arXiv subjects

Nicholas Wawrykow

Publications and source records attributed to Nicholas Wawrykow.

11 recordsLinked to original sources

arXiv2026

The 15 Puzzle and homological stability in the space direction

The ordered configuration space of $n$ open unit squares in the $w$ by $h$ rectangle exhibits homological stability in the space direction. That is, for fixed $n$ and fixed homological degree $k$, once the underlying rectangle is large enough, making it any larger does not change the $k$-th homology of the square configuration space. In this paper, we sharpen the stable range. Finding bounds for $w$ and $h$ in terms of $n$ and $k$, we prove that most rectangles can be almost entirely filled with squares and there still be an isomorphism between the $k$-th homology of the resulting square configuration space and the $k$-th homology of the ordered configuration space of $n$ points in the plane.

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arXiv2025

Representation asymptotics in the homology of pure graph braid groups

We give explicit formulas for the asymptotic Betti numbers, over an arbitrary field, of the ordered configuration spaces of a graph. In characteristic zero, we further give explicit formulas for the asymptotic multiplicities in homology of many irreducible representations of the symmetric group, in the spirit of representation stability.

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arXiv2025

The sequential (distributional) topological complexity of the ordered configuration space of disks in a strip

How hard is it to program $n$ robots to move about a long narrow aisle while making a series of $r-2$ intermediate stops, provided only $w$ of the robots can fit across the width of the aisle? In this paper, we answer this question by calculating the $r^{\text{th}}$-sequential topological complexity of $\text{conf}(n,w)$, the ordered configuration space of $n$ open unit-diameter disks in the infinite strip of width $w$, as well as its $r^{\text{th}}$-sequential distributional topological complexity. We prove that as long as $n$ is greater than $w$, the $r^{\text{th}}$-sequential (distributional) topological complexity of $\text{conf}(n,w)$ is $r\big(n-\big\lceil\frac{n}{w}\big\rceil\big)$. This shows that any non-looping program moving the $n$ robots between arbitrary initial and final configurations, with $r-2$ intermediate stops, must consider at least $r\big(n-\big\lceil\frac{n}{w}\big\rceil\big)$ cases.

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arXiv2025

Homology Generators and Relations for the Ordered Configuration Space of a Star Graph

We study the ordered configuration spaces of star graphs. Inspired by the representation stability results of Church--Ellenberg--Farb for the ordered configuration space of a manifold and the edge stability results of An--Drummond-Cole--Knudsen for the unordered configuration space of a graph, we determine how the ordered configuration space of a star graph with $k$ leaves behaves as we add particles at the leaves. We show that, as a module over the combinatorial category FI$_{k, o}$, the first homology of this ordered configuration space is finitely generated by $4$ particles for $k=3$, by $3$ particles for $k=4$, and by $2$ particles for $k\ge 5$. Additionally, we prove that every relation among homology classes can be described by relations on at most $6$ particles for $k=4$, at most $5$ particles when $k=5$, at most $4$ particles when $k=6$, and at most $3$ particles for $k\ge 7$, while proving that adding particles always introduces new relations when $k=3$. This proves that there is no finite universal presentation for the homology of ordered configuration spaces of graphs.

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arXiv2025

A discrete model for surface configuration spaces

One of the primary methods of studying the topology of configurations of points in a graph and configurations of disks in a planar region has been to examine discrete combinatorial models arising from the underlying spaces. Despite the success of these models in the graph and disk settings, they have not been constructed for the vast majority of surface configuration spaces. In this paper, we construct such a model for the ordered configuration space of $m$ points in an oriented surface $Σ$. More specifically, we prove that if we give $Σ$ a certain cube complex structure $K$, then the ordered configuration space of $m$ points in $Σ$ is homotopy equivalent to a subcomplex of $K^{m}$

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arXiv2024

The topological complexity of the ordered configuration space of disks in a strip

How hard is it to program $n$ robots to move about a long narrow aisle such that only $w$ of them can fit across the width of the aisle? In this paper, we answer that question by calculating the topological complexity of $\text{conf}(n,w)$, the ordered configuration space of open unit-diameter disks in the infinite strip of width $w$. By studying its cohomology ring, we prove that, as long as $n$ is greater than $w$, the topological complexity of $\text{conf}(n,w)$ is $2n-2\big\lceil\frac{n}{w}\big\rceil+1$, providing a lower bound for the minimum number of cases such a program must consider.

