Rips Complexes of Planar Point Sets

Fix a finite set of points in Euclidean $n$-space \euc^n, thought of as a point-cloud sampling of a certain domain D\subset\euc^n. The Rips complex is a combinatorial simplicial complex based on proximity of neighbors that serves as an easily-computed but high-dimensional approximation to the homotopy type of $D$. There is a natural ``shadow'' projection map from the Rips complex to \euc^n that has as its image a more accurate $n$-dimensional approximation to the homotopy type of $D$. We demonstrate that this projection map is 1-connected for the planar case $n=2$. That is, for planar domains, the Rips complex accurately captures connectivity and fundamental group data. This implies that the fundamental group of a Rips complex for a planar point set is a free group. We show that, in contrast, introducing even a small amount of uncertainty in proximity detection leads to `quasi'-Rips complexes with nearly arbitrary fundamental groups. This topological noise can be mitigated by examining a pair of quasi-Rips complexes and using ideas from persistent topology. Finally, we show that the projection map does not preserve higher-order topological data for planar sets, nor does it preserve fundamental group data for point sets in dimension larger than three.Comment: 16 pages, 8 figure

Optimally fast incremental Manhattan plane embedding and planar tight span construction

We describe a data structure, a rectangular complex, that can be used to represent hyperconvex metric spaces that have the same topology (although not necessarily the same distance function) as subsets of the plane. We show how to use this data structure to construct the tight span of a metric space given as an n x n distance matrix, when the tight span is homeomorphic to a subset of the plane, in time O(n^2), and to add a single point to a planar tight span in time O(n). As an application of this construction, we show how to test whether a given finite metric space embeds isometrically into the Manhattan plane in time O(n^2), and add a single point to the space and re-test whether it has such an embedding in time O(n).Comment: 39 pages, 15 figure

Distributed Coverage Verification in Sensor Networks Without Location Information

In this paper, we present three distributed algorithms for coverage verification in sensor networks with no location information. We demonstrate how, in the absence of localization devices, simplicial complexes and tools from algebraic topology can be used in providing valuable information about the properties of the cover. Our approach is based on computation of homologies of the Rips complex corresponding to the sensor network. First, we present a decentralized scheme based on Laplacian flows to compute a generator of the first homology, which represents coverage holes. Then, we formulate the problem of localizing coverage holes as an optimization problem for computing a sparse generator of the first homology. Furthermore, we show that one can detect redundancies in the sensor network by finding a sparse generator of the second homology of the cover relative to its boundary. We demonstrate how subgradient methods can be used in solving these optimization problems in a distributed manner. Finally, we provide simulations that illustrate the performance of our algorithms

Some problems in combinatorial topology of flag complexes

In this work we study simplicial complexes associated to graphs and their homotopical and combinatorial properties. The main focus is on the family of flag complexes, which can be viewed as independence complexes and clique complexes of graphs. In the first part we study independence complexes of graphs using two cofibre sequences corresponding to vertex and edge removals. We give applications to the connectivity of independence complexes of chordal graphs and to extremal problems in topology and we answer open questions about the homotopy types of those spaces for particular families of graphs. We also study the independence complex as a space of configurations of particles in the so-called hard-core models on various lattices. We define, and investigate from an algorithmic perspective, a special family of combinatorially defined homology classes in independence complexes. This enables us to give algorithms as well as NP-hardness results for topological properties of some spaces. As a corollary we prove hardness of computing homology of simplicial complexes in general. We also view flag complexes as clique complexes of graphs. That leads to the study of various properties of Vietoris-Rips complexes of graphs. The last result is inspired by a problem in face enumeration. Using methods of extremal graph theory we classify flag triangulations of 3-manifolds with many edges. As a corollary we complete the classification of face vectors of flag simplicial homology 3-spheres

