Knots and Galois actions.

Norbert A’Campo

Vortrag Arbeitstagung 2003 Bonn.

Introduction. Let f : C^{n+1} → C be a map defined by a polynomial. We assume that f(0) = 0 and that 0 ∈ C^{n+1} is an isolated critical point of f. For p ∈ C^{n+1} let ||p|| denote the square root of |z_{0}(p)|^{2} +|z_{1}(p)|^{2} +⋯+|z_{n}(p)|^{2}. Let B_{ϵ} := {p ∈ C^{n+1} | ||p|| ≤ ϵ}, 0 < ϵ, be a Milnor ball for the singularity of f and let Tube_{ϵ,δ} := {p ∈ B_{ϵ} | |f(p)| ≤ δ}, 0 < δ << ϵ, be a regular tubular neighborhood of {p ∈ B_{ϵ} | f(p) = 0} in B_{ϵ}. A monodromy vector field X for the singularity is a smooth vector field p ∈ Tube_{ϵ,δ} ↦ X_{p} ∈ C^{n+1} such that we have the following properties for p ∈ Tube_{ϵ,δ} (remember i = \sqrt{−1}):

- (df)_{p}(X_{p}) = 2πif(p),

- X_{p} is tangent to ∂B_{ϵ} if p ∈ ∂B_{ϵ},

- trajectories of X starting at p ∈ ∂B_{ϵ} are periodic with period 1.

Using partition of unity, one can construct monodromy vector fields. The flow at time 1 of a monodromy vector field X defines a monodromy diffeomorphism T = T_{X}: F → F, where the manifold with boundary (F, ∂(F)) := {p ∈ B_{ϵ} | f(p) = δ} is the Milnor fiber of the singularity. The relative isotopy class of the diffeomorphism T is independent from the chosen monodromy vector field and is the geometric monodromy of the singularity. The geometric monodromy is a topological invariant of the singularity.

From now on we will assume in addition, that the polynomial f is real meaning that its coefficients are real numbers. Let c : C^{n+1} → C^{n+1} denote the involution on complex space given by the complex conjugation of coordinate values. Hence with the above notations, we have c(Tube_{ϵ,δ}) = Tube_{ϵ,δ} and c(F) = F. We denote by c_{F} : F → F the restriction of the involution c to F.

Let X: Tube_{ϵ,δ} → C^{n+1} be a monodromy vector field for the isolated singularity of f. We may assume that we have constructed the vector field X with more care near the boundary of the Milnor ball in order to achieve that for some ϵ′ < ϵ we have the symmetry c(X_{p}) = −c(X_{c(p)}), p ∈ Tube_{ϵ,δ}, ||p|| > ϵ′.

Since f is real, we have

c((df)_{p}(X_{p})) = (df)_{c(p)}(c(X_{p})) = −2πif(c(p))

hence, we see (by substituting q for c(p) and accordingly c(q) for p) that the vector field X^{c} defined by:

q ∈ Tube_{ϵ,δ} ↦ X_{q}^{c} := −c(X_{c(q)}) ∈ C^{n+1},

is a monodromy vector field too. Let Y : Tube_{ϵ,δ} → C^{n+1} be the vector field Y := \frac{X+X^{c}}{2} , which due to the extra care is also a monodromy vector field. We have Y^{c} = Y. The following is an important symmetry of the geometric monodromy:

Lemma 1 Let T_{Y} be a monodromy diffeomorphism, which has been computed with a monodromy vector field Y satisfiyng Y^{c} = Y. We have the symmetry

c_{F} ∘ T_{Y} ∘ c_{F} = T_{Y}^{−1}.

The geometric monodromy T satisfies (up to relative isotopy) the symmetry

c_{F} ∘ T ∘ c_{F} = T^{−1}.

