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0 1 knot/Unknot - a simple un-knotted closed loop; 3 1 knot/Trefoil knot - (2,3)-torus knot, the two loose ends of a common overhand knot joined together; 4 1 knot/Figure-eight knot (mathematics) - a prime knot with a crossing number four
A three-dimensional model of a figure-eight knot.The figure-eight knot is a prime knot and has an Alexander–Briggs notation of 4 1.. Topology (from the Greek words τόπος, 'place, location', and λόγος, 'study') is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling ...
In topology, a branch of mathematics, the loop space ΩX of a pointed topological space X is the space of (based) loops in X, i.e. continuous pointed maps from the pointed circle S 1 to X, equipped with the compact-open topology. Two loops can be multiplied by concatenation. With this operation, the loop space is an A ∞-space.
As well as books and classes, there are YouTube tutorials and TikTok videos to help people who may need a clearer explanation to learn how to crochet. [ 23 ] Filet crochet , Tunisian crochet , tapestry crochet , broomstick lace , hairpin lace , cro-hooking , and Irish crochet are all variants of the basic crochet method.
Thus, a free loop, as opposed to a based loop used in the definition of the fundamental group, is a map from the circle to the space without the basepoint-preserving restriction. Assuming the space is path-connected , free homotopy classes of free loops correspond to conjugacy classes in the fundamental group.
The art of crochet has been used to demonstrate hyperbolic planes (pictured above) with the first being made by Daina Taimiņa, [28] whose book Crocheting Adventures with Hyperbolic Planes won the 2009 Bookseller/Diagram Prize for Oddest Title of the Year. [39] HyperRogue is a roguelike game set on various tilings of the hyperbolic plane.
Constructing the Čech complex of a set of points sampled from a circle. In algebraic topology and topological data analysis, the Čech complex is an abstract simplicial complex constructed from a point cloud in any metric space which is meant to capture topological information about the point cloud or the distribution it is drawn from.
A mapping : between total spaces of two fibrations : and : with the same base space is a fibration homomorphism if the following diagram commutes: . The mapping is a fiber homotopy equivalence if in addition a fibration homomorphism : exists, such that the mappings and are homotopic, by fibration homomorphisms, to the identities and . [2]: 405-406