TOPOLOGICAL STRUCTURE OF SPATIALLY-DISTRIBUTED NETWORK CODED INFORMATION

In this paper we generalize work using topological methods for testing wireless/sensor network coverage to the problem of covering a geographically-distributed wireless network with linear network coded data. We define the coverage complex, a new type of simplicial complex built on the nodes of the network which captures properties of the data coverage, and use tools from algebraic topology, persistent homology, and matroid theory to study it. The coverage complex shares properties with the Rips complex, however it also suffers from a more diverse variety of potential failures. We extend the standard coverage criteria to account for some of these situations using persistent homology, multi-sheeted localized covers of the space, and Mayer-Vietoris sequences. We also investigate the combinatorial properties of the coverage complex, determining the correspondence between it and the lattice of linear subspaces of a vector space. Finally we present algorithms for computing coverage complexes, present a software package designed to compute and experiment with coverage complexes, and provide a summary of ongoing and future work

Psr1p interacts with SUN/sad1p and EB1/mal3p to establish the bipolar spindle

Regular Abstracts - Sunday Poster Presentations: no. 382During mitosis, interpolar microtubules from two spindle pole bodies (SPBs) interdigitate to create an antiparallel microtubule array for accommodating numerous regulatory proteins. Among these proteins, the kinesin-5 cut7p/Eg5 is the key player responsible for sliding apart antiparallel microtubules and thus helps in establishing the bipolar spindle. At the onset of mitosis, two SPBs are adjacent to one another with most microtubules running nearly parallel toward the nuclear envelope, creating an unfavorable microtubule configuration for the kinesin-5 kinesins. Therefore, how the cell organizes the antiparallel microtubule array in the first place at mitotic onset remains enigmatic. Here, we show that a novel protein psrp1p localizes to the SPB and plays a key role in organizing the antiparallel microtubule array. The absence of psr1+ leads to a transient monopolar spindle and massive chromosome loss. Further functional characterization demonstrates that psr1p is recruited to the SPB through interaction with the conserved SUN protein sad1p and that psr1p physically interacts with the conserved microtubule plus tip protein mal3p/EB1. These results suggest a model that psr1p serves as a linking protein between sad1p/SUN and mal3p/EB1 to allow microtubule plus ends to be coupled to the SPBs for organization of an antiparallel microtubule array. Thus, we conclude that psr1p is involved in organizing the antiparallel microtubule array in the first place at mitosis onset by interaction with SUN/sad1p and EB1/mal3p, thereby establishing the bipolar spindle.postprin

Removal of antagonistic spindle forces can rescue metaphase spindle length and reduce chromosome segregation defects

Regular Abstracts - Tuesday Poster Presentations: no. 1925Metaphase describes a phase of mitosis where chromosomes are attached and oriented on the bipolar spindle for subsequent segregation at anaphase. In diverse cell types, the metaphase spindle is maintained at a relatively constant length. Metaphase spindle length is proposed to be regulated by a balance of pushing and pulling forces generated by distinct sets of spindle microtubules and their interactions with motors and microtubule-associated proteins (MAPs). Spindle length appears important for chromosome segregation fidelity, as cells with shorter or longer than normal metaphase spindles, generated through deletion or inhibition of individual mitotic motors or MAPs, showed chromosome segregation defects. To test the force balance model of spindle length control and its effect on chromosome segregation, we applied fast microfluidic temperature-control with live-cell imaging to monitor the effect of switching off different combinations of antagonistic forces in the fission yeast metaphase spindle. We show that spindle midzone proteins kinesin-5 cut7p and microtubule bundler ase1p contribute to outward pushing forces, and spindle kinetochore proteins kinesin-8 klp5/6p and dam1p contribute to inward pulling forces. Removing these proteins individually led to aberrant metaphase spindle length and chromosome segregation defects. Removing these proteins in antagonistic combination rescued the defective spindle length and, in some combinations, also partially rescued chromosome segregation defects. Our results stress the importance of proper chromosome-to-microtubule attachment over spindle length regulation for proper chromosome segregation.postprin