May22,2019 BenjaminBode Braidgroupactionsonthe n -adicintegers

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Real algebraic links

Braid group actions on the n-adic integers

Benjamin Bode

Osaka University

May 22, 2019

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Overview

1 Polynomials and braids

2 Braid group actions

3 Braid sequences

4 Normal subgroups

5 Real algebraic links

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3 Braid sequences

4 Normal subgroups

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Polynomials and braids

Consider the spaceX˜_n of monic complex polynomials f of fixed degreen and withn distinct roots (xi s.t. f(xi) = 0).

f 7→ {x₁,x₂, . . . ,x_n} gives a homeomorphism X˜n={(x₁,x2, . . . ,xn)∈Cⁿ:xi 6=xj ifi6=j}/S_n. and an isomorphismπ₁( ˜X_n) =Bn.

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Polynomials and braids Braid group actions Braid sequences Normal subgroups Real algebraic links

Critical values

Similarly, we could consider the space of monic complex polynomials of degreen, with n−1 distinct critical points (c_j s.t. f⁰(c_j) = 0) and constant term equal to 0.

group of this space isBn−1.

Or we could consider the space of monic complex polynomials of degreen, with n−1distinct critical values (v_j s.t. f(c_j) =v_j and f⁰(cj) = 0) and constant term equal to 0.

f 7→ {v₁,v₂, . . . ,vn−1} is not a homeomorphism, but it induces a homomorphism from the fundamental group of this subspace of the space of polynomials toBn−1.

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Critical values

f 7→ {c₁,c2, . . . ,cn−1} again gives a homeomorphism between a space of polynomials and a configuration space. The fundamental group of this space isBn−1.

f (cj) = 0) and constant term equal to 0.

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Critical values

f 7→ {c₁,c2, . . . ,cn−1} again gives a homeomorphism between a space of polynomials and a configuration space. The fundamental group of this space isBn−1.

Or we could consider the space of monic complex polynomials of degreen, with n−1distinct critical values (v_j s.t. f(c_j) =v_j and f⁰(cj) = 0) and constant term equal to 0.

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Something new

In all of these spaces there is some way to associate an unordered tuple of distinct complex numbers to a point in the space (i.e., a polynomial). Hence a way to associate braids to loops in the space of polynomials.

distinct (non-zero) critical values.

Since the critical values of a polynomialf ∈Xn are non-zero, the space of possible sets of critical values is

V_n={(v₁,v₂, . . . ,vn−1)∈(C\{0})ⁿ⁻¹:v_i 6=v_j if i6=j}/S_n−1,

π1(Vn) =B^affn−1. (1)

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Something new

Consider the spaceXn of monic complex polynomials of degree n with constant term equal to 0, withn distinct roots ANDn−1 distinct (non-zero) critical values.

π1(Vn) =B^affn−1. (1)

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Something new

Consider the spaceXn of monic complex polynomials of degree n with constant term equal to 0, withn distinct roots ANDn−1 distinct (non-zero) critical values.

Since the critical values of a polynomialf ∈X_n are non-zero, the space of possible sets of critical values is

π1(Vn) =B^affn−1. (1)

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Affine braids

Definition

Theaffine braid groupB^affn−1 is generated byx,σ2,σ3, . . . ,σn−1

subject to the relations σ2xσ2x=xσ2xσ2

σiσj =σjσi if|i−j|>1 σiσi+1σi =σi+1σiσi+1 if i= 2,3, . . . ,n−2

σ_ix=xσ_i if i>2. (2)

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Affine braids

There are now two ways in which a loopft⊂Xncorresponds to a n-strand braid, namely

A: (x1(t),x2(t), . . . ,xn(t)), (3) B: (v1(t),v2(t), . . . ,vn−1(t),0). (4) Question: What is the relation betweenA andB?

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3 Braid sequences

4 Normal subgroups

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A covering map

Theorem (Beardon-Carne-Ng)

The mapθ_n:X_n→V_n that sends a polynomial to its set of critical values is acovering map of degree nⁿ⁻¹.

