AAlgebraic links in lens spaces
Eva HorvatUniversity of Ljubljana, Faculty of EducationKardeljeva ploˇsˇcad 16, 1000 Ljubljana, [email protected] 10, 2020
Abstract
The lens space L p,q is the orbit space of a Z p -action on the three sphere. Weinvestigate polynomials of two complex variables that are invariant under thisaction, and thus define links in L p,q . We study properties of these links, andtheir relationship with the classical algebraic links. We prove that all algebraiclinks in lens spaces are fibered, and obtain results about their Seifert genus.We find some examples of algebraic knots in L p,q , whose lift in the 3-sphere isa torus link. Keywords : complex singularities, links in lens spaces, algebraic links
Classical algebraic links have been studied almost a century ago [1, 2, 8]. Complex alge-braic curves and links of their singularities present a field where complex algebraic geometrymeets topology and knot theory. Because of their rich (and at the same time rigid) struc-ture, they might still present a valuable insight into differential geometric structures (suchas contact structures) on the ambient 3-manifold.Our interest lies in exploring links in lens spaces, which is a relatively young branch ofknot theory. As the classical algebraic links in the 3-sphere represent a special class of links(so-called iterated torus links ), we would like to know which links play the correspondingrole in lens spaces. Do they share the topological properties of the classical algebraic links?Can we find examples of algebraic links in L p,q using our knowledge about their lifts in the a r X i v : . [ m a t h . G T ] A ug -sphere?The lens space L p,q is the orbit space of an action of the cyclic group G p,q ∼ = Z p on S . We define polynomials f ( x, y ) of two complex variables, whose corresponding variety V = f − (0) is invariant under this action. A classical link (cid:101) K of such a polynomial is thelift of a link K in the lens space L p,q , which we call an algebraic link . We show thatevery algebraic link in L p,q bounds a smooth surface (Proposition 3.9, Lemma 3.10) and istherefore nullhomologous. We investigate the relationship between the number of compo-nents of an algebraic link in a lens space and its lift in the 3-sphere, and show that the liftof every algebraic knot in L p,q has p -components (Corollary 3.13). In Theorem 3.16, weshow that every algebraic link K in a lens space L p,q defines a fibration Ψ : L p,q \ K → S .In Proposition 3.20, we obtain a formula that relates the Seifert genus of an algebraic knotin L p,q and that of its lift in S . In Corollary 4.2, we show that every torus link T ( a, b )with gcd( a, b ) = p is the lift of an algebraic knot in L p,q .The paper is organized as follows. In Section 2, we summarize the background materialwhich will be used in the paper. In Subsection 2.1, we recall some basic facts about linksof complex curve singularities. Subsection 2.2 contains the definition of lens spaces L p,q and recalls their construction using surgery. In Subsection 2.3, we briefly describe variousdiagrams of a link K in L p,q and recall the procedure that turns a diagram of K into thediagram of its lift in the 3-sphere. Section 3 is the core of the paper. We define ( p, q )-invariant polynomials of two complex variables and show that such a polynomial defines alink in the lens space L p,q . We show that every algebraic link in L p,q is nullhomologous,and obtain implications about the number of its components. Moreover, we show that thespace L p,q \ K is a smooth fiber bundle over S , and obtain the relationship between theSeifert genus of K and that of its lift (cid:101) K in S . In Section 4 we determine which algebraicknots/links in L p,q are lifted to torus links in S , and show some examples. We concludeby listing several open problems in Section 5. Throughout the paper, we will denote by S (cid:15) = { ( x, y ) ∈ C | | x | + | y | = (cid:15) } the 3-spherewith radius (cid:15) , centered at the origin of C . As usual, S will be denoted by S . Singularities of complex hypersurfaces in C n were explored by Milnor [15]. Here we recallsome classical results about singular points of complex curves in C .Let f : C → C be a non-constant polynomial in two complex variables with f (0 ,
0) = 0.We require that f is either irreducible or it is a product of distinct irreducible polynomials. hen f defines an algebraic curve V = f − (0) with a finite number of singular points.Denoting by r the number of local analytic branches of V passing through the origin, theintersection K = V ∩ S (cid:15) is a smooth compact 1-manifold; an r -component link in the 3-sphere S (cid:15) [15, Corollary2.9]. Moreover, the mapping Φ : S (cid:15) \ K → S , given by Φ( x, y ) = f ( x,y ) | f ( x,y ) | , is the projectionmap of a smooth fiber bundle [15, Theorem 4.8]. Definition 2.1.