Proof: The restriction of complex conjugation c_{Tube_{ϵ,δ}}: Tube_{ϵ,δ} → Tube_{ϵ,δ} maps the monodromy vector field Y to −Y and F to F. Hence, since c_{Tube_{ϵ,δ}} reverses the orientations of the trajectories, we have

T_{Y}^{−1} = (c_{F})^{−1} ∘ T_{Y} ∘ c_{F} = c_{F} ∘ T_{Y} ∘ c_{F} .

Since the geometric monodromy T is in the relative mapping class group of the Milnor fiber represented by T_{Y}, for T in the relative mapping class group we have the symmetry c_{F} ∘ T ∘ c_{F} = T^{−1}.

Symmetries of monodromies as in the lemma can occur in the more general context of so-called strongly invertible knots, see for instance [H-To],[T],[K].

The symmetry property c_{F} ∘ T ∘ c_{F} = T^{−1} expresses that the geometric monodromy T of a complex hypersurface with real defining equation is conjugate in the mapping class group by an element of order 2 to its inverse T^{−1}. This statement does not refer to any complex conjugation, so it can be stated for any complex hypersurface singularity with complex defining equation. We say that the singularity is strongly invertible if its geometric monodromy diffeomorphism T is conjugate by an element of order 2 in the relative mapping class group of the Milnor fiber to its inverse T^{−1}. The property of strong invertibility is a topological property for hypersurface singularities.

We can rewrite the symmetry property as follows: T_{Y} ∘c_{F} ∘T_{Y} ∘c_{F} = Id_{F}. We see that T_{Y} ∘c_{F} : F → F is an involution of F. It follows, see also [G-Z], the

Corollary 1 The geometric monodromy T of an isolated complex hypersurface singularity, which is defined by a real equation, is the composition of two involutions of the fiber F_{δ}, δ ∈ R, namely: T = (T ∘ c_{F}) ∘ c_{F}, where c_{F} is the restriction of the complex conjugation.

The above observations can be applied to plane curve singularities in general, since it follows from the Theory of Puiseux Pairs that every plane curve singularity is topologically equivalent to a singularity given by a rational equation and hence plane curve singularities are strongly invertible.

In the next picture we decompose a Dehn twist as composition of two involutions I_{a} ∘ I_{b}, where I_{a}, I_{b} are the orientation reversing diffeomorphism of the cylinder, that fixe pointwise the arc systems a or b.

+ +  a a P  a  a b b β βb′ b′ Divide for xy = 0  Fiber surface with arc systems a, b

Fig. 5. Dehn twist as composition of two involutions.

For complex hypersurface singularities of higher dimension the situation seems to be opposite. In C^{n+1}, n > 1, there exist isolated hypersurface singularities which are not topologically equivalent to a singularity with a real defining equation [T]. We expect that in general the geometric monodromy T of a complex hypersurface singularity fails to be strongly invertible.

For complex hypersurface singularities the eigenvalues of the monodromy T_{∗} acting on the homology E := ⨁_{k} H_{k}(F, Q) are roots of unity by the monodromy theorem and the characteristic polynomial of T_{∗} is a product of cyclotomic polynomials. It follows, that the complex homological monodromy ist strongly invertible. Mathias Schulze has proved strong invertibility of the homological monodromy with real coefficients and the following is a strengthening of his result.

Theorem 1 The rational homological monodromy of a complex hypersurface singularity is strongly invertible.

Proof: Let E = ⨁_{i} E_{i} a be finest possible direct sum decomposition in T_{∗}- invariant Q-subspaces of E. The characteristic polynomial of the restriction A_{i} : E_{i} → E_{i} of T_{∗} to a summand E_{i} is a power ψ(t) = ϕ(t)^{L} of a cyclotomic polynomial ϕ(t). We have ψ(A_{i}) = 0 by the Hamilton-Cayley theorem and we have ψ(A_{i}^{−1}) = 0 since a power of cyclotomic polynomial satisfies ψ(1/t) = ±t^{degree(ψ)}ψ(t).

