scholarly journals Injectively immersed tori in branched covers over the figure eight knot

1993 ◽  
Vol 36 (2) ◽  
pp. 231-259
Author(s):  
Kerry N. Jones
Keyword(s):  
Fundamental Group ◽  
Decomposition Theorem ◽  
The Other ◽  
Group Level ◽  
Branched Covers ◽  
Graph Theoretic ◽  
Branched Cover ◽  
Figure Eight Knot

An algorithm is given for determining presence or absence of injectively (at the fundamental group level) immersed tori (and constructing them, if present) in a branched cover of S3, branched over the figure eight knot, with all branching indices greater than 2. Such tori are important for understanding the topology of 3-manifolds in light of (for example) the Jaco-Shalen–Johannson torus decomposition theorem and the fact that the figure eight knot is universal, i.e., that all 3-manifolds are representable as branched covers of S3, branched over the figure eight knot.The algorithm is principally geometric in its derivation and graph-theoretic in its operation. It is applied to two examples, one of which has an incompressible torus and the other of which is atoroidal.

scholarly journals On left-orderability and double branched covers of Kanenobu’s knots

2015 ◽  
Vol 24 (14) ◽  
pp. 1550071
Author(s):  
Fabian Doria Medina ◽  
Michael Jackson ◽  
Joaquín Ruales ◽  
Hadas Zeilberger
Keyword(s):  
Fundamental Group ◽  
Infinite Family ◽  
Branched Covers ◽  
Rational Homology ◽  
Branched Cover

We show that the fundamental group of the double branched cover of an infinite family of homologically thin, non-quasi-alternating knots is not left-orderable, giving further support for a conjecture of Boyer, Gordon, and Watson that an irreducible rational homology 3-sphere is an L-space if and only if its fundamental group is not left-orderable.

scholarly journals Integrative statistical analyses of multiple liquid biopsy analytes in metastatic breast cancer

2021 ◽  
Vol 13 (1) ◽  
Author(s):  
Corinna Keup ◽  
Vinay Suryaprakash ◽  
Siegfried Hauch ◽  
Markus Storbeck ◽  
Peter Hahn ◽  
...  
Keyword(s):  
Breast Cancer ◽  
Overall Survival ◽  
Metastatic Breast Cancer ◽  
Mutual Information ◽  
Hierarchical Clustering ◽  
Liquid Biopsy ◽  
Metastatic Breast ◽  
The Other ◽  
Theoretic Analysis ◽  
Graph Theoretic

Abstract Background Single liquid biopsy analytes (LBAs) have been utilized for therapy selection in metastatic breast cancer (MBC). We performed integrative statistical analyses to examine the clinical relevance of using multiple LBAs: matched circulating tumor cell (CTC) mRNA, CTC genomic DNA (gDNA), extracellular vesicle (EV) mRNA, and cell-free DNA (cfDNA). Methods Blood was drawn from 26 hormone receptor-positive, HER2-negative MBC patients. CTC mRNA and EV mRNA were analyzed using a multi-marker qPCR. Plasma from CTC-depleted blood was utilized for cfDNA isolation. gDNA from CTCs was isolated from mRNA-depleted CTC lysates. CTC gDNA and cfDNA were analyzed by targeted sequencing. Hierarchical clustering was performed within each analyte, and its results were combined into a score termed Evaluation of multiple Liquid biopsy analytes In Metastatic breast cancer patients All from one blood sample (ELIMA.score), which calculates the contribution of each analyte to the overall survival prediction. Singular value decomposition (SVD), mutual information calculation, k-means clustering, and graph-theoretic analysis were conducted to elucidate the dependence between individual analytes. Results A combination of two/three/four LBAs increased the prevalence of patients with actionable signals. Aggregating the results of hierarchical clustering of individual LBAs into the ELIMA.score resulted in a highly significant correlation with overall survival, thereby bolstering evidence for the additive value of using multiple LBAs. Computation of mutual information indicated that none of the LBAs is independent of the others, but the ability of a single LBA to describe the others is rather limited—only CTC gDNA could partially describe the other three LBAs. SVD revealed that the strongest singular vectors originate from all four LBAs, but a majority originated from CTC gDNA. After k-means clustering of patients based on parameters of all four LBAs, the graph-theoretic analysis revealed CTC ERBB2 variants only in patients belonging to one particular cluster. Conclusions The additional benefits of using all four LBAs were objectively demonstrated in this pilot study, which also indicated a relative dominance of CTC gDNA over the other LBAs. Consequently, a multi-parametric liquid biopsy approach deconvolutes the genomic and transcriptomic complexity and should be considered in clinical practice.

