arXiv subjects

Mustafa Hajij

Publications and source records attributed to Mustafa Hajij.

At least 55 records · Page 3Linked to original sources

arXiv2020

TopoLines: Topological Smoothing for Line Charts

Line charts are commonly used to visualize a series of data values. When the data are noisy, smoothing is applied to make the signal more apparent. Conventional methods used to smooth line charts, e.g., using subsampling or filters, such as median, Gaussian, or low-pass, each optimize for different properties of the data. The properties generally do not include retaining peaks (i.e., local minima and maxima) in the data, which is an important feature for certain visual analytics tasks. We present TopoLines, a method for smoothing line charts using techniques from Topological Data Analysis. The design goal of TopoLines is to maintain prominent peaks in the data while minimizing any residual error. We evaluate TopoLines for 2 visual analytics tasks by comparing to 5 popular line smoothing methods with data from 4 application domains.

cs.HC↗

arXiv2019

Big Data Approaches to Knot Theory: Understanding the Structure of the Jones Polynomial

We examine the structure and dimensionality of the Jones polynomial using manifold learning techniques. Our data set consists of more than 10 million knots up to 17 crossings and two other special families up to 2001 crossings. We introduce and describe a method for using filtrations to analyze infinite data sets where representative sampling is impossible or impractical, an essential requirement for working with knots and the data from knot invariants. In particular, this method provides a new approach for analyzing knot invariants using Principal Component Analysis. Using this approach on the Jones polynomial data we find that it can be viewed as an approximately 3 dimensional manifold, that this description is surprisingly stable with respect to the filtration by the crossing number, and that the results suggest further structures to be examined and understood.

math.GT↗

arXiv2019

Persistent Homology Guided Force-Directed Graph Layouts

Graphs are commonly used to encode relationships among entities, yet their abstractness makes them difficult to analyze. Node-link diagrams are popular for drawing graphs, and force-directed layouts provide a flexible method for node arrangements that use local relationships in an attempt to reveal the global shape of the graph. However, clutter and overlap of unrelated structures can lead to confusing graph visualizations. This paper leverages the persistent homology features of an undirected graph as derived information for interactive manipulation of force-directed layouts. We first discuss how to efficiently extract 0-dimensional persistent homology features from both weighted and unweighted undirected graphs. We then introduce the interactive persistence barcode used to manipulate the force-directed graph layout. In particular, the user adds and removes contracting and repulsing forces generated by the persistent homology features, eventually selecting the set of persistent homology features that most improve the layout. Finally, we demonstrate the utility of our approach across a variety of synthetic and real datasets.

cs.GR↗

arXiv2019

$Q$-Series and Quantum Spin Networks

The tail of a quantum spin network in the two-sphere is a $q$-series associated to the network. We study the existence of the head and tail functions of quantum spin networks colored by $2n$. We compute the $q$-series for an infinite family of quantum spin networks and give the relation between the tail of these networks and the tail of the colored Jones polynomial. Finally, we show that the family of quantum spin networks under study satisfies a natural product structure, making these networks satisfy a natural product structure.

math.GT↗

arXiv2019

Propagate and Pair: A Single-Pass Approach to Critical Point Pairing in Reeb Graphs

With the popularization of Topological Data Analysis, the Reeb graph has found new applications as a summarization technique in the analysis and visualization of large and complex data, whose usefulness extends beyond just the graph itself. Pairing critical points enables forming topological fingerprints, known as persistence diagrams, that provides insights into the structure and noise in data. Although the body of work addressing the efficient calculation of Reeb graphs is large, the literature on pairing is limited. In this paper, we discuss two algorithmic approaches for pairing critical points in Reeb graphs, first a multipass approach, followed by a new single-pass algorithm, called Propagate and Pair.

cs.CG↗

arXiv2019

Mesh Learning Using Persistent Homology on the Laplacian Eigenfunctions

We use persistent homology along with the eigenfunctions of the Laplacian to study similarity amongst triangulated 2-manifolds. Our method relies on studying the lower-star filtration induced by the eigenfunctions of the Laplacian. This gives us a shape descriptor that inherits the rich information encoded in the eigenfunctions of the Laplacian. Moreover, the similarity between these descriptors can be easily computed using tools that are readily available in Topological Data Analysis. We provide experiments to illustrate the effectiveness of the proposed method.

stat.ML↗

arXiv2018

Inferring Quality in Point Cloud-based 3D Printed Objects using Topological Data Analysis

Assessing the quality of 3D printed models before they are printed remains a challeng- ing problem, particularly when considering point cloud-based models. This paper introduces an approach to quality assessment, which uses techniques from the field of Topological Data Analy- sis (TDA) to compute a topological abstraction of the eventual printed model. Two main tools of TDA, Mapper and persistent homology, are used to analyze both the printed space and empty space created by the model. This abstraction enables investigating certain qualities of the model, with respect to print quality, and identifies potential anomalies that may appear in the final product.

cs.GR↗

arXiv2018

Generating sets of Reidemeister moves of oriented singular links and quandles

We give a generating set of the generalized Reidemeister moves for oriented singular links. We use it to introduce an algebraic structure arising from the study of oriented singular knots. We give some examples, including some non-isomorphic families of such structures over non-abelian groups. We show that the set of colorings of a singular knot by this new structure is an invariant of oriented singular knots and use it to distinguish some singular links.

math.GT↗

arXiv2018

On Rational Knots and Links in the Solid Torus

We introduce the notion of rational links in the solid torus. We show that rational links in the solid torus are fully characterized by rational tangles, and hence by the continued fraction of the rational tangle. Furthermore, we generalize this by giving an infinite family of ambient isotopy invariants of colored diagrams in the Kauffman bracket skein module of an oriented surface.

