Octagonal Knotworks. A Mathematical Approach

Alexander Astrom, Christoffr Astrom

This work is intended as an approach on how to calculate the number of strands in Octagonal diagonal knot diagrams. The work is made with Troense diagrams as a starting point, which is one type of octagonal knot diagrams among many other. Our point of attack to the problem may seem impractical at rst, since it's not based on the structure or any geometrical characteristics on the diagrams. Our method and equations are based on the table that is constructed from a number of diagrams with di erent number of bights on the two variable sides of one type diagram. The equations in our method are based upon the similarities between the Troense table and the greatest common divisor function. The equations are not tested at any extent other than on Troense diagrams and one structure with three bights, compared to the Troense that has two bights. We are not presenting any proof or even an approach to a proof for the equations we describe. However, we have compared the result from the equations for the Troense method against about 150 drawn diagrams. Some diagrams with the structure of three bights are also compared with a very satisfying result. We have also made some attempts to classify the structure of bights to simplify future extensions of our work. Our ambition with this work is to present a new method or at least a new strategy for strand calculation that can help and assist others. We also believe that our method can be re ned and that our work can be carried on by others where we didn't nish.