A link K in S is called algebraic if there exists a complex polynomial f ( x, y ) with f (0 ,
0) = 0 and some (cid:15) > , such that the link f − (0) ∩ S (cid:15) is isotopic to K . The simplest family of algebraic links consists of torus links. A torus link is a linkthat lies on the boundary of an unknotted solid torus in S . It is specified by a pair ofintegers a and b : the torus link T ( a, b ) winds a times along the meridian and b timesalong the preferred longitude of the solid torus. The torus link T ( a, b ) is an algebraic link,associated with the complex polynomial f ( x, y ) = x a + y b .In general, links that arise at isolated singularities of complex curves are the so-called iterated torus links (tubular links), that were initially studied in [1], [8], [2]. To helpvisualization, the sphere S (cid:15) is usually represented by the union of two solid tori: (cid:8) ( x, y ) ∈ C | | x | ≤ | y | or | y | ≤ | x | (cid:9) = T ∪ T (1)and by choosing suitable coordinates, we may assume that the link corresponding to K lieswithin the solid torus T .Suppose that a branch of the algebraic curve V has the Puiseux expansion y = a x N m + a x N m + a x N m + . . . , where m ≤ N < N < N < . . . . The above expansion may be rewritten as y = a x n m + a x n m m + a x n m m m + . . . , (2)where m i and n i are coprime numbers with m ≤ n , n m < n , n m < n , . . . . Thecomponent of K , arising from this branch, is isotopic to the knot A k , where k is the smallestinteger for which m m . . . m k = m , and the knots A i are defined inductively as follows. A is the torus knot T ( n , m ) that lies on the boundary of the solid torus T . For i ≥ A i lies on the boundary of the regular neighborhood ν i − of A i − , winding n i times along the meridian and m i times along the prefered longitude of ν i − . The knot A k thus defined is sometimes denoted by the symbol { ( m , n ); ( m , n ); . . . ; ( m k , n k ) } [16]. Let p and q be relatively prime integers, and let ζ ∈ C be a primitive p th root of unity.Then ζ acts on the 3-sphere S (cid:15) by ζ · ( x, y ) = ( ζx, ζ q y ) . (3)Denote by G p,q the cyclic group (cid:104) ζ (cid:105) ∼ = Z p , generated by this action. The orbit space S (cid:15) / (cid:104) ζ (cid:105) is the lens space L p,q . The quotient map π : S (cid:15) → L p,q is a covering projection. Recall thefollowing well-known result. Theorem 2.2. [10, Theorem 7.13] Suppose (cid:102) M is a smooth manifold, and a discrete Liegroup Γ acts smoothly, freely, and properly on (cid:102) M . Then (cid:102) M / Γ is a topological manifold andhas a unique smooth structure such that π : (cid:102) M → (cid:102) M / Γ is a smooth covering map. Another useful representation of the lens space L p,q is the following. Decompose the 3-sphere S (cid:15) into two solid tori T and T as in (1). Denote by µ i (respectively λ i ) the meridian(respectively longitude) of the solid torus T i . Remove T from S (cid:15) and reglue it back alongthe boundary of T by a homeomorphism φ : ∂T → ∂T , for which φ ∗ ( µ ) = pλ − qµ . he resulting 3-manifold is precisely the lens space L p,q . In other words, L p,q is obtainedfrom the 3-sphere by a ( p, − q ) surgery on the solid torus T , and the corresponding Kirbydiagram of L p,q is an unknot with framing − pq . L p,q Knots and links in lens spaces were studied in [3, 6, 13]. We briefly recall some basicnotions which will be used in the paper.A link K inside a lens space L p,q may be represented in different ways. We may draw K inside the Kirby diagram of L p,q mentioned above, to obtain a so-called mixed linkdiagram. Since the surgery link is simply an unknot U , we may use a projection in which U intersects the plane of the diagram transversely in a single point, which we denote bya dot. In this way, we obtain a punctured disk diagram of K . By cutting the planeof the punctured disk diagram along a ray whose endpoint is the dot, we obtain a banddiagram of K . U K
