Topology of strongly polar weighted homogeneous links
aa r X i v : . [ m a t h . AG ] J a n TOPOLOGY OF STRONGLY POLAR WEIGHTEDHOMOGENEOUS LINKS
VINCENT BLANLŒIL AND MUTSUO OKA
Abstract.
We consider a canonical S action on S which is definedby ( ρ, ( z , z )) ( z ρ p , z ρ q ) for ρ ∈ S and ( z , z ) ∈ S ⊂ C . Weconsider a link consisting of finite orbits of this action, where some of theorbits are reversely oriented. Such a link appears as a link of a certaintype of mixed polynomials. We study the space of such links and showsmooth degeneration relations. Introduction
We consider a mixed polynomial f ( z , ¯ z ) = P ν,µ c ν,µ z ν ¯ z µ where z =( z , . . . , z n ), ¯ z = (¯ z , . . . , ¯ z n ), z ν = z ν · · · z ν n n for ν = ( ν , . . . , ν n ) (respec-tively ¯ z µ = ¯ z µ · · · ¯ z µ n n for µ = ( µ , . . . , µ n )). Definition 1.
We say f ( z , ¯ z ) is a mixed weighted homogeneous polynomialof radial weight type ( q , . . . , q n ; d r ) and of polar weight type ( p , . . . , p n ; d p )if n X j =1 q j ( ν j + µ j ) = d r , n X j =1 p j ( ν j − µ j ) = d p , if c ν,µ = 0 . Let f be a mixed weighted homogeneous polynomial. Using a polar coor-dinate ( r, η ) of C ∗ where r > η ∈ S with S = { η ∈ C | | η | = 1 } , wedefine a polar C ∗ -action on C n by( r, η ) ◦ z = ( r q η p z , . . . , r q n η p n z n ) , ( r, η ) ∈ R + × S ( r, η ) ◦ ¯ z = ( r, η ) ◦ z = ( r q η − p ¯ z , . . . , r q n η − p n ¯ z n ) . More precisely, it is a R + × S -action. Then f satisfies the functional equality f (( r, η ) ◦ ( z , ¯ z )) = r d r η d p f ( z , ¯ z ) . (1)This notion was introduced by Ruas-Seade-Verjovsky [12] and Cisneros-Molina [3].A mixed polynomial f ( z , ¯ z ) is called strongly polar weighted homogeneous if the polar weight and the radial weight coincide, i.e., p j = q j , ≤ j ≤ n .In this case, the C ∗ action is simply defined by ζ ◦ z = ( z ζ p , . . . , z n ζ p n ) , ζ ∈ C ∗ . In this paper, we study the geometry of the links defined by strongly polarweighted homogeneous mixed polynomials. rr K S n +2 × { } rr K S n +2 × { } S n +2 × [0 , Figure 1.
A cobordism between K and K Cobordism of links
First of all we have to point out that the topology of mixed links is veryparticular and we recall some classical results and definitions in the case ofknots and algebraic links.Let K be a closed 2 k − k + 1)-dimensional sphere S k +1 . We suppose that K is ( k − ≥ K is orientable, we further assume that it is oriented. Then we call K or its (oriented) isotopy class an 2 k − -knot Definition 2.
Two 2 k − K and K in S k +1 are said to be cobor-dant if there exists a properly embedded (2 k )-dimensional manifold X of S k +1 × [0 ,
1] such that(1) X is diffeomorphic to K × [0 , ∂X = ( K × { } ) ∪ ( K × { } ) . The manifold X is called a cobordism between K and K . When the knotsare oriented, we say that K and K are oriented cobordant (or simply cobordant ) if there exists an oriented cobordism X between them such that ∂X = ( − K × { } ) ∪ ( K × { } ) , where − K is obtained from K by reversing the orientation.It is clear that isotopic knots are always cobordant. However, the converseis not true in general (see Fig. 2).For a classification of high dimensional knots up to cobordism we refer to[2].Let us study one example of dimensional one links. We denote by T + and T − respectively the one dimensional right and the left trefoil knots (whichare both mixed links). We know that T + and T − are cobordant, see [11] p.219 ; but let us give here the idea of the proof. Recall that a manifold with boundary Y embedded in a manifold X with boundaryis said to be properly embedded if ∂Y = ∂X ∩ Y and Y is transverse to ∂X . OPOLOGY OF STRONGLY POLAR WEIGHTED HOMOGENEOUS LINKS 3 r K r K Figure 2.
