7 citations · 10 across the 19 of their papers we have counts for
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The complex of discrete Morse matchings of the -simplex: homotopy types and structural results
Nicholas A. Scoville
The complex of discrete Morse matchings $\M(K)$, introduced by Chari and Joswig, is a simplicial complex whose simplices are the acyclic matchings on the Hasse diagram of . Its…
The Face Group of a Simplicial Complex
Gregory Lupton, Nicholas A. Scoville, P. Christopher Staecker
The edge group of a simplicial complex is a well-known, combinatorial version of the fundamental group. It is a group associated to a simplicial complex that consists of equivalenc…
A McCord-type theorem for pseudotopological spaces and directed graphs
Nikola Milićević, Nicholas A. Scoville
We construct weak homotopy equivalences between the geometric realizations of directed Vietoris-Rips complexes and their underlying directed graphs, seen as pseudotopological space…
Discrete Morse theory for open complexes
Kevin P. Knudson, Nicholas A. Scoville
We develop a discrete Morse theory for open simplicial complexes where is a simplicial complex and a subcomplex of . A discrete Morse function on $K…
A Second Homotopy Group for Digital Images
Gregory Lupton, Oleg Musin, Nicholas A. Scoville +2
We define a second (higher) homotopy group for digital images. Namely, we construct a functor from digital images to abelian groups, which closely resembles the ordinary second hom…
On cycles and merge trees
Julian Brüggemann, Nicholas A. Scoville
In this paper, we extend the notion of a merge tree to that of a generalized merge tree, a merge tree that includes 1-dimensional cycle birth information. Given a discrete Morse fu…
Star clusters in the Matching, Morse, and Generalized Morse complex
Connor Donovan, Nicholas A. Scoville
In this paper, we determine the homotopy type of the Morse complex and matching complex of multiple families of complexes by utilizing star cluster collapses and the Cluster Lemma.…
Merge trees in discrete Morse theory
Benjamin Johnson, Nicholas A. Scoville
In this paper, we study merge trees induced by a discrete Morse function on a tree. Given a discrete Morse function, we provide a method to constructing an induced merge tree and d…
The Digital Hopf Construction
Gregory Lupton, John Oprea, Nicholas A. Scoville
Various concepts and constructions in homotopy theory have been defined in the digital setting. Although there have been several attempts at a definition of a fibration in the digi…
Digital Fundamental Groups and Edge Groups of Clique Complexes
Gregory Lupton, Nicholas A. Scoville
In previous work, we have defined---intrinsically, entirely within the digital setting---a fundamental group for digital images. Here, we show that this group is isomorphic to the…
On the homotopy and strong homotopy type of complexes of discrete Morse functions
Connor Donovan, Maxwell Lin, Nicholas A. Scoville
In this paper, we determine the homotopy type of the Morse complex of certain collections of simplicial complexes by studying dominating vertices or strong collapses. We show that…
A Fundamental Group for Digital Images
Gregory Lupton, John Oprea, Nicholas Scoville
We define a fundamental group for digital images. Namely, we construct a functor from digital images to groups, which closely resembles the ordinary fundamental group from algebrai…
Subdivision of Maps of Digital Images
Gregory Lupton, John Oprea, Nicholas A. Scoville
With a view towards providing tools for analyzing and understanding digitized images, various notions from algebraic topology have been introduced into the setting of digital topol…
Homotopy Theory in Digital Topology
Gregory Lupton, John Oprea, Nicholas Scoville
Digital topology is part of the ongoing endeavour to understand and analyze digitized images. With a view to supporting this endeavour, many notions from algebraic topology have be…
On the automorphism group of the Morse complex
Maxwell Lin, Nicholas A. Scoville
Let be a finite, connected, abstract simplicial complex. The Morse complex of , first introduced by Chari and Joswig, is the simplicial complex constructed from all gradient…
The realization problem for discrete Morse functions on trees
Yuqing Liu, Nicholas A. Scoville
We introduce a new notion of equivalence of discrete Morse functions on graphs called persistence equivalence. Two functions are considered persistence equivalent if and only if th…
Strong discrete Morse theory and simplicial L-S category: A discrete version of the Lusternik-Schnirelmann Theorem
Desamparados Fernández-Ternero, Enrique Macías-Virgós, Nicholas A. Scoville +1
We prove a discrete version of the Lusternik-Schnirelmann theorem for discrete Morse functions and the recently introduced simplicial Lusternik-Schnirelmann category of a simplicia…
Cosheaf Theoretical Constructions in Networks and Persistent Homology
Nicholas A. Scoville, Karthik Yegnesh
Persistent homology has recently emerged as a powerful technique in topological data analysis for analyzing the emergence and disappearance of topological features throughout a fil…
On the Lusternik-Schnirelmann category of a simplicial map
Nicholas A. Scoville, Willie Swei
In this paper, we study the Lusternik-Schnirelmann category of a simplicial map between simplicial complexes, generalizing the simplicial category of a complex to that of a map. Se…
Categorical sequences
Rob Nendorf, Nick Scoville, Jeffrey Strom
We define and study the categorical sequence of a space, which is a new formalism that streamlines the computation of the Lusternik-Schnirelmann category of a space X by induction…