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A chain homotopy offers a way to relate two chain maps that induce the same map on homology groups, even though the maps may be different. Given two chain complexes A and B, and two chain maps f, g : A → B, a chain homotopy is a sequence of homomorphisms h n : A n → B n+1 such that hd A + d B h = f − g. The maps may be written out in a ...
The boundary of a chain is the linear combination of boundaries of the simplices in the chain. The boundary of a k-chain is a (k−1)-chain. Note that the boundary of a simplex is not a simplex, but a chain with coefficients 1 or −1 – thus chains are the closure of simplices under the boundary operator.
A centroid of a tree is a vertex v such that if rooted at v, no other vertex has subtree size greater than half the size of the tree. chain 1. Synonym for walk. 2. When applying methods from algebraic topology to graphs, an element of a chain complex, namely a set of vertices or a set of edges. Cheeger constant See expansion. cherry
Frequently the word link is used to describe any submanifold of the sphere diffeomorphic to a disjoint union of a finite number of spheres, .. In full generality, the word link is essentially the same as the word knot – the context is that one has a submanifold M of a manifold N (considered to be trivially embedded) and a non-trivial embedding of M in N, non-trivial in the sense that the 2nd ...
A reduced diagram is a knot diagram in which there are no reducible crossings (also nugatory or removable crossings), or in which all of the reducible crossings have been removed. [ 3 ] [ 4 ] A petal projection is a type of projection in which, instead of forming double points, all strands of the knot meet at a single crossing point, connected ...
Faces are reduced to half as many sides, and square faces degenerate into edges. For example, the tetrahedron is an alternated cube, h{4,3}. Diminishment is a more general term used in reference to Johnson solids for the removal of one or more vertices, edges, or faces of a polytope, without disturbing the other vertices.
Topological geometry deals with incidence structures consisting of a point set and a family of subsets of called lines or circles etc. such that both and carry a topology and all geometric operations like joining points by a line or intersecting lines are continuous.
A standard definition of an ellipse is the set of points for which the sum of a point's distances to two foci is a constant; if this constant equals the distance between the foci, the line segment is the result. A complete orbit of this ellipse traverses the line segment twice. As a degenerate orbit, this is a radial elliptic trajectory.