A cobordism which is not an isotopyPrecisely, we denote by S (resp. S − ) the upper (resp. lower) hemisphereof the unit 3-sphere ∂D = S ֒ → R . Set E be the equatorial hyperplaneof D , and let π : R → E the orthogonal projection onto E .One can suppose that T + and T − , which is the mirror image of T + , areembedded in S and S − respectively such that T − = − (cid:0) π ( T + ) × [0 , (cid:1) ∩ S − . Then we construct the connected sum O = T + T − of T + and T − in S ;we illustrate this construction in Fig. 3.Set ˜ T + (resp. ˜ T − ) the intersection ˜ T + = O ∩ S (resp. ˜ T = O ∩ S − ).One can assume that the connected sum O is made in order to have˜ T − = − (cid:0) π ( ˜ T + ) × [0 , (cid:1) ∩ S − . Now, if we denote D = (cid:0) π ( ˜ T + ) × [0 , (cid:1) ∩ D , then D is homeomorphic to a 2-disk since π ( ˜ T + ) is a 1-disk. Moreover ∂ D = O = T + T − . Since O bounds a 2-disk embedded in D then O isnull cobordant, and, T + and T − are cobordant. In [4] D. T. Lˆe proved that the Alexander polynomial determines thetopological type of the link of an isolated singularity of a complex analyticcurve and moreover he proved that cobordant links are isotopic since theproduct of their Alexander polynomials is a square.In the case of mixed links things are different. For example the two trefoilknots T + and T − are cobordant but not isotopic mixed links. Recall thatthey are not isotopic since they have distincts Jones polynomials. Remark 3.
Moreover, since the trivial knot O is a mixed link, then theconnected sum of mixed one dimensional links can be a mixed link contraryto the classical case as proved by N. A’Campo [1]. The sign is necessary to have the right orientation. Remark that the trefoil knot is homeomorphic to a sphere, then to prove that T + and T − are cobordant it is sufficient to prove that their connected sum bounds a disk [5]. VINCENT BLANLŒIL AND MUTSUO OKA ✲ T + ✛ T − E S Figure 3.
The connected sum of the trefoil knot and itsinverse in S Strongly polar weighted homogeneous links
Hereafter we consider strongly polar weighted homogeneous polynomial f ( z , ¯ z ) of two variables i.e., n = 2 with weight vector P = t ( p, q ). Herewe assume that gcd( p, q ) = 1. We assume that f is convenient and non-degenerate so that the link L = f − (0) ∩ S is smooth. Let M ( P ; d p ) bethe space of strongly polar weighted homogeneous mixed polynomials of thepolar degree d p , which is non-degenerate convenient and let L ( P ; d p ) be theassociated oriented links. Hereafter we denote simply M , L for M ( P ; d p )and L ( P ; d p ) respectively. We have a canonical mapping π : M → L definedby π ( f ) the link defined by f − (0) ∩ S . A difficulty in the mixed polyno-mial situation is that for a fixed link, there exist an infinitely many mixedpolynomials which define the link.Let d r , d p be the radial and polar degrees respectively. As f is assumedto be convenient, f contains monomials z a ¯ z b and z a ¯ z b such that p ( a + b ) = q ( a + b ) = d r , p ( a − b ) = q ( a − b ) = d p . Therefore pq = a a = b b and we see that p | a , b and q | a , b and thus pq | d r , d p . As our link is S invariant, its component is a finite union of orbits of the action. Recall thatthe associated S -action is defined by S × S → S , ( ρ, ( z , z )) ( z ρ p , z ρ q ) , ρ ∈ S Let P = ( p, q ) be the primitive weight vector of f . P is fixed throughoutthis paper. Note that L is stable under the action, by the Euler equiality f ( ρ ◦ z ) = ρ d p f ( z ) . Two orbit z = 0 and z = 0 are singular but by the covenience assumption,our link has only regular orbits. OPOLOGY OF STRONGLY POLAR WEIGHTED HOMOGENEOUS LINKS 5
Coordinates of the orbits.
