Repetitions of Pak-Stanley Labels in G-Shi Arrangements
Abstract: Given a simple graph G, one can define a hyperplane arrangement called the G-Shi arrangement. The Pak-Stanley algorithm labels the regions of this arrangement with G*-parking functions. When G is a complete graph we recover the full Shi arrangement, and the Pak-Stanley labels give a bijection with ordinary parking functions. However, for proper subgraphs, some appear more than once. These repetitions of Pak-Stanley labels are a topic of interest in the study of G-Shi arrangements and G*-parking functions. The key insight of our work is the introduction of a combinatorial model called the Three Rows Game. Analyzing the histories of this game and the ways in which they can induce the same outcomes allows us to characterize the multiplicities of the Pak-Stanley labels. Using this model, we develop a classification theorem for the multiplicities of the Pak-Stanley labels of the regions in T-Shi arrangements, where T is any tree. Finally, we discuss the possibilities and difficulties in applying our method to arbitrary graphs.
This research was completed at the 2022 SUMMER@ICERM REU in Providence, Rhode Island under the mentorship of Dr. Gordon Rojas Kirby and graduate student Lucy Martinez. Our paper preprint is available on the ArXiv and currently being submitted for publication.
Knot Colorability and Maximum Knot Determinants
Abstract: Knot theory is the study of embeddings of circles in R3, called knots and links. Invariants are a means of differentiating knots and links of different types. We focus on two closely related such invariants, called p-colorability and the knot determinant. Grid diagrams are rectilinear representations of knots and links which provide novel methods for understanding and computing these invariants. The grid diagram perspective on determinants raises the natural question of determining the maximal knot or link determinant amongst links represented by grids of a fixed size. We connect this problem to the long-standing Hadamard’s maximal determinant problem regarding determinants of matrices for which the entries are restricted to 0 or 1. After exploring these connections, we present our findings on lower and upper bounds for the maximal determinants of knots and links on grid diagrams of a fixed size.
This research was completed at the 2021 SURIEM REU at Michigan State University under the mentorship of Dr. Matthew Hedden and Dr. Teena Gerhardt. My group’s final report is available here. I have presented our work at the following conferences:
- 2021 Summer Undergraduate Michigan Mathematics Research Conference [slides]
- 2022 Joint Mathematics Meeting – Pi Mu Epsilon Poster Session [poster]
- 2022 Nebraska Conference for Undergraduate Women in Mathematics [slides]