Immersed curves, knots and singularities

Norbert A'Campo

Vortrag Arbeitstagung 1999 Bonn.

A divide is a relative generic immersion of a 1-manifold $(M,\partial{M})$ in the unit 2-disk $(D,\partial{D})$. 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 $P \subset D$ is associated a classical link in the 3-sphere. Consider the tangent space $T{\bf R}^2$ as ${\bf R}^2 \times {\bf R}^2={\bf R}^2+i{\bf R}^2={\bf C}^2$. The link L(P) of the picture P is the following subset in the unit sphere S³ of ${\bf C}^2$:

$$L(P):=\{(x,u) \in T{\bf R}^2={\bf R}^4 \mid x \in P, u \in T_xP, \vert\vert x\vert\vert^2+\vert\vert u\vert\vert^2=1 \}$$

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 $D-{1 \over \sqrt{2}}D$ 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 TS³. It follows that the link L(P) has natural orientation. Links L(P) of pictures P have special properties. For instance, the involution $(x,u) \to (x,-u)$ 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 1   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 $r,\delta > 0$ there are infinitely many links with r components and with unknotting number $\delta$, but only a finite number of transversal isotopy classes of pictures with r branches and $\delta$ 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 10₁₃₉, 10₁₄₅, 10₁₅₂ 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 10₁₃₉ 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 10₁₃₉.

Links of pictures tend to be fibered.

Theorem 2   The link of a connected picture is a fibered link.

The proof uses a complex function f_P on $TD \subset {\bf C}^2$ which satisfies the Cauchy-Riemann equations along the zero section D in TD, 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 ${\bf R}^2$.

Let the polynomial function $f:{\bf C}^2 \to {\bf C}$ has at $0 \in {\bf C}^2$ an isolated singularity. We may assume without restricting the local topology of the singularity that the factorization of f into iReducible local branches has only real factors. Let $B(f) \subset S^3$ be the local link of the singularity of f at 0. Let $P(\bar{f})$ be the picture of some small real deformation $\bar{f}$ with the maximal number $\delta$ of local double points of the singularity. Hence, the Milnor number of the singularity is $2\delta-r+1$, 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 3   The links B(f) and $L(P(\bar{f}))$ are equivalent.

An embedded tree B in the unit disk D, such that the intersection $B \cap \partial{D}$ 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 $P_B \subset D$ of the interval [0,1] with the following properties:

(i) The immersion is relative, i.e. the endpoints are embedded in $\partial{D}$.

(ii) The immersion is generic, i.e. there are only transversal crossing points, only the endpoints lie on $\partial{D}$ and the immersion is transversal to $\partial{D}$.

(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 $D \setminus P_B$ 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 E₁₀ and slalom.

The Dynkin diagram $\Delta_B$ 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 D₆ but the Dynkin diagram $\Delta_B$ of P_B has 10 vertices and we can denote it by E₁₀. The Dynkin diagram $\Delta_B$ 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 $\Delta_B$ has the property that the terminal vertices of $\Delta_B$ different from the root, are never new.

Theorem 4   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 $\Delta_B$ is neither the diagram $A_{2k},\ 1 \leq k,$ nor the diagrams E₆ or E₈.

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 $\Delta_B$ is among $A_{2k},\ 1 \leq k,\ E_6,\ E_8$, 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 4: Fundamental domain for the complement of the slalom knot E₁₀.

With the help of SnapPea and Snap, one gets that the complement M of the slalom knot E₁₀ with its hyperbolic structure is triangulated by 3 isometric ideal simplices with cross ratios $[0,1,B,\infty]=-z^2+z, [0,1,A,B]=-z^2+z, [0,B,C,\infty]=-z+1$, where z is the root with negative imaginairy part of x³-x²+1=0 (see Fig. 3, $Z=\infty$). Hence B=-z² + z, A=-2/5z²+3/5z+1/5, C=-z²+z-1. The fundamental group of M is generated by 2 elements, which can be chosen such that the coResponding hyperbolic motions are the fractional transformations

$$\left(\matrix{-z^2 + z& z^2\cr z& z^2 - 1\cr}\right)\ \left(\matrix{z+1& -1\cr 2z^2+z+1& -z^2 - 1\cr}\right)$$

Figure 5: The slalom knot E₁₀.

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 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.

[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 10₁₃₉ and 10₁₅₂ are 4, Osaka J. Math. 35 (1998), 3, 539-546.