Project leader: Professor James Hughes
Project manager: Jason Ma
Team members: Vincent Chen, Patton Galloway, Luciana Wei
If you take several strands, cross adjacent strands arbitrarily, we get what’s called a braid. Joining the ends of the strands creates a knot or a link, represented as a braid closure. Then, we can use several operations on them to create a surface whose cross sections are braids, called weaves. We can encode the crossing data combinatorially simply by associating them with colors, creating a colored graph. Our project is focused on when these graphs exhibit certain rotational symmetry. To accomplish this, we employ a combinatorial strategy that focuses on plabic tilings. Generally speaking, plabic tilings are tilings of n-gons generated by creating cliques from certain kinds of sets that are unchanged under addition. These tilings in turn create rotationally symmetric weaves through a process called T-shift, detailed by Casals, Le, Sherman-Bennett, and Weng. The end goal of our project is to create an algorithm that can encode this combinatorial data, given specific numeric conditions.
In particular, the type of sets we want to generate are symmetric maximal weakly separated collections, which are parameterized by the natural number input variables n, k, and l. These are collections of k-element subsets of the first n natural numbers fixed under addition by l modulo n and of size k(n-k)+1. Our algorithm follows the work of Pasquali, Thörnblad, and Zimmerman, generalizing their construction of symmetric maximal weakly separated collections for l = k to the cases where l ≠ k. Accordingly, we also generalize their necessary and sufficient condition on n, k, and l. In particular, we must have that k is congruent to -1, 0, or 1 modulo n/gcd(n,l). Then, to guarantee that these symmetric weakly separated collections give the weaves we want, we also prove the existence of an injection from maximal symmetric weakly separated collections to well-behaved, embedded, and symmetric weaves via our algorithm and T-shift.
The plabic tiling for n = 100, k = 26, l = 4
The step-by-step T-shift process resulting in a torus link T(3,3) boundary
The step-by-step T-shift process resulting in a torus link T(3,3) boundary
This project has culminated in the following preprint which is available at arXiv:2509.19095