In particular, we have the homotopy lifting property. Therefore the braid type of the braid that is formed by the roots (A) depends only on the braid type of the braid formed by the critical values (B), not on the particular parametrisation.

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An infinite tower

Iff ∈Xn, then one of its roots is zero. We canembedXn into Vn

by sending a polynomial to itsn−1 non-zero roots. We define X_n¹=X_n and X_n^j⁺¹=θ_n⁻¹(X_n^j).

X_n V_n

X_n²:=θ_n⁻¹(X_n) V_n

X_n³:=θ_n⁻¹(X_n²) V_n

θn

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An infinite tower

We can embedXn into Vnby sending a polynomial to its n−1 non-zero roots. We defineX_n¹=X_n andX_n^j⁺¹=θ_n⁻¹(X_n^j).

. . .X_n^j⁺¹→X_n^j →. . .→X_n²→Xn→Vn (6)

Action ofπ1(Vn) =B^affn−1 on the fibers.

The fiber inXn^j consists of (nⁿ⁻¹)^j points, i.e. we have homomorphismsB^affn−1→S_(nⁿ⁻¹₎^j. The actions arecompatible, meaning that for any loopγ∈V_n with basepointv∈V_n we have θn(x.γ) =θn(x).γ for all x∈(θ_n⁻¹)^j(v).

We obtain an action on the fiber in lim←−

j

X_n^j = lim←−

j

Z/(nⁿ⁻¹)^jZ.

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Figure

V

_n

X

n n

1 2

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The n-adic integers

Definition

The set of n-adic integersZn is the inverse limit lim←−

j

Z/n^jZ.

In other words, ann-adic integer is a sequence of numbers (a1,a2,a3, . . .) such that aj∈Z/n^jZwith aj+1≡aj modn^j. E. g. (0,2,2,10, . . .)∈Z2

Remarks: Z_∏^`

ipⁿⁱ_i ∼=Zp1×Zp2×. . .×Zp`

Thefiber in lim←−

j

Xn^j over any basepoint in Vnis bijective to Znⁿ⁻¹∼=Zn.

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The n-adic integers

Definition

The set of n-adic integersZn is the inverse limit lim←−

j

Z/n^jZ.

In other words, ann-adic integer is a sequence of numbers (a1,a2,a3, . . .) such that aj∈Z/n^jZwith aj+1≡aj modn^j. E. g. (0,2,2,10, . . .)∈Z2

Zn is a topological group and metric space.

ordn(a) = min{k≥1 :ai = 0 for all i<k},|a|_n=n^−ordⁿ^(a). Remarks: Z_∏^`

ipⁿⁱ_i ∼=Zp1×Zp2×. . .×Zp`

Thefiber in lim←−

j

Xn^j over any basepoint in Vnis bijective to

∼

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Braid group actions

Definition

We say a pointv= (v1,v2, . . . ,vn−1)∈Vn has0 in the jth position if exactlyj−1of the v_i have negative real part.

Wepick n basepointswj ∈Vn,j= 1,2, . . . ,n such thatwj has 0 in thejth position.

atw_k. By constructionk=π(B)(j), where π:Bn→S_n is the permutation representation.

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Braid group actions

Definition

We say a pointv= (v1,v2, . . . ,vn−1)∈Vn has0 in the jth position if exactlyj−1of the v_i have negative real part.

Wepick n basepointswj ∈Vn,j= 1,2, . . . ,n such thatwj has 0 in thejth position.

Given a braid wordB onn strands, we considern parametrisations ofB as paths in Vn, each one starting at one of thewj and ending atw_k. By constructionk=π(B)(j), where π:Bn→S_n is the permutation representation.

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Braid group actions

compatiblewith each other and we obtain an actionϕn ofBn on Z/nZ×Znⁿ⁻¹∼=Zn.