Figure 2: The mixed link diagram (left), punctured disk diagram (middle) and theband diagram (right) of a knot K in L p,q Starting from a band diagram of a link K in L p,q , it is possible to obtain a diagram forthe lift of K in the 3-sphere [13]. The Garside braid ∆ n on n strands is given by∆ n = ( σ n − σ n − . . . σ )( σ n − . . . σ ) . . . σ , where σ i are the standard Artin’s generators of the braid group B n . Proposition 2.3. [13, Proposition 6.4] Let K be a link in the lens space L p,q and let B K be a band diagram of K with n boundary points. Then a diagram for the lift (cid:101) K in the3-sphere S can be found by juxtaposing p copies of B K and closing them with the braid ∆ qn . xample 2.4. Consider the knot K in the lens space L , , which is given by the banddiagram on the right of Figure 2. The band diagram has 2 boundary points, and ∆ = σ .By Proposition 2.3, the lift of K in the 3-sphere is given by the diagram in Figure 3. Figure 3: The diagram of (cid:101) K , the lift of K in S (see Example 2.4) We are going to introduce complex polynomials whose singular points define links in lensspaces. Throughout this Section, we denote by p, q two coprime integers, and by ζ ∈ C aprimitive p th root of unity.The radius function ρ ( x, y ) = | x | + | y | induces a lamination of C \{ } into concentricspheres. Denoting by τ : C \{ } → S the normalization map, given by τ ( z ) = z ρ ( z ) , thepair ( ρ, τ ) defines an orientation preserving diffeomorphism( ρ, τ ) : C \{ } → (0 , ∞ ) × S . Via this diffeomorphism, the 3-sphere S (cid:15) is identified by { (cid:15) } × S . The action of the Liegroup G p,q = (cid:104) ζ (cid:105) , defined by (3), may be imposed on (0 , ∞ ) × S to yield the orbit space(0 , ∞ ) × S /G p,q ≈ (0 , ∞ ) × L p,q . Consider complex polynomials whose zero-set is invariantunder this action. Definition 3.1.
A polynomial f ( x, y ) in two complex variables will be called ( p, q ) -invariant if there exists an integer ≤ k < p , such that f ( ζ · ( x, y )) = ζ k f ( x, y ) for any ( x, y ) ∈ C . enoting by π : S → L p,q the quotient projection, we obtain the following diagram: C \{ (0 , } ( ρ,τ ) (cid:47) (cid:47) f (cid:39) (cid:39) (0 , ∞ ) × S × π (cid:47) (cid:47) (0 , ∞ ) × L p,q C Our definition immediately yields:
Proposition 3.2. If f is a ( p, q ) -invariant polynomial, then the algebraic link (cid:101) K = f − (0) ∩ S (cid:15) is the lift of a smooth link in L p,q .Proof. Denote by V = f − (0) the algebraic curve defined by f . Since f is ( p, q )-invariant,the link (cid:101) K = V ∩ S (cid:15) is invariant under the action of G p,q . It follows that π − ( π ( (cid:101) K )) = (cid:101) K .Since (cid:101) K is a smooth manifold [15, Corollary 2.9], also π ( (cid:101) K ) is a smooth manifold byTheorem 2.2. Definition 3.3.
A link K in the lens space L p,q will be called algebraic if there exists a ( p, q ) -invariant polynomial f , such that the lift of K in S is isotopic to the algebraic link f − (0) ∩ S (cid:15) for some (cid:15) > . Example 3.4.