Take a regular orbit L . We can take apoint X = ( β , β ) ∈ L ⊂ S ⊂ C such that β is a positive number. β and | β | are unique by L but β is not unique. The umbiguity is the actionof Z /p Z . Thus | β | = p − β and the argument of β is unique mudulo2 π/p . Thus the space of the regular orbits is isomorphic to the punctureddisk ∆ ∗ := { ξ = rρ ∈ C | < r < , ρ ∈ S } , by the correspondence β β p ∈ ∆ ∗ . For u = r p e iθ ∈ ∆ ∗ p , we associate the regular orbit K ( u ) := { ( ρ p p − r , ρ q re iθ/p ) | ρ ∈ S } , u = re iθ ∈ ∆ ∗ . (2)Consider a strongly polar weighted homogeneous polynomial for arbitrarynon-negative integer k : ( ℓ u,k ( z ) := z q + kq ¯ z kq − α u,k z p + kp ¯ z kp = z q k z q k k − α u,k z p k z p k k ¯ ℓ u,k ( z ) := ¯ z q k z q k k − α u,k ¯ z p k z p k k where(3) α u,k = (1 − r ) q (1 / k ) r p (1+2 k ) e iθ . (4)Note that the polar degrees of α u,k are pq but the radial degrees are differentand they are given as rdeg ℓ u,k = (2 k + 1) pq. Observation 1.
The polynomials ℓ u,k define K ( u ) and ¯ ℓ u,k defines K ( u ) with reversed orientation for any k = 0 , , . . . Hereafter we simply use the notation: ℓ u ( z ) := ℓ u, ( z ) = z q − (1 − r ) q/ r p e iθ z p , u = r p e iθ . Let L ( P ; dpq, r ) be the subspace of L ( P ; dpq ) which has d + 2 r componentswhere r components are negatively oriented. First we prepare the nextlemma: Lemma 2.
The moduli space L ( P ; dpq, r ) is connected and therefore anytwo links of this moduli has the same topology.Proof. Note that L ( P ; dpq, r ) are parametrized by M d,r := (∆ ∗ ) d +2 r \ Ξwhere Ξ = { u = ( u , . . . , u d +2 r ) ∈ ∆ ∗ ( d +2 r ) | u i = u j ( ∃ i, j, i = j ) } . Thus itis easy to see that M d +2 r is connected. u corresponds to the link ∪ d +2 ri =1 K ( u i )where K ( u j ) are reversely oriented for j = d + r + 1 , . . . , d + 2 r . (cid:3) Typical degeneration.
We consider an important degeneration oflinks L ( t ) , t ∈ C which is defined by the family of strongly polar weightedhomogeneous polynomials: f ( z , ¯ z , t ) = − z p ¯ z p + z q ¯ z q + tz p ¯ z q . VINCENT BLANLŒIL AND MUTSUO OKA
Using Proposition 1 ([7]), we see that the degeneration locus is given as thefollowing real semi-algebraic varietyΣ := { t ∈ C | t = 2 s − s , ∃ s ∈ S } Figure 1 shows the graph of Σ. Let Ω be the bounded region surrounded byΣ. By Example 59 in [8], we can see the following.
Proposition 3.
For any t ∈ Ω , L ( t ) has one link component, while for t ∈ C \ ¯Ω (= the outside of Σ ), L ( t ) has three components.Proof. Let us consider the weighted projective space P ( P ) := C \ { O } / C ∗ by the above C ∗ -action. For U := P ( P ) ∩ { z z = 0 } , it is easy to see that u := z p /z q is a coordinate function. Our link corresponds to the solutions(=zero points) of − u ¯ u + tu + 1 = 0and there exists one solution (respectively 3 solutions) for each t ∈ Ω (resp. t / ∈ ¯Ω). See Example 59, [8] or [9]. (cid:3) We consider the point − ∈ Σ which is a smoot point of Σ. There are twocomponents for L ( −
3) ( u = 1 / u = −
1) and the component passingthrough (1 , e iπ/p ) is a doubled component. Here we are considering the linkon the sphere of radius √ S √ for simplicity. Let us consider the variety: W = { ( z , z , t ) ∈ S √ × R | − − ε ≤ t ≤ − ε, f ( z , ¯ z , t ) = 0 } , ε ≪ . The following is the key assertion.
