example all the colored conways have a right hand screw thread, and all the white conways have the opposite handedness. Figures for six conways were colored with two different colors for forty two families in order to show the dissection of the knot in two tangles corresponding to a particular factorization of the Conway's function. The Conway's function of each family is expressed as the internal product of two vectors corresponding to each of two colored family 2-tangles, and with a full factorization which is not unique.