For example, if one computes the homology of the Klein bottle (see Section 7.1.1) with coefficients in the field Fp with p elements, where p is a prime, then β0(K)=1 for all primes p. However, β1(K)=2 and β2(K)=1 if p=2, but β1(K)=1 and β2(K)=0 for all ...
6.4 Klein bottle To study our approach for less rich persistence diagrams and to also consider a point set sampled from an object other than an embedded smooth manifold, we ran a second set of experiments on a dataset sampled from an immersion of the Klein bottle in \(\mathbb {R}^3\) ...
The euler characteristic is a first attempt at this and has the advantage of being quite easy to compute, but it fails to distinguish between the torus and the Klein bottle, which both have X = 0. See Figure 6.1.doi:10.1007/978-1-4612-0899-0_6L. Christine Kinsey...
There is a canard that every textbook of algebraic topology either ends with the definition of the Klein bottle or is a personal communication to J. H. C. ... JJ Rotman - PERGAMON 被引量: 707发表: 1957年 A generating function for the Euler characteristic of out(Fn) Let A be a group...
The euler characteristic is a first attempt at this and has the advantage of being quite easy to compute, but it fails to distinguish between the torus and the Klein bottle, which both have X = 0. See Figure 6.1.T. GojoboriNational Institute of Genetics, Shizuoka, Japan...
An Introduction to Algebraic Topology There is a canard that every textbook of algebraic topology either ends with the definition of the Klein bottle or is a personal communication to J. H. C. ... JJ Rotman - PERGAMON 被引量: 707发表: 1957年 ...
sidedsurfacethat is closed, that is, without any one-dimensional boundaries. This example, now called theKlein bottle, cannot exist in three-dimensional space without intersecting itself and, thus, was of interest to mathematicians who previously had considered surfaces only in three-dimensional space...