Since the decomposition has no refinement, we may choose a vector e_{1} ∈ E_{i}, that is cyclic for A_{i} and for A_{i}^{−1}. The systems

e_{1}, e_{2} := A_{i}(e_{1}), ⋯, e_{dim E_{i}} := A^{dim E_{i}−1}(e_{1}),
f_{1} := e_{1}, f_{2} := A_{i}^{−1}(e_{1}), ⋯, f_{dim E_{i}} := A^{− dim E_{i}+1}(e_{1})

are bases for the space E_{i}. Let b_{i} : E_{i} → E_{i} be the linear map defined by b_{i}(e_{j}) = f_{j}, 1 ≤ j ≤ dim E_{i}.

We have A_{i}^{−1}b_{i}(e_{j}) = b_{i}A_{i}(e_{j}), 1 ≤ j < dim E_{i}. The polynomial ψ_{1}(t) := −ψ(t) + t^{dim E_{i}} satisfies ψ_{1}(A_{i}^{±1}) = A_{i}^{± dim E_{i}}. Since the degree of the polynomial ψ_{1}(t) is less than dim E_{i}, we have

A_{i}^{−1}b_{i}(e_{dim E_{i}}) = A^{− dim E_{i}}b_{i}(e_{1}) = ψ_{1}(A_{i}^{−1})b_{i}(e_{1}) =
b_{i}ψ_{1}(A_{i})(e_{1}) = b_{i}A_{i}^{dim E_{i}}(e_{1}) = b_{i}A_{i}(e_{dim E_{i}}).

We conclude that A_{i}^{−1}b_{i} = b_{i}A_{i} and A_{i}^{−1} = b_{i}A_{i}b_{i}^{−1} hold.

The polynomial ψ_{0}(t) := \frac{ψ(t)−ψ(0)}{−tψ(0)} satisfies A_{i}^{−1} = ψ_{0}(A_{i}) and A_{i} = ψ_{0}(A_{i}^{−1}). We deduce b_{i}A_{i}^{−1} = b_{i}ψ_{0}(A_{i}) = ψ_{0}(A_{i}^{−1})b_{i} = A_{i}b_{i} and conclude A_{i}^{−1} = b_{i}^{−1}A_{i}b_{i}. We observe at this point that both the conjugates of A_{i} by b_{i} and by b_{i}^{−1} are equal to the inverse A_{i}^{−1}.

For 0 ≤ j < dim E_{i} we have (remember e_{1} = f_{1} = b_{i}(e_{1}) = b_{i}(f_{1}))

b_{i}(f_{j+1}) = b_{i}A_{i}^{−j}(f_{1}) = A_{i}^{j}b_{i}(f_{1}) = A_{i}^{j}(e_{1}) = e_{j+1}.

Hence b_{i} is of order two, which shows that A_{i} is strongly invertible over Q. The sum b := ⨁ b_{i} is a rational strong inversion for T_{∗}.

Question: Is the integral homological monodromy of a complex hypersurface singularity strongly invertible? The answer is yes for plane curve singularities. We would like to know the answer for the isolated surface singularity f(x, y, z) = L_{1}L_{2}^{2}L_{3}^{3}L_{4}^{4}L_{5}^{5}L_{6}^{6}L_{7}^{7}L_{8}^{8}L_{9}^{9} + x^{46} + y^{46} + z^{46} where L_{j}, 1 ≤ j ≤ 9 are linear forms on C^{3}, such that the 9 lines {L_{j} = 0} in the complex projective plane span the 9_{3} configuration of flex tangents to a nonsingular cubic. No real equation for this singularity can exist, since the configuration 9_{3} cannot be realized in the real projective plane.

The autor does not know any general feasible algorithmic method for computing the integral homological monodromy of isolated surface singularities. For instance, compute for the surface singularity f(x, y, z) = x^{3} + x^{2}y^{2}z^{2} + yz^{10} + zy^{11} of very low multiplicity 3 the integral monodromy. With SINGULAR you can discover a Jordan block of length 3 for the eigenvalue −1 in the monodromy of f..