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Braid group actions

Thelifts of thesenpaths in V_npermute the points in the fibers over {w₁,w2, . . . ,wn} in anyXn^j. Again these permutations are

compatiblewith each other and we obtain an actionϕ_n ofBn on Z/nZ×Znⁿ⁻¹∼=Zn.

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Braid group actions

V

_n

X

n n

1 2

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Transitivity and continuity

Zn is uncountable, while Bn is countable, sonone of the defined actions can be transitive.

Proposition

Xn^j is path-connectedfor all n, j.

Corollary

Restricting the actionϕ_n to the fiber in X_n^j for a fixed j results in an action on n×(nⁿ⁻¹)^j points. This actionis transitive for all j.

for all x,y∈Z/nZ×Znⁿ⁻¹,γ∈Bn). The action onZnis continuous.

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Transitivity and continuity

Zn is uncountable, while Bn is countable, sonone of the defined actions can be transitive.

Proposition

Xn^j is path-connectedfor all n, j.

Corollary

Restricting the actionϕ_n to the fiber in X_n^j for a fixed j results in an action on n×(nⁿ⁻¹)^j points. This actionis transitive for all j. Proposition

The actiononZ/nZ×Znⁿ⁻¹ is by isometries (|x.γ−y.γ|=|x−y| for all x,y∈Z/nZ×Znⁿ⁻¹,γ∈Bn). The action onZnis

continuous.

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3 Braid sequences

4 Normal subgroups

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Braid sequences

V

_n

X

n n

1 2

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Braid sequences

V

_n

X

n n

1 2

B B B B B B

1,1 1,3 1,4 1,2

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Braid sequences

Letzi, i= 1,2, . . . ,nⁿ, denote the points inθ_n⁻¹({w₁,w2, . . . ,wn}) in X_n. The lifted paths inX_n correspond to braids onn strands (by sending a polynomial to its set of roots). Denote the braid that corresponds to the lifted path that starts atzi byB1,i. Then B1,i are invariants of the original braidB (corresponding ton paths in V_n).

The whole sequence(B,{B_1,i}_i_=1,2,...,nn,{B_2,i}i=1,2,...,n²ⁿ⁻¹, . . .) is an invariant ofB.

Say we have a braid invariantI_n:Bn→X valued in some set X. Then (I_n(B),{I_n(B_1,i)},{I_n(B_2,i)}, . . .) is an invariant ofB, presumably a lotstronger thanIn(B), but not much harder to compute.

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Braid sequences

Similarly, the lifted paths inX_n^j correspond to braids, say B_j,i, i= 1,2, . . . ,n×(nⁿ⁻¹)^j, which are invariants of B.

Say we have a braid invariantI_n:Bn→X valued in some set X. Then (I_n(B),{I_n(B_1,i)},{I_n(B_2,i)}, . . .) is an invariant ofB, presumably a lotstronger thanIn(B), but not much harder to compute.

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Braid sequences

The whole sequence(B,{B_1,i}i=1,2,...,nⁿ,{B_2,i}i=1,2,...,n²ⁿ⁻¹, . . .) is an invariant ofB.

presumably a lotstronger thanIn(B), but not much harder to compute.

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Braid sequences

The whole sequence(B,{B_1,i}i=1,2,...,nⁿ,{B_2,i}i=1,2,...,n²ⁿ⁻¹, . . .) is an invariant ofB.

Say we have a braid invariantI_n:Bn→X valued in some set X. Then (I_n(B),{I_n(B_1,i)},{I_n(B_2,i)}, . . .) is an invariant ofB, presumably a lotstronger thanIn(B), but not much harder to

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Braid sequences

Algebraically, the lifting process corresponds to a homomorphism Bn→B^n×(n

n−1)^j

n oS_n×(nⁿ⁻¹₎^j. (7)

The action is constructed from the projection to S_n×(nⁿ⁻¹₎^j and the braid sequences come from the projection toB^n×(n

n−1)^j

n .

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Braid sequences

Conjugate braids lift to conjugate braidsin the sense that if A and B are conjugate braids, then for everyj there is a ρ_j ∈S_n×(nⁿ⁻¹₎j

such thatAj,i is conjugate to B_j_,ρ_j_(i).

