Immersed curves, knots and singularities

Norbert A’Campo

A divide is a relative generic immersion of a 1-manifold (M, ∂M) in the unit 2-disk (D, ∂D). In this talk a divide will be an immersion of a finite union of copies of [0, 1] and will be called a picture. To a picture P ⊂ D is associated a classical link in the 3-sphere. Consider the tangent space T ℝ^{2} as ℝ^{2} ×ℝ^{2} = ℝ^{2} +iℝ^{2} = ℂ^{2}. The link L(P) of the picture 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 link L(P) does not change its type if we move P with an isotopy in the space of generic relative immersions. So we may assume that in the collar D − \frac{1}{\sqrt{2}}D the picture P consists of radial line segments, and we see that the link L(P) is transversal to the contact structure of complex planes in T S^{3}. It follows that the link L(P) has natural orientation. Links L(P) of pictures P have special properties. For instance, the involution (x, u) → (x, −u) induces a rigid symmetry of the link L(P), which induces for each component a strong inversion.

Figure 1: A picture with 2 branches

The unknotting number of the link of a picture is computed by:

Theorem. The unknotting number and the 4-ball genus of the link L(P) of the picture P is the number of double points of the immersion P.

The proof relies on the positive answer of Kronheimer and Mrowka to the Thom conjecture.

For given natural numbers r, δ > 0 there are infinitely many links with r components and with unknotting number δ, but only a finite number of transversal isotopy classes of pictures with r branches and δ double points. So we see that links L(P) of pictures are as rare among links as orchids among flowers. It is a surprise that the knots 10_{139}, 10_{145}, 10_{152} of the table of Rolfsen, for which the unknotting number has been determined only recently by Tomomi Kawamura [K], are knots of pictures. The knot 10_{139} is very rare since it is a knot of two pictures which can not be related by an transversal isotopy (see Fig. 2).

Figure 2: Two pictures for the knot 10_{139}.

Links of pictures tend to be fibered.

Theorem. The link of a connected picture is a fibered link.

The proof uses a complex function f_{P} on T D ⊂ ℂ^{2} which satisfies the Cauchy-Riemann equations along the zero section D in T D, such that the 0-level of f_{P} intersects D along P.

The geometric monodromy can be read from the picture in a similar way as the monodromy of a plane curve singularity from a picture provided by a small local real deformation of the singularity with the maximal possible number of double points in ℝ^{2}.

Let the polynomial function f : ℂ^{2} → ℂ has at 0 ∈ ℂ^{2} an isolated singularity. We may assume without restricting the local topology of the singularity that the factorization of f into irreducible local branches has only real factors. Let B(f) ⊂ S^{3} be the local link of the singularity of f at 0. Let P (f̄) be the picture of some small real deformation f̄ with the maximal number δ of local double points of the singularity. Hence, the Milnor number of the singularity is 2δ−r+1, where r is the number of local branches (see AT1974). The following theorem gives new insight for the local topology of plane curve singularities.

Theorem. The links B(f) and L(P (f̄)) are equivalent.

An embedded tree B in the unit disk D, such that the intersection B ∩ ∂D consists of one terminal vertex r of B, is called a rooted planar tree. For a rooted planar tree B there exists an immersed copy P_{B} ⊂ D of the interval [0, 1] with the following properties:

(i) The immersion is relative, i.e. the endpoints are embedded in ∂D. (ii) The immersion is generic, i.e. there are only transversal crossing points, only the endpoints lie on ∂D and the immersion is transversal to ∂D. (iii) The double points of P_{B} lie in the interior of the edges of B, such that the local branches are transversal to the edge of B. (iv) Each connected component of D \ P_{B} contains exactly one vertex of B.

(v) The only intersection points of P_{B} with B are the double points of P_{B}.

The immersed curve P_{B} is well-defined up to regular relative isotopy and is called the slalom curve of the rooted planar tree B, see Fig. 3. The left picture of Fig. 2 is a slalom curve.

Figure 3: Rooted planar tree, its Dynkin diagram E_{10} and slalom.

