Markov’s theorem

Braids, Knots, and Links

2.5 Markov’s theorem

We state a fundamental theorem that allows us to describe all braids with isotopic closures inR³. This theorem, due to A. Markov, is based on so-called Markov moves on braids.

2.5.1 Markov moves

The presentation of an oriented link inR³ as a closed braid is far from being unique. As we know, if two braids β, β ∈ B_n are conjugate (we record it as β ∼c β), then their closures β, β are isotopic in the solid torus and a fortiori inR³. In general, the converse is not true. For instance, the closures of the 2-string braidsσ1, σ⁻₁¹ are trivial knots although these braids are not conjugate in B2 ∼= Z. There is another simple construction of braids with isotopic closures. Forβ∈Bn, consider the braidsσnι(β) andσ⁻_n¹ι(β), whereι is the natural embeddingBn →Bn+1. Drawing pictures, one easily observes that the closures ofσnι(β) andσ_n⁻¹ι(β) are isotopic toβin R³.

For β, γ∈ Bn, the transformationβ →γβγ⁻¹ is called thefirst Markov moveand is denoted by M1. The transformationβ →σ_n^ει(β) withε=±1 is called thesecond Markov move and is denoted by M2. Note that the inverse to an M₁-move is again an M₁-move. We shall say that two braidsβ, β(possibly with different numbers of strings) are M-equivalent if they can be related by a finite sequence of moves M₁,M₂,M⁻₂¹, where M⁻₂¹ is the inverse of an M₂-move. We record it as β ∼ β. It is clear that the M-equivalence ∼ is an equivalence relation on the disjoint union n≥1B_n of all braid groups.

For example, the braids σ1, σ₁⁻¹ ∈ B2 are M-equivalent. Indeed, using the equalitiesσ₂⁻¹σ₁⁻¹σ₂⁻¹=σ₁⁻¹σ⁻₂¹σ⁻₁¹andσ⁻₁¹σ⁻₂¹σ1=σ2σ⁻₁¹σ⁻₂¹, we obtain

σ1∼σ₂⁻¹σ1∼c(σ1σ2)⁻¹(σ⁻₂¹σ1)(σ1σ2)

=σ₂⁻¹σ⁻₁¹σ⁻₂¹σ²₁σ2=σ⁻₁¹σ⁻₂¹σ⁻₁¹σ₁²σ2

=σ₁⁻¹σ⁻₂¹σ₁σ₂=σ₂σ₁⁻¹σ₂⁻¹σ₂

=σ2σ⁻₁¹∼σ⁻₁¹.

As we saw, the closures of M-equivalent braids are isotopic as oriented links inR³. The following deep theorem asserts that conversely, any two braids with isotopic closures are M-equivalent.

Theorem 2.8 (A. Markov). Two braids (possibly with different numbers of strings) have isotopic closures in Euclidean space R³ if and only if these braids are M-equivalent.

The following fundamental corollary yields a description of the set of iso- topy classes of oriented links inR³in terms of braids.

Corollary 2.9.Let L be the set of all isotopy classes of nonempty oriented links inR³. The mappingn≥1Bn→ Lassigning to a braid the isotopy class of its closure induces a bijection from the quotient set(n≥1Bn)/∼ontoL.

Here the surjectivity follows from Alexander’s theorem, while the injectiv- ity follows from Markov’s theorem.

The proof of Theorem 2.8 starts in Section 2.5.3 and occupies the rest of the chapter.

2.5.2 Markov functions

Corollary 2.9 allows one to identify isotopy invariants of oriented links inR³ with functions onn≥1B_n constant on the M-equivalence classes. This leads us to the following definition.

Definition 2.10.A Markov function with values in a setE is a sequence of set-theoretic maps{fn :Bn→E}n≥1, satisfying the following conditions:

(i) for all n≥1 and all α,β ∈B_n,

f_n(αβ) =f_n(βα) ; (2.1)

(ii) for all n≥1 and all β∈Bn,

fn(β) =fn+1(σnβ) and fn(β) =fn+1(σ⁻_n¹β). (2.2) For example, for any e ∈ E, the constant maps Bn → E sending Bn

toe for alln form a Markov function. More interesting examples of Markov functions will be given in Chapters 3 and 4.

Any Markov function{fn :Bn→E}n≥1determines anE-valued isotopy invariantfof oriented links inR³as follows. LetLbe an oriented link inR³. Pick a braidβ∈Bn whose closure is isotopic toLand setf(L) =fn(β)∈E.

Note that f(L) does not depend on the choice of β. Indeed, if β ∈ Bn is another braid whose closure is isotopic toL, thenβ andβ are M-equivalent (Theorem 2.8). It follows directly from the definition of M-equivalence and the definition of a Markov function thatfn(β) =fn(β). The function fis an isotopy invariant of oriented links: ifL,Lare isotopic oriented links inR³ andβ∈Bn is a braid whose closure is isotopic to L, then the closure ofβ is also isotopic toL andf(L) =fn(β) =f(L).

