Classification of Knots in a Thickened Torus with Minimal Octahedron Diagrams which are not Contained in an Annulus

Abstract

The aim of this research is to tabulate knots in a thickened torus T×I having minimal diagrams which are not contained in an annulus and correspond to the octahedron graph. Tabulation consists of three steps. First, a table of knot projections on T was compiled. Then, every projection was converted into a set of corresponding diagrams. Finally, using a generalized version of the Kauffman bracket as an invariant, duplicates were removed and all the knots obtained were proved to be different.