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Math Notes

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TheoremComplete

Markov's Theorem

Theorem5.2Markov's Theorem

Two braids β1∈Bn1\beta_1 \in B_{n_1} and β2∈Bn2\beta_2 \in B_{n_2} have isotopic closures β^1≅β^2\hat{\beta}_1 \cong \hat{\beta}_2 (as oriented links) if and only if β1\beta_1 can be transformed into β2\beta_2 by a finite sequence of the following Markov moves:

  1. (Type I -- Conjugation) β↦αβα−1\beta \mapsto \alpha\beta\alpha^{-1} for any α∈Bn\alpha \in B_n (conjugation within BnB_n).
  2. (Type II -- Stabilization) β∈Bn↦βσn±1∈Bn+1\beta \in B_n \mapsto \beta\sigma_n^{\pm 1} \in B_{n+1} or βσn±1∈Bn+1↦β∈Bn\beta\sigma_n^{\pm 1} \in B_{n+1} \mapsto \beta \in B_n (adding or removing a strand with one crossing).

The equivalence classes of braids under Markov moves are in bijection with isotopy classes of oriented links.

Proof (Outline)

Proof

Step 1: Markov moves preserve link type. Type I: conjugation αβα−1\alpha\beta\alpha^{-1} amounts to choosing a different starting point on the braid closure, which does not change the link. Type II: stabilization β↦βσn±1\beta \mapsto \beta\sigma_n^{\pm 1} adds a small loop to one strand of the closure, which can be removed by a Reidemeister I move. Both directions preserve isotopy type.

Step 2: Sufficiency (the hard direction). Suppose β^1≅β^2\hat{\beta}_1 \cong \hat{\beta}_2. We must show β1\beta_1 and β2\beta_2 are related by Markov moves. The isotopy between β^1\hat{\beta}_1 and β^2\hat{\beta}_2 can be decomposed into a sequence of Reidemeister moves (on the closure diagrams) and planar isotopies.

Step 3: Reidemeister moves to Markov moves. Each Reidemeister move on a braid closure can be realized by Markov moves:

  • RI move (adding/removing a kink): Corresponds to Type II stabilization β↔βσn±1\beta \leftrightarrow \beta\sigma_n^{\pm 1}.

  • RII move (adding/removing two opposite crossings): If both crossings occur within the braid (not crossing the closure region), this is a relation in BnB_n. If one crossing involves the closure arc, it can be achieved by first stabilizing (Type II), then conjugating (Type I), and destabilizing.

  • RIII move (triangle move): This corresponds to the braid relation σiσi+1σi=σi+1σiσi+1\sigma_i\sigma_{i+1}\sigma_i = \sigma_{i+1}\sigma_i\sigma_{i+1}, which holds in BnB_n.

Step 4: Moving crossings past the closure. Planar isotopies that slide crossings around the closure (from the "bottom" of the braid to the "top" or vice versa) correspond exactly to Type I conjugation moves.

Step 5: Combining. Any sequence of Reidemeister moves and planar isotopies on braid closures decomposes into a sequence of braid group relations, Type I conjugations, and Type II stabilizations/destabilizations. This establishes the theorem. □\square

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ExampleMarkov Equivalences

Unknot representations: 1∈B11 \in B_1, σ1σ1−1∈B2\sigma_1\sigma_1^{-1} \in B_2, and σ1∈B2\sigma_1 \in B_2 (by destabilization of 1⋅σ11 \cdot \sigma_1) all have unknot closures. Trefoil equivalence: σ13∈B2\sigma_1^3 \in B_2 and σ1σ2σ1σ2∈B3\sigma_1\sigma_2\sigma_1\sigma_2 \in B_3 are Markov equivalent (the latter is obtained by stabilization and conjugation). Computing: σ^13\hat{\sigma}_1^3 has one component with 3 crossings, yielding the trefoil.

RemarkApplications to Link Invariants

Markov's theorem provides a machine for constructing link invariants: find a function ff on braids that is invariant under both types of Markov moves, and f(β)=f(β^)f(\beta) = f(\hat{\beta}) defines a link invariant. The Jones polynomial arises this way: the Temperley-Lieb algebra representation ρ:Bn→TLn\rho: B_n \to TL_n followed by the Markov trace trM:TLn→C\mathrm{tr}_M: TL_n \to \mathbb{C} yields a Markov-invariant function. Similarly, the HOMFLYPT polynomial uses the Hecke algebra Hn(q)H_n(q) and its Markov trace. This algebraic approach to link invariants, pioneered by Jones (1984), revolutionized knot theory and led to connections with statistical mechanics and quantum field theory.