Consider the polynomial f ( x, y ) = x + y . The algebraic link f − (0) ∩ S is the 2-component torus link T (8 , . Let ζ ∈ C be a primitive third root of unity. Since f ( ζx, ζy ) = ζ f ( x, y ) for any ( x, y ) ∈ C , f is (3 , -invariant. It follows by Proposition3.2 that T (8 , is the lift of an algebraic link in L , . Indeed: consider the 2-componentlink K in L , , given by the diagrams on Figure 4. Applying Proposition 2.3, the lift of K in the 3-sphere is given by the diagram on Figure 6, which clearly represents the torus link T (8 , . Figure 4: The algebraic link in L , from Example 3.4 (punctured disk diagram onthe left and band diagram on the right) Recall that a link K in a 3-manifold M is called fibered if its complement M − K isthe total space of a fiber bundle over S . We would like to show that every algebraic linkin a lens space is fibered. S of the link in L , (see Example 3.4) For the remainder of this Section, let f ( x, y ) be a ( p, q )-invariant polynomial thatdefines an algebraic link K in L p,q , whose lift is an algebraic link (cid:101) K = f − (0) ∩ S (cid:15) in the3-sphere.By a classical result of Milnor, every algebraic link in the 3-sphere is fibered. Define amap Φ : S (cid:15) \ (cid:101) K → S by Φ( x, y ) = f ( x, y ) | f ( x, y ) | . (4) Lemma 3.5. [15, Corollary 4.5] Given any non-constant polynomial f ( x, y ) which vanishesat the origin, there exists an (cid:15) > such that, for (cid:15) ≤ (cid:15) , the map Φ : S (cid:15) \ (cid:101) K → S has nocritical points at all. Theorem 3.6. [15, Theorem 4.8] Let (cid:101) K be an algebraic link in S , associated to a complexpolynomial f . For (cid:15) ≤ (cid:15) , the space S (cid:15) \ (cid:101) K is a smooth fiber bundle over S with projectionmapping Φ . Since f is ( p, q )-invariant, we have f ( ζx, ζ q y ) = ζ k f ( x, y ) for some 0 ≤ k < p . Eventhough the zero-set of the polynomial f is invariant under the action of G p,q , its nonzerovalues are not (unless k = 0). Denote p = p gcd( k,p ) and define a covering map µ : S → S by µ ( z ) = z p . Proposition 3.7.
The map
Φ : S (cid:15) \ (cid:101) K → S induces a well-defined map Ψ : L p,q \ K → S ,such that the following diagram commutes: S (cid:15) \ (cid:101) K π (cid:47) (cid:47) Φ (cid:15) (cid:15) L p,q \ K Ψ (cid:15) (cid:15) S µ (cid:47) (cid:47) S roof. Denote π ( x, y ) = [ x, y ]. Following the diagram, the map Ψ is given by Ψ([ x, y ]) =( µ ◦ Φ)( x, y ). To see that this is well-defined, compute( µ ◦ Φ)( ζx, ζ q y ) = µ (cid:18) f ( ζx, ζ q y ) | f ( ζx, ζ q y ) | (cid:19) = µ (cid:18) ζ k f ( x, y ) | f ( x, y ) | (cid:19) = µ (cid:18) f ( x, y ) | f ( x, y ) | (cid:19) = ( µ ◦ Φ)( x, y ) . By [15, Lemma 6.1], the closure of each fiber of the fibration Φ : S (cid:15) \ (cid:101) K → S is a Seifertsurface of the link (cid:101) K . To obtain a similar statement about the map Ψ, we make use ofLemma 3.5 together with the following result: Theorem 3.8. [10, Theorem 5.22] Let M and N be smooth manifolds, and let φ : M → N be a smooth map with constant rank k . Each level set of φ is a closed embedded submanifoldof codimension k in M . Proposition 3.9.
Let (cid:15) ≤ (cid:15) and consider { (cid:15) } × L p,q as a quotient of S (cid:15) under the action (3) . For any t ∈ [0 , π ) , the inverse image F t = Ψ − ( e it ) ⊂ ( { (cid:15) } × L p,q ) \ K is a smoothsurface.Proof. By Lemma 3.5, the map Φ has no critical points on S (cid:15) \ (cid:101) K . By Theorem 2.2, thequotient maps π : S (cid:15) → L p,q and µ : S → S are both smooth covering maps. Thus theinduced map Ψ : ( { (cid:15) } × L p,q ) \ K → S is a smooth map without critical points and byTheorem 3.8, each level set F t = Ψ − ( e it ) is a smooth surface. Lemma 3.10.
