Math 2.0

Manifolds and cell complexes Knot theory Theorem

Markov theorem

Builds on LinkBraid groupConjugacy classAlexander's theorem on closed braidsReidemeister's theorem

Statement

For n≥1n \ge 1, let BnB_n be the braid group with generators σ1,…,σn−1\sigma_1, \ldots, \sigma_{n-1}. View BnB_n as a subgroup of Bn+1B_{n+1} by adding a straight strand on the right. For a braid β\beta, let β^\hat\beta be its closure, an oriented link in ℝ3\R^3. The two Markov moves on braids are:

  1. Conjugation: replace β∈Bn\beta \in B_n by γβγ−1\gamma \beta \gamma^{-1} for some γ∈Bn\gamma \in B_n, or the converse.
  2. Stabilization: replace β∈Bn\beta \in B_n by βσn\beta \sigma_n or by βσn−1\beta \sigma_n^{-1} in Bn+1B_{n+1}, or the converse.

Let β∈Bn\beta \in B_n and β′∈Bm\beta' \in B_m. Then β^\hat\beta and β^′\hat\beta' are ambient isotopic as oriented links if and only if a finite sequence of Markov moves turns β\beta into β′\beta'.

Remarks

One direction is a direct check. Closing up γβγ−1\gamma \beta \gamma^{-1} lets γ\gamma slide around the closure and cancel γ−1\gamma^{-1}, so the closure does not change. Closing up βσn±1\beta \sigma_n^{\pm 1} adds one strand with a single kink, which a first Reidemeister move removes. The hard direction follows an isotopy of closed braids step by step and shows that each step is a combination of Markov moves. For example, σ1∈B2\sigma_1 \in B_2 and the identity braid in B1B_1 both close up to the unknot, and they differ by one stabilization. Conjugation alone does not suffice, since conjugation never changes the number of strands, while braids with different numbers of strands can close to the same link.

Going deeper

For a student

By Alexander's theorem, every oriented link is a closed braid. The Markov theorem then says that oriented links up to isotopy are the same as braids in all BnB_n up to the two Markov moves. So a family of functions fnf_n on BnB_n gives a link invariant when fnf_n is constant on conjugacy classes and fn+1(βσn±1)f_{n+1}(\beta \sigma_n^{\pm 1}) equals fn(β)f_n(\beta). Traces on the Temperley–Lieb and Hecke algebras with this property give the Jones polynomial and the HOMFLY polynomial.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

Show the path as a graph
Braid group, concept, no evidence levelBraid groupconceptConjugacy class, concept, no evidence levelConjugacy classconceptKnot, concept, no evidence levelKnotconceptLink, concept, no evidence levelLinkconceptAlexander's theorem on closed braids, statement, no evidence levelAlexander's theorem o…statementReidemeister's theorem, statement, no evidence levelReidemeister's theoremstatementMarkov theorem, statement, no evidence levelMarkov theoremstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Braid group concept
  2. Conjugacy class concept
  3. Knot concept
  4. Link concept
  5. Alexander's theorem on closed braids statement
  6. Reidemeister's theorem statement
  7. Markov theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Markov_theorem
About this page
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Kindstatement, theorem
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AboutLink, Braid group
UsesLink, Braid group, Conjugacy class, Alexander's theorem on closed braids, Reidemeister's theorem
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Changed2 November 2026

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