# Thesis Defense: Crossing number bounds for mosaic knot diagrams

Posted on: April 11, 2017

# Crossing number bounds for mosaic knot diagrams

### Location: MAN 121

We begin our work by tabulating some previously undocumented link mosaic diagrams, and continue by proving some bounds on crossing number of mosaic diagrams of knots in terms of winding number. This work is specifically for knots that make make only counterclockwise turns, but we wanted to generalize our work to bounds that did not require our knots to have only counterclockwise turns. As we searched for our new bounds for all knots we began to draw our diagrams outside of our tiling scheme and began to make them resemble a more grid-like structure. We found that this grid structure of a knot had been studied under the term “grid diagram”, and was equivalent to the arc presentation of a knot. We found that we could generate grid diagrams using pairs of permutations from the group $((S_{n-1}\times S_n)/\mathbb{Z}_2)$.