The boundary of the closure of F t in L p,q is precisely the link K .Proof. Let (cid:101) F t = π − ( F t ) be the preimage of F t in S (cid:15) . Denote the boundary of the closureof F t in L p,q (respectively closure of (cid:101) F t in S (cid:15) ) by ∂F t (respectively ∂ (cid:101) F t ). By [15, Lemma6.1] we have ∂ (cid:101) F t = (cid:101) K .To show that K ⊂ ∂F t , choose an x ∈ K and let U ⊂ L p,q be any neighbourhood of x .Then π − ( U ) ⊂ S (cid:15) is a neighbourhood of the set π − ( x ) ⊂ (cid:101) K = ∂ (cid:101) F t , thus there exists apoint z ∈ π − ( U ) ∩ (cid:101) F t . It follows that π ( z ) ∈ U ∩ F t and x ∈ U \ F t , therefore x ∈ ∂F t .To check that ∂F t ⊂ K , let y ∈ ∂F t . Since F t is an open set, y / ∈ F t , therefore π − ( y ) ∩ (cid:101) F t = ∅ . Choose an x ∈ π − ( y ) and let U ⊂ S (cid:15) be a neighbourhood of x in S (cid:15) suchthat π | U : U → π ( U ) is a diffeomorphism. Then π ( U ) is a neighbourhood of y , therefore π ( U ) ∩ F t (cid:54) = ∅ . Choose an element z ∈ π ( U ) ∩ F t and let w ∈ U be the unique element forwhich π ( w ) = z . It follows that w ∈ U ∩ (cid:101) F t , therefore U ∩ (cid:101) F t (cid:54) = ∅ and U \ (cid:101) F t (cid:54) = ∅ . We haveshown that x ∈ ∂ (cid:101) F t = (cid:101) K , which implies y = π ( x ) ∈ K . Corollary 3.11.
Every algebraic link in L p,q is nullhomologous. or an algebraic link K in L p,q with r components K , . . . , K r , denote by δ i ∈ H ( L p,q ) ∼ = Z p the homology class of the component K i . By Corollary 3.11, we have r (cid:88) i =1 δ i = 0 . The following result relates the number of components of a link in L p,q and of its lift in S . Lemma 3.12. [12, Proposition 2] Let K be an r -component link in L p,q , whose lift is an (cid:101) r -component link in S . Denote by δ i ∈ H ( L p,q ) ∼ = Z p the homology class of the i -thcomponent of K for i = 1 , . . . , r . Then we have r (cid:88) i =1 gcd( δ i , p ) = (cid:101) r . Corollary 3.13. (a) If K is an algebraic knot in L p,q , then (cid:101) K is a p -component link in S .(b) If K is a 2-component algebraic link in L p,q , then (cid:101) K has δ , p ) components.Proof. (a) Since K has one component, Corollary 3.11 implies that δ = 0 and by Lemma3.12, its lift has p components.(b) By Corollary 3.11 we have δ = − δ and thus gcd( δ , p ) = gcd( δ , p ). Lemma 3.12implies the result. Example 3.14.
The zero set of the complex polynomial f ( x, y ) = x + y intersects the3-sphere in the 2-component torus link T (8 , . In Example 3.4 we have shown that T (8 , is the lift of a 2-component algebraic link in L , , whose components represent homologyclasses ± ∈ H ( L , ) .The polynomial f ( x, y ) is also (2 , -invariant, thus T (8 , is the lift of an algebraiclink in L , . Indeed: consider the knot in L , , that is given by diagrams in Figure 6. Inthe lens space L , , the knot is equivalent to the knot (see Figure 8) by [6, AppendixB]. Applying Proposition 2.3, we can see that the lift of in the 3-sphere is equivalent tothe torus link T (8 , . We have thus shown that T (8 , is at the same time the lift of analgebraic knot in L , , and the lift of a 2-component algebraic link in L , . Recall the commutative diagram of smooth maps S (cid:15) \ (cid:101) K π (cid:47) (cid:47) Φ (cid:15) (cid:15) L p,q \ K Ψ (cid:15) (cid:15) S µ (cid:47) (cid:47) S Firstly, we combine the maps µ and Φ to obtain: in L , from Example 3.14 (punctured disk diagramon the left and band diagram on the right) Lemma 3.15.
