Crossing Number Bounds of Mosaic Knot Diagrams

Abstract

In this thesis, we tabulate some previously undocumented link mosaic diagrams. Next we prove an upper and lower bound on crossing number of certain mosaic diagrams of knots in terms of winding number for knot diagrams that make only counterclockwise turns. Next we begin drawing mosaic diagrams that have a more grid-like structure and no crossings. This grid structure of a knot is similar to a ``grid diagram" which is equivalent to the arc presentation of a knot diagram. We generate grid diagrams using pairs of permutations from the group $((S_{n-1}\times S_n)$. These permutations form a set of coordinate pairs that locate the position of a turn tile in the diagram.