Introduction to Knot Theory: Basic Concepts of Knots, Links, and Braids in Algebraic Topology
By
marysminefnuf
Crisp on the outside, thoughtful on the inside. A keeper.
Summary
This article provides an introduction to algebraic topology concepts, specifically focusing on knots, links, and braids. It defines a knot as a simple closed curve in Euclidean 3-space and discusses knot equivalence through orientation-preserving homeomorphisms. The content covers wild embeddings, Schoenflies' theorem from 1908 about homeomorphisms in the plane, and contrasts these properties with higher dimensions. The article appears to be an educational or textbook-style introduction to fundamental concepts in knot theory and algebraic topology.
Key quotes
· 5 pulledA knot is a simple closed curve (homeomorphic image of S(1)) in Euclidean 3-space E(3).
Two knots are called equivalent when there is an orientation-preserving homeomorphism of E(3) onto itself sending one knot to the other.
Schoenflies proved in 1908 that any homeomorphism from a simple closed curve in the plane E(2) onto the unit circle S(1) can be extended to a homeomorphism of the plane onto itself.
Similar things do not hold in higher dimensions.
For example, there exist wild embeddings of simple arcs into E(3): homeomorphic images of t
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