Let (cid:15) be as in Lemma 3.5, and let (cid:15) ≤ (cid:15) . The map µ ◦ Φ : S (cid:15) \ (cid:101) K → S isthe projection of a smooth fiber bundle, whose fiber consists of p disjoint fibers of the fiberbundle Φ : S (cid:15) \ (cid:101) K → S .Proof. Choose a point e it ∈ S , then µ − ( e it ) = { e i ( t +2 πjp ) | j = 1 , . . . , p } . Since µ is acovering map, there exists a neighbourhood U ⊂ S of e it , such that µ − ( U ) = U (cid:116) . . . (cid:116) U p ,and µ | U j is a diffeomorphism for j = 1 , . . . , p . We have ( µ ◦ Φ) − ( U ) = Φ − ( U ) (cid:116) . . . (cid:116) Φ − ( U p ). By Theorem 3.6, the map Φ is the projection of a smooth fiber bundle. Denoteby (cid:101) F j = Φ − (cid:16) e i ( t +2 πjp (cid:17) its fibers at the points of µ − ( e it ); then there exist diffeomorphisms h j : U j × (cid:101) F j → Φ − ( U j ) for j = 1 . . . , p . Define a map h : U × (cid:16)(cid:70) pj =1 (cid:101) F j (cid:17) → (cid:70) pj =1 Φ − ( U j )by h ( u, z ) = h j (cid:0) ( µ | U j ) − ( u ) , z (cid:1) for z ∈ (cid:101) F j . Then h is a diffeomorphism with inverse h − ( w ) = ( µ, id) ◦ h − j ( w ) for w ∈ Φ − ( U j ).Now we are prepared to prove the following. Theorem 3.16.
For (cid:15) ≤ (cid:15) , the space ( { (cid:15) } × L p,q ) \ K is a smooth fiber bundle over S with projection mapping Ψ .Proof. By Theorem 2.2, the map π : S (cid:15) \ (cid:101) K → ( { (cid:15) } × L p,q ) \ K is a smooth covering map.Denote by Z , Z . . . , Z p ⊂ S (cid:15) \ (cid:101) K the distinct leaves of this covering. Then π | Z : Z → ( { (cid:15) } × L p,q ) \ K is a diffeomorphism.Choose a point e it ∈ S . Denote by (cid:101) F t = ( µ ◦ Φ) − ( e it ) and F t = Ψ − ( e it ) its fibers.By Lemma 3.15, the map µ ◦ Φ : S (cid:15) \ (cid:101) K → S is the projection of a smooth fiber bundle, sothere exists a neighbourhood U ⊂ S of e it and a diffeomorphism h : U × (cid:101) F t → ( µ ◦ Φ) − ( U ).Define a map g : U × F t → ψ − ( U ) by g ( u, π ( x, y )) = π (cid:0) h (cid:0) u, ( π | Z ) − ( π ( x, y )) (cid:1)(cid:1) . Since π | Z is a diffeomorphism, g is a well-defined smooth map. Moreover, g is a diffeo-morphism with inverse g − ( π ( x, y )) = (id , π ) ◦ h − (cid:0) ( π | Z ) − ( π ( x, y )) (cid:1) . orollary 3.17. Every algebraic link in a lens space is fibered.
The relationship between the fibre bundles S (cid:15) \ (cid:101) K and L p,q \ K can also be describedin terms of monodromy. Let g : (cid:101) F → (cid:101) F denote the monodromy map of the fiber bundle µ ◦ Φ : S (cid:15) \ (cid:101) K → S , thus S (cid:15) \ (cid:101) K is diffeomorphic to ( (cid:101) F × I ) / ( x, ∼ ( g ( x ) , . Denote by F the fiber of the fibre bundle Ψ : L p,q \ K → S . The surface (cid:101) F may be decomposed into p fundamental parts (cid:101) F , . . . , (cid:101) F p , so that π | (cid:101) F i : (cid:101) F i → F is a diffeomorphism for each i . Thenthe monodromy map of the fiber bundle Ψ : L p,q \ K → S is given by π ◦ g ◦ ( π | (cid:101) F ) − : F → F .Recall that for a fibered link in S , a fiber surface is the unique surface that realizesthe minimal Seifert genus of the link [9], [17]. This fact generalizes to links in an arbitrary3-manifold: Proposition 3.18. [5, Proposition 4.1] If L is a fibered link in a 3-manifold M , then anyfiber is a minimal Seifert surface and conversely, any minimal Seifert surface is isotopicto a fiber. Corollary 3.19.
