This is a list of some things I wrote in undergrad. I am moderately-to-severely embarrassed by everything here, but it may still be of some value to some people.
Small Quotients of Braid Groups (preprint, joint with Kevin Kordek) - We prove that the symmetric group S_n is the smallest non-cyclic quotient of the braid group B_n for n = 5, 6 and that the alternating group A_n is the smallest non-trivial quotient of the commutator subgroup B′ n for n = 5, 6, 7, 8. We also give an improved lower bound on the order of any non-cyclic quotient of B_n.
A Geometric Derivation of the Automorphism Group of the Braid Group - This short paper gives a geometric proof of a result due to Dyer and Grossman that the automophism group of the braid group on n strands is B_n/Z(B_n) ⋊ Z_2.
Simple Colorings of Pretzel Knots (joint with Sudipta Kolay)- A simple n-coloring is an assignment of transpositions in S_n to strands in a link diagram subject to certain conditions. In this paper, we give a complete characterization of the simple n-colorability of pretzel links, and provide a method for counting the number of such colorings.
Branched Covers of 2- and 3-Manifolds (term paper for MATH 6441 Algebraic Topology)- Branched covers are continuous maps between manifolds which fail to be coverings on a small set. This expository paper gives a few basic results, and shows how branched covers give a surprising bridge between 3-manifolds and knot theory.
Generalizing Knot Colorings with Groups (term paper for MATH 4803 Knot Theory)- Classic knot colorings are some of the easiest knot invariants to understand, although they are not very powerful. This paper is an introduction to a different way to color knots using groups which generalizes the classic notion of a knot coloring.
Chess Algorithms - This book introduces the fundamental algorithms of chess programming with an emphasis on thorough explanations. It is written with a novice programmer in mind—someone with a basic understanding of recursion but without any formal training in computer science. After finishing the book, readers should feel comfortable writing their own expert level (~2000 ELO) chess engine and be able to read technical documents containing advanced chess programming techniques.
The following two papers are from the 2021 UMN REU in algebraic combinatorics.
F-polynomial Ratios in the r-Kronecker (joint with Ariana Chin, Nyah Davis and Swapnil Garg ) - We provide an algebraic proof for the limit of the ratio of consecutive F-polynomials of the 2-Kronecker. We do this by expanding a proof by Reading [4] which involves cluster scattering diagrams. Furthermore we use this result to recursively define the limit for general r-Kronecker
Lattice Models and Puzzles for Dual Weak symmetric Grothendieck Polynomials (joint with Elisabeth Bullock, Ariana Chin, Nyah Davis and Gahl Shemy) - We construct a solvable lattice model for the dual weak symmetric Grothendieck polynomials given by Pylyavskyy and Lam in [6] in hopes of using such a model to prove related properties of these polynomials, including Cauchy identities and branching rules. We also consider a similar lattice model construction for the weak symmetric Grothendieck polynomials in hopes of proving a Cauchy identity given by Yeliussizov [12], concluding with a negative result. Moreover, we expand on the work done by Pylyavskyy and Yang in [10] and by Zinn-Justin in [13] by giving boundary conditions for a proposed lattice model for the Littlewood Richardson coefficients of the dual weak symmetric Grothendieck polynomials, via an MS puzzle construction.