Let f(z_{0}, z_{1}, ⋯, z_{n}) ∈ K[z_{0}, z_{1}, ⋯, z_{n}] be a polynomial with coefficients in a number field K. We assume that f has an isolated singularity at 0 ∈ K^{n+1}. How does the local topology at 0 ∈ C^{n=1} of the mapping f^{σ} : C^{n+1} → C change depending upon Galois twists σ? For n = 1 the local topology depends upon the Puiseux pairs of f^{σ}, hence does not change. For n = 2 many interesting changes occur in the local monodromy diffeomorphism. The rational homological monodromy can be computed using the Gauss-Manin connection and the algorithm of Brieskorn, which is implemented in SINGULAR. The interesting changes stay hidden to us, since monodromy computation beyond the rational homological monodromy are not known yet.

So we try global monodromy of mappings f^{σ} : C^{2} → C for f(z_{0}, z_{1}) ∈ K[z_{0}, z_{1}].

We recall from at1999:

A divide P is the image of a relative generic immersion of a 1-manifold (M, ∂M) in the unit 2-disk (D, ∂D). To a divide P ⊂ D is associated a classical link in the 3-sphere, as follows. Consider the tangent space T R^{2} as R^{2} × R^{2} = R^{2} + iR^{2} = C^{2}. The link L(P) of the divide P is the following subset in the unit sphere S^{3} of C^{2}:

L(P) := {(x, u) ∈ T R^{2} = R^{4} | x ∈ P, u ∈ T_{x}P, ||x||^{2} + ||u||^{2} = 1}

The involution c : (x, u) ↦ (x, −u) induces a rigid symmetry of the link L(P), which induces for each component a strong inversion.

Figure 1: A divide with 2 branches

Links of divides tend to be fibered:

Theorem 2 The link of a connected divide is a fibered link. The fibration can be choosen to be invariant by the involution c : (x, u) ↦ (x, −u) with c(F) = F for the fiber F over 1. Moreover, the monodromy diffeomorphism T can be choosen, such we have c_{F} ∘ T ∘ c_{F} = T^{−1}.

The fibration of the link of a divide is very similar to the Milnor fibration of links of plane curve singularities. This holds too for the construction of the fibration, that we explain now.

First we make by a small isotopy of P the divide to be linear with right angles near its crossing points. For a connected divide P ⊂ D(0, ρ), let f_{P} : D(0, ρ) → R be a generic C^{∞} function, such that P is its 0-level and that each region has exactly one local maximum or minimum. Such a function exist for a connected divide and is well defined up to sign and isotopy. In particular, there are no critical points of saddle type other then the crossing points of the divide. We assume moreover that the function f_{P} is quadratic and euclidean in a neighborhood of its critical points, i.e. for euclidean coordinates (X, Y) with center at a critical point c of f_{P} we have in a neighborhood of c the expression f_{P} (X, Y) = f_{P} (c) + XY or f_{P} (X, Y) = f_{P} (c) + X^{2} + Y^{2}. Let χ : D(0, ρ) → [0, 1] be a C^{∞}, positive function, which evaluates to zero outside of the neighborhoods where f_{P} is quadratic and to 1 in some smaller neighborhood of the critical points of f_{P} . Let θ_{P} : ∂B(0, ρ) → C be given by: θ_{P,η}(J(v)) := f_{P} (x) + iηdf_{P} (x)(u) − \frac{1}{2}η^{2}χ(x)H_{f_{P}} (v) for J(v) = (x, u) ∈ T D(0, ρ) = D × R^{2} and η ∈ R, η > 0. Observe that the Hessian H_{f_{P}} is locally constant in a the neighborhood of the critical points of f_{P} . The function θ_{P,η} is C^{∞}. Let π_{P,η} : ∂B(0, ρ) \ L(P) → S^{1} be defined by:π_{P,η}(J(v)) := θ_{P,η}(J(v))/|θ_{P,η}(J(v))|.