The Dynkin diagram Δ_{B} of the slalom curve P_{B} is deduced from the rooted tree B as follows: First make a new tree B′ by subdividing each edge of B with a new vertex, which is placed at the crossing point of P_{B} on the edge; next, remove from B′ the root vertex r and the terminal edge of B′ pointing to r. In Fig. 3 the tree B has the shape of the classical Dynkin diagram D_{6} but the Dynkin diagram Δ_{B} of P_{B} has 10 vertices and we can denote it by E_{10}. The Dynkin diagram Δ_{B} of a rooted tree B is a bicolored rooted tree with an embedding in the plane. The root is the new vertex which lies on the edge of B originating from the root point of B and the bicoloring is such that the new vertices are of the same color. Moreover, the Dynkin diagram Δ_{B} has the property that the terminal vertices of Δ_{B} different from the root, are never new.

Theorem. Let B be a rooted tree. The complement of the slalom knot K_{B} admits a complete hyperbolic metric of finite volume if and only if the Dynkin diagram Δ_{B} is neither the diagram A_{2k}, 1 ≤ k, nor the diagrams E_{6} or E_{8}.

Using in addition to rooted trees also “disk-wide-webs” the theorem has a formulation which includes also extended Dynkin diagrams.

If the Dynkin diagram Δ_{B} is among A_{2k}, 1 ≤ k, E_{6}, E_{8}, the knot K_{B} is the torus knot (2, 2k + 1), (3, 4) or (3, 5) and appears as the local knot of a simple plane curve singularity; the monodromy diffeomorphism (with free boundary) of the knot K_{B} can be chosen to be of finite order in those cases and its complement does not carry a complete hyperbolic metric.

Figure 3: Fundamental domain for the complement of the slalom knot E_{10}.

With the help of SnapPea and Snap, one gets that the complement M of the slalom knot E_{10} with its hyperbolic structure is triangulated by 3 isometric ideal simplices with cross ratios [0, 1, B, ∞] = −z^{2} + z, [0, 1, A, B] = −z^{2} + z, [0, B, C, ∞] = −z + 1, where z is the root with negative imaginairy part of x^{3} − x^{2} + 1 = 0 (see Fig. 4). Hence B = −z^{2} + z, A = −2/5z^{2} + 3/5z + 1/5, C = −z^{2} + z − 1. The fundamental group of M is generated by 2 elements, which can be chosen such that the corresponding hyperbolic motions are the fractional transformations

( \begin{matrix}−z^{2} + z & z^{2} \\ z & z^{2} − 1 \end{matrix} ) ( \begin{matrix}z + 1 & −1 \\ 2z^{2} + z + 1 & −z^{2} − 1 \end{matrix} )

From the above theorem we get many examples of hyperbolic fibered knots, whose monodromy diffeomorphism and gordian number are known explicitly. The monodromy diffeomorphism of a slalom knot can be realized as a product of right

Figure 4: The slalom knot E_{10}.

Dehn twists of a system of simple closed curves on the fiber surface, such that the union of the curves is a spline in the fiber surface and the dual graph of the system is the Dynkin diagram of the rooted tree; the gordian number of a slalom knot equals the number of crossings of the slalom divide. We call the isotopy class of the monodromy diffeomorphism of the slalom knot of a rooted tree the Coxeter diffeomorphism of the Dynkin diagram of the rooted tree. It follows from the theorem and a theorem of Thurston that a Coxeter diffeomorphism of the Dynkin diagram of a rooted tree is pseudo-anosov if and only if the Dynkin diagram is not a classical Dynkin diagram.

Slalom knots can be characterized by the following properties:

- the knot is prime and fibered,

- the monodromy has minimal complexity 4g − 1, where g is the genus of the knot,

- the monodromy is a product of Dehn twists, which all belong to the same conjugacy class in the mapping class group.

The complexity of an element T of the mapping class group, which we use here, is the minimum of the quantity a + b over all the product decompositions of T as product of Dehn twists, where a is the length of the product and where b is the number of mutual intersection points of the core curves of the twists involved in the product decomposition.

References

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

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

[AC3] Norbert A’Campo, Planar trees, slalom curves and hyperbolic knots, Publ. Math. IHES, to appear.

[K] Tomomi Kawamura, The unknotting numbers of 10_{139} and 10_{152} are 4, Osaka J. Math. 35 (1998), 3, 539–546.