2.5.3 A pivotal lemma

We formulate an important lemma needed in the proof of Theorem 2.8. We begin with some notation. Given two braidsα ∈ B_m and β ∈ B_n, we form theirtensor product α⊗β ∈ B_m+n by placing β to the right ofα without any mutual intersection or linking; see Figure 2.13. Here the vertical lines represent bunches of parallel strands with the number of strands indicated near the line.

A diagram of α⊗β is obtained by placing a diagram ofβ to the right of a diagram ofα without mutual crossings. For example, 1m⊗1n = 1m+n, where 1m is the trivial braid onmstrands. Clearly,

α⊗β= (α⊗1n)(1m⊗β) = (1m⊗β)(α⊗1n). Note also that

(α⊗β)⊗γ=α⊗(β⊗γ)

for any braidsα, β, γ. This allows us to suppress the parentheses and to write simplyα⊗β⊗γ.

α⊗β α β

n m+n

=

m

m+n m n

Fig. 2.13.The tensor product of braids

For a sign ε = ± and any integers m, n ≥ 0 with m+n ≥ 1, we de- fine a braid σ_m,n^ε ∈ Bm+n as follows. Consider the standard diagram of σ1 ∈ B2 consisting of two strands with one crossing. Replacing the over- crossing strand with m parallel strands running very closely to each other and similarly replacing the undercrossing strand withn parallel strands, we obtain a braid diagram withm+n strands andmncrossings. This diagram representsσ⁺_m,n∈Bm+n. Transforming all overcrossings in the latter diagram into undercrossings, we obtain a diagram ofσ⁻_m,n∈Bm+n. The braids σ_m,n⁺ andσ_m,n⁻ are schematically shown in Figure 2.14. In particular,

σ_m,0⁺ =σ_m,0⁻ =σ⁺_0,m=σ_0,m⁻ = 1m

for allm≥1. It is clear that (σ_m,n^ε )⁻¹=σ_n,m^−ε for allm, n, andε.

It is convenient to introduce the symbols σ_0,0⁺ , σ⁻_0,0, and 1₀; they all rep- resent an “empty braid on zero strings” ∅, which satisfies the identities

∅ ⊗α=α⊗ ∅=αfor any genuine braidα.

σ_m,n⁻ σ_m,n⁺

m n m

n m n

n

m

Fig. 2.14.The braidsσ_m,n⁺ ,σ_m,n⁻ ∈Bm+n

Lemma 2.11.For any integersm, n≥0,r, t≥1, signsε, ν=±, and braids α∈Bn+r,β∈Bn+t,γ∈Bm+t,δ∈Bm+r, consider the braid

α, β, γ, δ|ε, ν= (1_m⊗α⊗1_t)(1_m+n⊗σ_t,r^ν )(1_m⊗β⊗1_r)(σ_n,m^−ε ⊗1_t+r)

×(1n⊗γ⊗1r)(1n+m⊗σ_r,t^−ν)(1n⊗δ⊗1t)(σ_m,n^ε ⊗1r+t)∈Bm+n+r+t. Then the M-equivalence class ofα, β, γ, δ|ε, νdoes not depend onε,ν, and

α, β, γ, δ|ε, ν ∼ δ, γ, β, α|ε, ν. (2.3) The reader is encouraged to draw the braidα, β, γ, δ|ε, νforε=ν= +.

We shall draw the closure of this braid using the following conventions. Let us think of braid diagrams as lying in a squareI×I⊂R×Iwith inputs on the top sideI×{0}and outputs on the bottom sideI×{1}. The standard orientation on the strands of a braid diagram runs from the inputs to the outputs. We can rotate the squareI×I around its center by the angleπ/2. RotatingI×I by the angleπ/2 counterclockwise (resp. clockwise), we transform any picture ain I×I into a picture in I×I denoted by a+ (resp. a₋). Ifa is a braid diagram, then the inputs and outputs ofa+, a₋ lie on the vertical sides of the square. Note also thata++=a₋₋, where a++= (a+)+ anda₋₋= (a₋)₋.

Pick certain diagrams of the braids α, β, γ, δ, which we denote by the same lettersα, β, γ, δ, respectively. A little contemplation should persuade the reader that Figure 2.15 represents the closure of the braidα, β, γ, δ|+,+.

β₋ α₋

γ₊ m n r

t t r

m n

δ+

Fig. 2.15.The closure ofα, β, γ, δ|+,+

The rest of the proof of Theorem 2.8 goes as follows. In Section 2.6 we deduce this theorem from Lemma 2.11. In Section 2.7 we prove Lemma 2.11.

These two sections use different techniques and can be read in any order.

Exercise 2.5.1.Verify that the braidsα, β, γ, δ|ε, νandα, β, γ, δ| −ε,−ν have isotopic closures. Verify that

α, β, γ, δ|ε, ν ∼cγ, δ, α, β| −ε,−ν. (2.4) (Hint: Rotate the closed braid in Figure 2.15 through 180^◦.)

Exercise 2.5.2.Verify (2.3) form=n= 0.