## Knot theory and arc index

Adam Lowrance (Mathematics and Statistics)

A knot is a closed path in 3-space, and two knots are considered the same if one can be continuously deformed into the other by stretching or moving the path through 3-space. A common way to study knots is via their diagrams, i.e. projections of the path in 3-dimensions to a picture in the plane. The fewest number of crossings in any diagram of a knot is a natural measure of complexity on knots. Another way to study knots is via their grid diagrams, square grids where segments of the knot diagram are parallel to the coordinate axes. The smallest size of any grid for which a given knot has a grid diagram is another natural measure of complexity on knots called the arc index of the knot. In this project, we will compare these two different measures of complexity for specific families of knots.

**Prerequisites:
** Required prerequisite: Math 220 and Math 221. Useful (but not required): some proof-writing experience.

**How should students express interest in this project?
** Students should email me to set up an appointment to discuss the project in more detail.

*This is a 10 week project running from May 27-July 31*