2025 Project: Mosaic Knot Theory
8 weeks: May 27-July 18 (flexible)
We will study diagrams of knots built from mosaic tiles. Knot theory is the study of knotted loops in three space up to isotopy. These tilings give a combinatorial approach to describing knots, which can help approach some questions in knot theory as well as raise new questions arising from the tile structure. Students will help choose questions of interest on topics ranging from efficiency of square and hexagonal tilings to relationships of tilings and invariants in contact topology.
Knot theory is the study of knotted loops in three dimensions -- how to untangle a knotted loop without breaking the string. This can seem like an intractable problem as knots get large and complicated (think knotted proteins or strands of DNA), but many hands-on techniques reveal useful insights about knots.
Drs. Kate Kearney and Eric Hogle lead research focuses on knot mosaics, which are diagrams of knots created as a mosaic of tiles with simple arcs or crossings on them. These can be studied with a combinatorial approach, examining different ways to combine tiles on a limited size grid to discover relationships between the properties of the mosaic design and properties of the knot.