scholarly journals COMBINATORIAL KNOT FLOER HOMOLOGY AND DOUBLE BRANCHED COVERS

2013 ◽  
Vol 22 (06) ◽  
pp. 1350014
Author(s):  
FATEMEH DOUROUDIAN
Keyword(s):  
Floer Homology ◽  
Combinatorial Proof ◽  
Branched Covers ◽  
Knot Floer Homology ◽  
Branched Cover ◽  
Heegaard Diagram

Using a Heegaard diagram for the pullback of a knot K ⊂ S3 in its double branched cover Σ2(K), we give a combinatorial proof for the invariance of the associated knot Floer homology over ℤ.

scholarly journals Accepting the truth: A step toward reconciliation

Temida ◽  
2002 ◽  
Vol 5 (4) ◽  
pp. 15-18
Author(s):  
Zoran Ilic
Keyword(s):  
National Identity ◽  
Defense Mechanism ◽  
Psychological Explanation ◽  
The Other ◽  
Group Level ◽  
Moral Norms ◽  
Extreme Form ◽  
Individual Level ◽  
Psychological Mechanisms

The truth is not only about accepting the facts. The truth includes emotional components - longing for acknowledgment of mistakes and validation of painful losses and experiences. While one side in the conflict considers itself as the only victim and experiences determination and acknowledgment of the truth as a possibility for healing its own trauma through satisfaction of justice and compensation, the other side is not accepting the truth. For this other side, confrontation with the facts is a painful trauma, which endangers individual moral norms, threatens the national identity and requires, on individual level, plugging in the defense mechanism, in order to prevent penetration of painful emotions into the consciousness. Primitive psychological mechanisms of defense, focused around splitting (projection, denial), are the main obstacles in accepting the truth both on individual and group level. There is also another extreme form of reaction manifested through "hypertrophy" of the truth and one-sided self-accusation. In this paper the author presents psychological explanation of the process of accepting the truth as the prerequisite for reconciliation and transformation of the conflict.

scholarly journals A Lemma on Continuous Functions

1960 ◽  
Vol 3 (2) ◽  
pp. 186-187
Author(s):  
J. Lipman
Keyword(s):  
Topological Space ◽  
Fundamental Group ◽  
Closed Interval ◽  
Continuous Functions ◽  
The Other ◽  
Special Consideration ◽  
Boundary Lines

The point of this note is to get a lemma which is useful in treating homotopy between paths in a topological space [1].As explained in the reference, two paths joining a given pair of points in a space E are homotopic if there exists a mapping F: I x I →E (I being the closed interval [0,1] ) which deforms one path continuously into the other. In practice, when two paths are homotopic and the mapping F is constructed, then the verification of all its required properties, with the possible exception of continuity, is trivial. The snag occurs when F is a combination of two or three functions on different subsets of I x I. Then the boundary lines between these subsets have to be given special consideration, and although the problems resulting are routine their disposal can involve some tedious calculation and repetition. In the development [l] of the fundamental group of a space, for example, this sort of situation comes up four or five times.

Motivation and the Human Monitor: I. The Effect of Contingent Credit

1969 ◽  
Vol 28 (2) ◽  
pp. 623-629 ◽  
Author(s):  
Charles G. Halcomb ◽  
Peggy Blackwell
Keyword(s):  
The Other ◽  
Group Level ◽  
Visual Monitoring ◽  
Monitoring Task ◽  
College Population ◽  
Improved Performance ◽  
High Level ◽  
Over Time ◽  
Course Credit

This research was designed to test the hypothesis that relevant incentives would result in improved performance on a visual monitoring task. Course credit was used as an incentive due to its apparent relevance for the college population. Two groups of Ss were employed. One group received credit made contingent on performance; the other group received credit for participation. The contingent group performed at a higher level than did the non-contingent group. Level of performance for both groups was high, suggesting that a relevant incentive can be effective in maintaining a high level of performance over time.

