Theorems in algebraic topology

Markov theorem

In mathematics the Markov theorem gives necessary and sufficient conditions for two braids to have closures that are equivalent knots or links. The conditions are stated in terms of the group structures on braids. Braids are algebraic objects described by diagrams; the relation to topology is given by Alexander's theorem which states that every knot or link in three-dimensional Euclidean space is the closure of a braid. The Markov theorem, proved by Russian mathematician Andrei Andreevich Markov Jr. describes the elementary moves generating the equivalence relation on braids given by the equivalence of their closures. More precisely Markov's theorem can be stated as follows: given two braids represented by elements in the braid groups , their closures are equivalent links if and only if can be obtained from applying to a sequence of the following operations: 1. * conjugating in ; 2. * replacing by (here are the standard generators of the braid groups; geometrically this amounts to adding a strand to the right of the braid diagram and twisting it once with the (previously) last strand); 3. * the inverse of the previous operation (if with replace with ). (Wikipedia).

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(ML 14.2) Markov chains (discrete-time) (part 1)

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From playlist Machine Learning

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(ML 14.3) Markov chains (discrete-time) (part 2)

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From playlist Machine Learning

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From playlist Machine Learning

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From playlist Machine Learning

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From playlist iLecturesOnline: Probability & Stats 3: Markov Chains & Stochastic Processes

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From playlist Machine Learning

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From playlist Mathematics

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From playlist iLecturesOnline: Probability & Stats 3: Markov Chains & Stochastic Processes

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From playlist Machine Learning

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From playlist HIM Lectures 2015

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From playlist Mathematics

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Hamza Fawzi: "Sum-of-squares proofs of logarithmic Sobolev inequalities on finite Markov chains"

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25. Putting It All Together

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From playlist MIT 6.262 Discrete Stochastic Processes, Spring 2011

Related pages

Knot (mathematics) | Mathematics | Link (knot theory) | Conjugacy class | Alexander's theorem | Group (mathematics)