**Project leader**: Professor James Hughes

**Project manager**: Zijun Li

**Team members**: Holly Keegan, Helen Pertsemlidis, June Wood

Take a shoelace, knot it with itself, and glue the ends of the shoelace together. The object that you have created is a topological knot. In our project, we consider two knots that are isotopic to each other, which means that you can wiggle one knot around without breaking it to get it to look like the other. Tracing the process of wiggling the knot produces a cylinder Φ that can be thought of as living in five-dimensional space. Our two knots are the boundaries of two surfaces, Σ_{1} and Σ_{2}, that also live in five-dimensional space. We can glue Φ to Σ_{2} to create a new surface Σ_{2}+Φ. The goal of our project is to show that Σ_{2}+Φ is equivalent to Σ_{1}.

The surfaces we study are called Legendrian weaves, which means they must satisfy geometric properties induced by a specific contact structure on five-dimensional space. As a result, we obtain a projection (x_{1}, x_{2}, y_{1}, y_{2}, z) → (x_{1}, x_{2}, z) that allows us to study these surfaces in 3-dimensional space while being able to recover the y_{1} and y_{2 }coordinates. The 3-dimensional “wavefronts” that we use to study these surfaces can also be encoded using combinatorial objects called N-graphs. We use these N-graphs to represent Φ and then search for a series of isotopy moves that would simplify the N-graph representing Σ_{2}+Φ into the N-graph representing Σ_{1}. Throughout this process, we identify and justify local pictures critical to the overall isotopy. We also discuss a new isotopy move between known singularities in the weaves. Throughout our project, we focus on the specific case of Σ_{1}, Σ_{2, }and Φ. Our hope is that finding an equivalence in this case will foreshadow how equivalences work for more general choices of these surfaces.