Let K be an algebraic link in L p,q , and let Ψ : L p,q \ K → S be the corres-ponding fibration. Then the closure of each fiber F t = Ψ − ( e it ) in L p,q is a Seifert surfaceof minimal genus for K .Proof. It follows immediately from Theorem 3.16, Proposition 3.18 and Lemma 3.10.The genus of the fiber F t is related to the genus of π − ( F t ) = (cid:101) F t . In case of an algebraicknot in a lens space, we obtain: Proposition 3.20.
Let K be an algebraic knot in L p,q , and let (cid:101) K be its lift in S . Denoteby g (respectively (cid:101) g ) the Seifert genus of K (respectively (cid:101) K ). Then we have g = 2 (cid:101) g + p + gcd( p, k ) −
22 gcd( p, k ) . Proof.
Denote by (cid:101) F t = ( µ ◦ Φ) − ( e it ) (respectively F t = Ψ − ( e it )) the fibers that representinteriors of the Seifert surfaces for (cid:101) K (respectively K ) . The covering π : (cid:101) F t → F t is cyclicof order p , so χ ( (cid:101) F t ) = p · χ ( F t ). By Lemma 3.15, (cid:101) F t is a disjoint union of p diffeomorphiccopies of the fiber Φ − ( e it ), which in turn is connected by [15, Corollary 6.3]. The boundaryof (cid:101) F t in S (cid:15) has p components by Corollary 3.13. Since p = p gcd( p,k ) , we may compute p (2 − (cid:101) g − p ) = p (1 − g ) p gcd( p, k ) (2 − (cid:101) g − p ) = p (1 − g )2 − (cid:101) g − p = gcd( p, k )(1 − g )and solving for g yields the expression above. Examples
In this Section, we find some examples of algebraic knots and links in lens spaces. Weconcentrate on the case when the lift of the algebraic knot/link in the 3-sphere is a toruslink.
Proposition 4.1.
Let a, b, p and q be integers, with p and q relatively prime. The toruslink T ( a, b ) is the lift of an algebraic link in L p,q ⇔ a ≡ qb (mod p).Proof. The torus link T ( a, b ) is the link of the singularity (0 , ∈ C of the complexpolynomial f ( x, y ) = x a + y b . Let ζ ∈ C be a primitive p -th root of unity. Then we have f ( ζx, ζ q y ) = ζ a x a + ζ qb y b and ζ k f ( x, y ) = ζ k ( x a + y b ). The polynomial f is ( p, q )-invariantif and only if there exists an integer 0 ≤ k < p , such that the equality( ζ a − ζ k ) x a + ( ζ qb − ζ k ) y b = 0holds for every ( x, y ) ∈ C , which is equivalent to a ≡ qb (mod p). Corollary 4.2.
Let a, b, p and q be integers, with p and q relatively prime. The torus link T ( a, b ) is the lift of an algebraic knot in L p,q if and only if gcd( a, b ) = p . In this case, theSeifert genus of the knot equals g = (cid:101) g + p − p , where (cid:101) g is the Seifert genus of T ( a, b ) .Proof. Let T ( a, b ) be the lift of an algebraic knot in L p,q . Since the torus link T ( a, b ) hasgcd( a, b ) components, Corollary 3.13 implies that gcd( a, b ) = p .Conversely, if gcd( a, b ) = p , then a ≡ qb (mod p ) for every q , so T ( a, b ) is the liftof an algebraic link in L p,q by Proposition 4.1. The algebraic set x a + y b = 0 consistsof p nonsingular branches, and the action of G p,q induces a cyclic permutation of thesebranches. Under the quotient map π : S (cid:15) → L p,q , all the p components of T ( a, b ) identify.Therefore the algebraic link in L p,q is actually a knot.To prove the last statement of the Corollary, let a = a p and b = b p with gcd( a , b ) =1. Since f ( x, y ) = x a p + y b p = f ( ζx, ζ q y ) for every ( x, y ) ∈ C , we have k = 0 andgcd( p, k ) = p . Using Proposition 3.20, we obtain g = 2 (cid:101) g + p + p − p = (cid:101) g + p − p . Example 4.3.