Theorem 3 Let P ⊂ D(0, ρ) be a divide, such that the system of immersed curves is connected. The link L(P) in ∂B(0, ρ) = {J(v) | v ∈ T (D(0, ρ)) and ∥J(v)∥ = ρ} is a fibered link. The map π_{P} := π_{P,η} is for η sufficiently small, a fibration of the complement of L(P) over S^{1}. The geometric monodromy is the product of μ positive Dehn twist, where μ is the first Betti number of the fiber.

Connected divides P such that the auxilary Morse function f_{P} has only two critical values are in the world of divides a natural generalization of simple singularities, see [A’C3].

A natural problem is the classification of the knots, that come from simple divides. Since simple divides are “slalom” curves around trees we call the corresponding knots also slalom knots or divides.

Since the Alexander polynomial suffices for the classification of the knots of irreducible plane curve singularities, I had as first naive guess that the Alexander polynomial suffices too for the classification of slalom knots. This turned out to be very naive.

The following two slalom divides P_{1}, P_{2} have equal abstract underlying trees. The knots K(P_{1}) and K(P_{2}) are not equivalent, but related by a mutation. So, many invariants coincide. They have equal Khovanov homology as computed by Alexander Shumakovitch, although Khovanov homology is not invariant under mutation, as shown by Stefan Wehrli [W]. The knots K(P_{1}) and K(P_{2}) are hyperbolic and can be distinguised by the isometry groups of the their complements with the Mostow metrics. Yvonne Gerber has constructed the Thurston foliation on the fibers of hyperbolic slalom links [S]. Her method allows to distinguish the knots K(P_{1}) and K(P_{2}) too. Soo the classification of slalom knots is not typically easy.

Instead of formulating properties, that the knots K(P_{1}) and K(P_{2}) do not have in common, I ask the following question: Do the fundamental groups of the complements of the knots K(P_{1}) and K(P_{2}) have isomorphic profinite completions?

A positive answer would explain why the two knots K(P_{1}) and K(P_{2}) are not easy to separate by knot invariants.

If the answer is yes, a reason could be that the monodromies of the knots K(P_{1}) and K(P_{2}) have the same orbit under the action of G_{Q} := Gal(Q̄/Q) on the profinite completion of the mapping class group Mod_{5,1}.

The second part of the talk is devoted to construct by using the Theorem of Belyi a polynomial function f_{Γ} : C^{2} → C for an finite planar graph. The coefficients of f_{Γ} belong to a number field. We are especially interested to detect the monodromy of the slalom link of the given graph as a global monodromy of the mapping f_{Γ} : C^{2} → C.

As work in progress we propose a construction, such that a Galois transformation of the number field acts. The aim is to relate by a Galois action the monodromies of the slalom knots of trees, that have Galois related Belyi functions.

[AC1] Norbert A’Campo, Real deformations and complex topology of plane curve singularities, Annales de la Faculté des Sciences de Toulouse, (1999).

[AC2] Norbert A’Campo, Generic immersions of curves, knots, monodromy and gordian number, Publ. Math. IHES, (1999).

[AC3] Norbert A’Campo, Planar trees, slalom curves and hyperbolic knots, Publ. Math. IHES, (1999).

[AC4] Norbert A’Campo, ICM1974.

[G] Yvonne Gerber, preprint.

[G-Z] S. M. Gusein-Zade, Index of a singular point of a gradient vector field, Funkcional. Anal. i Prilozen 18 (1984), 6–10. [S] Alexander Shumakovitch,

http://www.geometrie.ch/KhoHo

[W] Stefan Wehrli, http://arXiv.org/

[T] B. Teissier, Un exemple de classe d’équisingularité irrationelle, C. R. Acad. Sci. Paris, 311 2 (1990), 111–113.