We can thereforeapply invariants of conjugacy classes of braids to the sequences (B,{B_1,i}, . . .) and obtain stronger invariants of conjugacy classes.

Question: How do the braid sequences change under

(de)stabilization moves? Can we use them to define link invariants?

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3 Braid sequences

4 Normal subgroups

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Normal subgroups

By construction the actionsϕn correspond to sequences of homomorphismshj :Bn→S_n×(nⁿ⁻¹₎^j. LetHj:= ker(hj). Then

H_jDH_j+1. . . is a descending series of normal subgroups ofBn.

H_n=H_N for all n≥N.

Question: Is ^T^∞_j=1H_j={e}? If and only ifϕn is faithful.

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Normal subgroups

By construction the actionsϕn correspond to sequences of homomorphismshj :Bn→S_n×(nⁿ⁻¹₎^j. LetHj:= ker(hj). Then

H_jDH_j+1. . . is a descending series of normal subgroups ofBn.

Proposition

The sequenceHj does not stabilize, i.e., there is no N such that H_n=H_N for all n≥N.

Question: Is^T^∞_j=1H_j ={e}? If and only ifϕn is faithful.

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3 Braid sequences

4 Normal subgroups

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Real algebraic links

Definition

Consider a polynomial mapf :R⁴→R² such that f(0) = 0,∇f(0) = 0,

There is a nbhdU of 0∈R⁴ such that0 is the only point inU, where∇f does not have full rank.

Then we call 0an isolated singularity off. Definition

If0is an isolated singularity of f :R⁴→R², the intersection f⁻¹(0)∩S_ρ³ is a linkL for all small enough radiiρ. We callLthe link of the singularity. A link Lis real algebraic if there is a

polynomialf :R⁴→R² with an isolated singularity andLis the link of that singularity.

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Real algebraic links

Figure:José Seade: On the topology of isolated singularities in analytic spaces

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Real algebraic links

Theorem (Milnor)

Lreal algebraic =⇒ L fibered.

Known real algebraic links: All algebraic links, 41 (Perron, Rudolph),

K#K ifK is a fibered knot (Looijenga).

The closure of B², whereB is homogeneous. (B)

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Real algebraic links

Theorem (Milnor)

Lreal algebraic =⇒ L fibered.

Conjecture (Benedetti, Shiota) Lreal algebraic ⇐⇒ Lfibered.

Known real algebraic links:

All algebraic links, 4₁ (Perron, Rudolph),

K#K ifK is a fibered knot (Looijenga).

The closure of B², whereB is homogeneous. (B)

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Real algebraic links from lifted affine braids

Proposition (B)

Letf_t be a loop inX˜_n such that itscritical values

(v1(t),v2(t), . . . ,vn−1(t))are distinct, non-zero for all t and such that ^∂^arg_∂t^v^j(h)6= 0for all j = 1,2, . . . ,n−1 and h∈[0,1]. Let B denote the braid that is formed bythe roots of ft. Then the closure of B² is real algebraic.

Therefore, to construct real algebraic links we look for affine braids such that

it can be parametrised by (0,v₁(t),v₂(t), . . . ,vn−1(t)) as in the proposition and

one of its lifts in X_n⊂X˜_n is a loop.

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Solar systems

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Solar systems

-2 -1 1 2

-3 -2 -1 1 2 3

-2 -1 1 2

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Real algebraic links

Corollary (B)

Letε∈ {±1}and let B=∏^`_j=1w_i^ε

j be a 3-strand braid with w1=σ2, w2=σ₂⁻¹σ₁²σ2, w3= (σ1σ2σ1)² w4= (σ2σ1σ₂⁻¹σ1σ2)², w5=σ₂⁻¹σ1σ₂²σ1, (8) and such that either there is a j with i_j = 3or there is only one residue class k mod3such that ij6=k for all j.

Thenthe closure of B² is real algebraic.