Title: Coloring Carpets

Abstract: In 2023, Barry Cipra and Paul Zorn noticed interesting patterns in carpets obtained by applying rules from knot theory to color them. Over the last three years, students and I have been working to understand these patterns. We will show many pretty pictures and do our best to explain why they look the way they do.

Title: Kauffman bracket skein algebra of a surface

Abstract: The Kauffman bracket skein algebra of a surface is a construction that generalizes the Kauffman bracket for knots and links in S^3 to knots and links in a thickened surface. With few exceptions, the algebra is noncommutative with frustratingly complicated multiplicative structure.  But at the same time it has an easy combinatorial description and so lends itself well to undergraduate research.  I’ll discuss a student project on the skein algebra of a 1-hole torus that started off as a theoretical project but ended up as an experimental/algorithmic paper.  And if time allows, I’ll also discuss a version of the skein algebra for punctured surfaces that includes arcs.