A decomposition theorem for locally compact groups

2019 ◽  
Vol 26 (1) ◽  
pp. 29-33
Author(s):  
Sanjib Basu ◽  
Krishnendu Dutta
Keyword(s):  
Compact Group ◽  
Lebesgue Point ◽  
Decomposition Theorem ◽  
Locally Compact Group ◽  
The Other ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Topological Groups ◽  
Locally Compact ◽  
Interesting Connection

Abstract We prove that, under certain restrictions, every locally compact group equipped with a nonzero, σ-finite, regular left Haar measure can be decomposed into two small sets, one of which is small in the sense of measure and the other is small in the sense of category, and all such decompositions originate from a generalised notion of a Lebesgue point. Incidentally, such class of topological groups for which this happens turns out to be metrisable. We also observe an interesting connection between Luzin sets in such spaces and decompositions of the above type.

PREDICTIVE DECOMPOSITION OF SEISMIC TRACES

Geophysics ◽  
1957 ◽  
Vol 22 (4) ◽  
pp. 767-778 ◽  
Author(s):  
Enders A. Robinson
Keyword(s):  
Time Series ◽  
Regression Analysis ◽  
Spectral Decomposition ◽  
Past History ◽  
Decomposition Theorem ◽  
Time Sequence ◽  
The Other ◽  
Volume Control ◽  
Minimum Phase ◽  
Arrival Times

The generalized harmonic analysis, or spectral decomposition, of a time series results in its representation in terms of its harmonic, or sinusoidal, components. This paper, on the other hand, develops in an expository manner the generalized regression analysis, or predictive decomposition, of a time series. This decomposition results in the representation of the time series at any moment in terms of its own observable past history plus an unpredictable, random‐like innovation. For the purposes of this paper, it is assumed that a seismic trace (recorded with automatic volume control) is additively composed of many overlapping seismic wavelets which arrive as time progresses. It is assumed that each wavelet has the same stable, one‐sided, minimum‐phase shape and that the arrival times and strengths of these wavelets may be represented by a time sequence of uncorrelated random variables. By applying the predictive decomposition theorem, it is shown how the wavelet shape may be extracted from the trace, leaving as a residual the strengths of the wavelets at their respective arrival times.

scholarly journals Parameterizations of 1-Bridge Torus Knots

2003 ◽  
Vol 12 (04) ◽  
pp. 463-491 ◽  
Author(s):  
Doo Ho Choi ◽  
Ki Hyoung Ko
Keyword(s):  
Normal Form ◽  
Fundamental Group ◽  
Explicit Formula ◽  
Normal Forms ◽  
Double Cover ◽  
Torus Knots ◽  
Number Of Components ◽  
Torus Knot ◽  
Branched Cover ◽  
The Family

A 1-bridge torus knot in a 3-manifold of genus ≤ 1 is a knot drawn on a Heegaard torus with one bridge. We give two types of normal forms to parameterize the family of 1-bridge torus knots that are similar to the Schubert's normal form and the Conway's normal form for 2-bridge knots. For a given Schubert's normal form we give algorithms to determine the number of components and to compute the fundamental group of the complement when the normal form determines a knot. We also give a description of the double branched cover of an ambient 3-manifold branched along a 1-bridge torus knot by using its Conway's normal form and obtain an explicit formula for the first homology of the double cover.

ALEXANDER POLYNOMIALS AND ORDERS OF HOMOLOGY GROUPS OF BRANCHED COVERS OF KNOTS

2009 ◽  
Vol 18 (07) ◽  
pp. 973-984 ◽  
Author(s):  
SE-GOO KIM
Keyword(s):  
Prime Power ◽  
Alexander Polynomial ◽  
Branched Covers ◽  
Splitting Property ◽  
Homology Groups ◽  
Alexander Polynomials ◽  
Knot Concordance ◽  
Branched Cover ◽  
Linearly Independent

Fox showed that the order of homology of a cyclic branched cover of a knot is determined by its Alexander polynomial. We find examples of knots with relatively prime Alexander polynomials such that the first homology groups of their q-fold cyclic branched covers are of the same order for every prime power q. Furthermore, we show that these knots are linearly independent in the knot concordance group using the polynomial splitting property of the Casson–Gordon–Gilmer invariants.

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