math.GT↗

arXiv2018

An Efficient Algorithm to Compute the Colored Jones Polynomial

The colored Jones polynomial is a knot invariant that plays a central role in low dimensional topology. We give a simple and an efficient algorithm to compute the colored Jones polynomial of any knot. Our algorithm utilizes the walks along a braid model of the colored Jones polynomial that was refined by Armond from the work of Huynh and Lê. The walk model gives rise to ordered words in a $q$-Weyl algebra which we address and study from multiple perspectives. We provide a highly optimized Mathematica implementation that exploits the modern features of the software. We include a performance analysis for the running time of our algorithm. Our implementation of the algorithm shows that our method usually runs in faster time than the existing state-of the-art method by an order of magnitude.

math.QA↗

arXiv2018

Graph Based Analysis for Gene Segment Organization In a Scrambled Genome

DNA rearrangement processes recombine gene segments that are organized on the chromosome in a variety of ways. The segments can overlap, interleave or one may be a subsegment of another. We use directed graphs to represent segment organizations on a given locus where contigs containing rearranged segments represent vertices and the edges correspond to the segment relationships. Using graph properties we associate a point in a higher dimensional Euclidean space to each graph such that cluster formations and analysis can be performed with methods from topological data analysis. The method is applied to a recently sequenced model organism \textit{Oxytricha trifallax}, a species of ciliate with highly scrambled genome that undergoes massive rearrangement process after conjugation. The analysis shows some emerging star-like graph structures indicating that segments of a single gene can interleave, or even contain all of the segments from fifteen or more other genes in between its segments. We also observe that as many as six genes can have their segments mutually interleaving or overlapping.

stat.ML↗

arXiv2017

The Shape of an Image: A Study of Mapper on Images

We study the topological construction called Mapper in the context of simply connected domains, in particular on images. The Mapper construction can be considered as a generalization for contour, split, and joint trees on simply connected domains. A contour tree on an image domain assumes the height function to be a piecewise linear Morse function. This is a rather restrictive class of functions and does not allow us to explore the topology for most real world images. The Mapper construction avoids this limitation by assuming only continuity on the height function allowing this construction to robustly deal with a significant larger set of images. We provide a customized construction for Mapper on images, give a fast algorithm to compute it, and show how to simplify the Mapper structure in this case. Finally, we provide a simple procedure that guarantees the equivalence of Mapper to contour, join, and split trees on a simply connected domain.

cs.CV↗

arXiv2017

Visual Detection of Structural Changes in Time-Varying Graphs Using Persistent Homology

Topological data analysis is an emerging area in exploratory data analysis and data mining. Its main tool, persistent homology, has become a popular technique to study the structure of complex, high-dimensional data. In this paper, we propose a novel method using persistent homology to quantify structural changes in time-varying graphs. Specifically, we transform each instance of the time-varying graph into metric spaces, extract topological features using persistent homology, and compare those features over time. We provide a visualization that assists in time-varying graph exploration and helps to identify patterns of behavior within the data. To validate our approach, we conduct several case studies on real world data sets and show how our method can find cyclic patterns, deviations from those patterns, and one-time events in time-varying graphs. We also examine whether persistence-based similarity measure as a graph metric satisfies a set of well-established, desirable properties for graph metrics.

cs.GR↗

arXiv2017

Foundations of the Colored Jones Polynomial of singular knots

This article gives the foundations of the colored Jones polynomial for singular knots. We extend Masbum and Vogel's algorithm to compute the colored Jones polynomial for any singular knot. We also introduce the tail of the colored Jones polynomial of singular knots and use its stability properties to prove a false theta function identity that goes back to Ramanujan.

math.GT↗

arXiv2017

The Bubble Skein Element and Applications

We study a certain skein element in the relative Kauffman bracket skein module of the disk with some marked points, and expand this element in terms linearly independent elements of this module. This expansion is used to compute and study the head and the tail of the colored Jones polynomial and in particular we give a simple $q-$series for the tail of the knot $8_5$. Furthermore, we use this expansion to obtain an easy determination of the theta coefficients.

math.GT↗

arXiv2017

Twist Regions and Coefficients Stability of the Colored Jones Polynomial

We prove that the coefficients of the colored Jones polynomial of alternating links stabilize under increasing the number of twists in the twist regions of the link diagram. This gives us an infinite family of $q$-power series derived from the colored Jones polynomial parametrized by the color and the twist regions of the alternating link diagram.

math.GT↗

arXiv2016

The colored Kauffman skein relation and the head and tail of the colored Jones polynomial

Using the colored Kauffman skein relation, we study the highest and lowest $4n$ coefficients of the $n^{th}$ unreduced colored Jones polynomial of alternating links. This gives a natural extension of a result by Kauffman in regard with the Jones polynomial of alternating links and its highest and lowest coefficients. We also use our techniques to give a new and natural proof for the existence of the tail of the colored Jones polynomial for alternating links.

math.GT↗

arXiv2016

Segmenting a Surface Mesh into Pants Using Morse Theory

A pair of pants is a genus zero orientable surface with three boundary components. A pants decomposition of a surface is a finite collection of unordered pairwise disjoint simple closed curves embedded in the surface that decompose the surface into pants. In this paper we present two Morse theory based algorithms for pants decomposition of a surface mesh. Both algorithms operates on a choice of an appropriate Morse function on the surface. The first algorithm uses this Morse function to identify handles that are glued systematically to obtain a pant decomposition. The second algorithm uses the Reeb graph of the Morse function to obtain a pant decomposition. Both algorithms work for surfaces with or without boundaries. Our preliminary implementation of the two algorithms shows that both algorithms run in much less time than an existing state-of-the-art method, and the Reeb graph based algorithm achieves the best time efficiency. Finally, we demonstrate the robustness of our algorithms against noise.

cs.GR↗