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arXiv2023

Representation Stability for Disks in a Strip

We consider the ordered configuration space of $n$ open unit-diameter disks in the infinite strip of width $w$. In the spirit of Arnol'd and Cohen, we provide a finite presentation for the rational homology groups of this ordered configuration space as a twisted algebra. We use this presentation to prove that the ordered configuration space of open unit-diameter disks in the infinite strip of width $w$ exhibits a notion of first-order representation stability similar to Church--Ellenberg--Farb and Miller--Wilson's first-order representation stability for the ordered configuration space of points in a manifold. In addition, we prove that for large $w$ this disk configuration space exhibits notions of second- (and higher) order representation stability.

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arXiv2022

On the symmetric group action on rigid disks on a strip

In this paper we decompose the rational homology of the ordered configuration space of $p$ open unit-diameter disks on the infinite strip of width $2$ as a direct sum of induced $S_{n}$-representations. Alpert proved that the $k^{\text{th}}$-integral homology of the ordered configuration space of $n$ open unit-diameter disks on the infinite strip of width $2$ is an FI$_{k+1}$-module by studying certain operations on homology called "high-insertion maps." The integral homology groups $H_{k}(\text{cell}(n,2))$ are free abelian, and Alpert computed a basis for $H_{k}(\text{cell}(n,2))$ as an abelian group. In this paper, we study the rational homology groups as $S_{n}$-representations. We find a new basis for $H_{k}(\text{cell}(n,2);\mathbb{Q}),$ and use this, along with results of Ramos, to give an explicit description of $H_{k}(\text{cell}(n,2);\mathbb{Q})$ as a direct sum of induced $S_{n}$-representations arising from free FI$_{*}$-modules. We use this decomposition to calculate the dimension of the rational homology of the unordered configuration space of $p$ open unit-diameter disks on the infinite strip of width $2$.

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arXiv2021

Secondary representation stability and the ordered configuration space of the once-punctured torus

In this paper we study stability patterns in the homology of the ordered configuration space of the once-punctured torus. In the last decade Church and Church-Ellenberg-Farb proved that the homology groups of the ordered configuration space of a connected noncompact orientable manifold stabilize in a representation theoretic sense as the number of points in the configuration grows, with respect to a map that adds each new point "at infinity." Miller and Wilson proved that there is a secondary representation stability pattern among the unstable homology classes, with respect to adding a pair of orbiting points "near infinity." This pattern is formalized by considering sequences of homology classes as FIM$^{+}$-modules. We prove that, as FIM$^{+}$-modules, the sequence of "new" homology generators in the n-th homology of the ordered configuration space of 2n-2 points on the once-punctured torus is neither "free" nor "stably zero." We also show that this sequence is generated by homology classes on at most 4 points. Our proof uses Pagaria's work on the Betti numbers of the ordered configuration space of the torus to calculate the growth rate of the Betti numbers of the ordered configuration space of the once-punctured torus. Our computations are the first to demonstrate that secondary representation stability is a non-trivial phenomenon in positive-genus surfaces.

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arXiv2017

Realization of groups with pairing as Jacobians of finite graphs

We study which groups with pairing can occur as the Jacobian of a finite graph. We provide explicit constructions of graphs whose Jacobian realizes a large fraction of odd groups with a given pairing. Conditional on the generalized Riemann hypothesis, these constructions yield all groups with pairing of odd order, and unconditionally, they yield all groups with pairing whose prime factors are sufficiently large. For groups with pairing of even order, we provide a partial answer to this question, for a certain restricted class of pairings. Finally, we explore which finite abelian groups occur as the Jacobian of a simple graph. There exist infinite families of finite abelian groups that do not occur as the Jacobians of simple graphs.

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arXiv2017

Chip-firing on trees of loops

Cools, Draisma, Payne, and Robeva proved that generic metric graphs that are "paths of loops" are Brill-Noether general. We show that Brill-Noether generality does not hold for "trees of loops": the only trees of loops that are Brill-Noether general are paths of loops. We study various notions of generality and examine which of these graphs satisfy them.

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