Lemma 4. W is a smooth manifold with boundary L ( − − ε ) ∪ − L ( − ε ) .Proof. Let f ( z , ¯ z , t ) = g ( z , ¯ z , t ) + i h ( z , ¯ z , t ). We assert that W is a completeintersection variety. For this purpose, we show that three 1-forms dg, dh, dρ are independent on L ( − ρ ( z ) = k z k . As the polynomial f isstrongly polar weighted homogeneous, it is enough to check the assertion on apoint ˜ z = (1 , α, − ∈ W where α = e iπ/p . For the calculation’s simplicity,we use the base { dz , d ¯ z , dz , d ¯ z , dt } of the complexified cotangent space.Using the equalities g = ( f + ¯ f ) / , h = ( f − ¯ f ) / (2 i ), we get dg (˜ z ) dh (˜ z ) dρ (˜ z ) = A dz d ¯ z dz d ¯ z dt where A = − iq iq ip ¯ α − ipα
01 1 ¯ α α Thus it is easy to see that rank A = 3. (cid:3)
OPOLOGY OF STRONGLY POLAR WEIGHTED HOMOGENEOUS LINKS 7 –2–1012–3 –2 –1 1
Figure 4.
Σ3.3.
Milnor fibrations.
Take u = ( u , . . . , u d +2 r ) ∈ M d,r and consider thecorresponding link L ( u ) = ∪ d +2 rj =1 K ( u j ) with d + 2 r components and thelast r components are negatively oriented. Let f ( z ) be a strongly polarweighted homogeneous polynomial which defines L ( u ) with pdeg f = dpq and rdeg f = ( d + 2 s ) pq with s ≥ r . For example, we can take g ( z ) = ℓ u ,s − r ( z ) d + r Y j =2 ℓ u j ( z ) d +2 r Y j = d + r +1 ¯ ℓ u j ( z ) . Let F be the Milnor fiber of f : F = { z ∈ S | f ( z ) > } . As we assume that L ( u ) has no singular orbit, f ( z ) is a convenient mixed polynomial. Thus itcontains monomials z ( d + s ) q ¯ z qs and z ( d + s ) p ¯ z ps . The monodromy h : F → F is defined by h ( z ) = e πi/dpq ◦ z and it is the restriction of S -action to Z dpq ⊂ S . Thus we have a commutative diagram: F ֒ → S \ L ( u ) ց ξ y π P ( P ) \ W where W is d + 2 r points corresponding to the components of L ( u ). π, ξ are canonical quotient mapping by S and Z dpq respectively. As F is a Z dpq cyclic covering over P \ W , with two singular points (0 ,
1) and (1 , q, p points respectively. Thuswe have Proposition 5. (cf. Theorem 65, [8] ) The Euler charactersitic of F is givenas χ ( F ) = − ( d + 2 r ) dpq + p + q Note that χ ( F ) depends on the number of components d + 2 r but it doesnot depend on the radial degree ( d + 2 s ) pq . Thus we see that, under fixedpolar and radial degrees, there are s + 1 different topologies among theirMilnor fibrations. The components types can be d + 2 r, r = 0 , . . . , s . VINCENT BLANLŒIL AND MUTSUO OKA main result Consider a smooth family of strongly polar weighted homogeneous links L ( t ) ∈ L ( P ; dpq ) , ≤ t ≤ P = t ( p, q ) such that(1) the variety W = { ( z , t ) ∈ S × [0 , z ∈ L ( t ) } is a smooth variety ofcodimension two.(2) There exists t , < t < ( L ( t ) ∈ L ( P ; dpq, r − t < t L ( t ) ∈ L ( P ; dpq, r ) t > t The link L ( t ) is singular. One component is the limit of two componentswith opposite orientations. We call such a family a smooth elimination of apair of links . Theorem 6.
For any link L ∈ L ( P ; dpq, r ) with r > , there exists asmooth elimination family L ( t ) of a pair of links with L (0) = L and L (1) ∈L ( P ; dpq, r − . Corollary 7.
For any link L ∈ L ( P ; dpq, r ) with r > , r pairs of linkswith opposite orientations can be eliminated successively to a link L ′ ∈L ( P ; dpq, of positive link. L ′ is isomorphic to a holomorphic torus linkdefined by z qd − z pd = 0 . References [1] N. A’Campo. Le nombre de Lefschetz d’une monodromie. In
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OPOLOGY OF STRONGLY POLAR WEIGHTED HOMOGENEOUS LINKS 9 (V. Blanlœil)
IRMA, UFR Math´ematiques et Informatique, Universit´e de Stras-bourg, 7, rue Ren´e Descartes, F-67084 STRASBOURG
E-mail address : [email protected] (M. OKA) Department of Mathematics, Tokyo University of Science, 26Wakamiya-cho, Shinjuku-ku, Tokyo 162-8601
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