It follows from Corollary 4.2 that the torus link T (9 , is the lift of analgebraic knot in L (3 , q ) for q = 1 , . The link T (9 , is the closure of a braid on 3strands, that is given by the braid word ( σ σ ) . We use Proposition 2.3 with n = 3 andnote that the braid relation σ i σ i +1 σ i = σ i +1 σ i σ i +1 implies (∆ ) = σ σ σ σ σ σ = ( σ σ ) and (∆ ) = ( σ σ ) . It follows that T (9 , is the lift of a knot K in L , , whose banddiagram corresponds to the braid word ( σ σ ) . T (9 , is also the lift of the knot K in L , , whose band diagram corresponds to the braid word σ σ (see Figure 7). It followsfrom the Corollary 3.17 that K and K are both fibered knots. K in L , (left) and K in L , (right) Example 4.4.
By Corollary 4.2, the torus link T (3 , is the lift of a knot in L , . Figure8 shows the knot from the knot atlas in [6]. Figure 9 shows that the lift in S of theknot in L , is equivalent to the link T (3 , . It follows from the Corollary 3.17 that is a fibered knot in L , . Figure 8: The knot 2 from the knot atlas in [6] (punctured disk diagram on the leftand band diagram on the right) ∼ ∼ Figure 9: The lift in S of the knot 2 in L , , see Example 4.4 Example 4.5.
By Corollary 4.2, the torus link T (4 , (also called the Solomon’s knot) isthe lift of a knot in L , . Indeed: Figure 10 shows the knot from the knot atlas in [6]. e obtain the diagram of its lift in S by Proposition 2.3. Figure 11 shows that the lift isequivalent to the Solomon’s knot. It follows by Corollary 3.17 that is a fibered knot in L , . Figure 10: The knot 3 from the knot atlas in [6] ∼ ∼ Figure 11: The lift in S of the knot 3 in L , , see Example 4.5 More generally, an ample supply of algebraic knots in L p,q is guaranteed by the followingconstruction: Lemma 4.6.
Let f ( x, y ) be an irreducible complex polynomial with f (0 ,
0) = 0 . For anypair of positive integers p and q with gcd ( p, q ) = 1 , the polynomial f p ( x, y ) = f ( x p , y p ) defines an algebraic knot in L p,q .Proof. The polynomial f p is obviously ( p, q )-invariant. Since f is irreducible, the algebraiccurve { ( x, y ) ∈ C | f p ( x, y ) = 0 } consists of p branches. The action of G p,q induces acyclic permutation of these branches. The link of the singularity at the origin is thereforea p -component link in S , whose quotient in L p,q is an algebraic knot.
1. Torus links represent the simplest class of classical algebraic links. Proposition 4.1and Corollary 4.2 classify torus knots and links that are lifts of algebraic links in p,q . It would be interesting to find a topological characterization of the algebraiclinks in L p,q that lift to torus links.2. More generally, can we find a topological characterization of all algebraic links inlens spaces?3. For a ( p, q )-invariant polynomial f , one might investigate the complex algebraiccurve V = f − (0) ⊂ (0 , ∞ ) × S and its quotient in π ( V ) ⊂ (0 , ∞ ) × L p,q . Howdo techniques that we use to study plane algebraic curves, translate to the study of π ( V )? Applying them, we might gain new insight about the topology of the quotientcurve.4. It is known that a smooth fibration Ψ : M \ K → S of a closed, orientable 3-manifold M defines an open book decomposition of M . By [7], there is a one-to-one corres-pondence between isotopy classes of oriented contact structures on M and open bookdecompositions up to positive stabilizations. By Theorem 3.16, every algebraic knot K in L p,q defines a fibration Ψ : L p,q \ K → S . Can we determine which contactstructures on L p,q correspond to these fibrations? Does every contact structure on L p,q correspond to the fibration of some algebraic knot in L p,q ? Are there algebraicknots/links that determine the same contact structure?5. The smooth 4-genus of a classical algebraic link K is known to be realized by thealgebraic curve whose boundary is K . Is it possible to generalize this result to thealgebraic links in lens spaces? The first problem we meet is a meaningful definitionof the smooth 4-genus of lens space knots. As opposed to the 3-sphere, a typical lensspace is not the boundary of the 4-ball (which may be conveniently completed tothe complex projective plane). Here we need to move from the complex to the moregeneral symplectic setting. The symplectic Thom conjecture, proven by Ozsv´athand Szab´o, states that a symplectic surface in a symplectic 4-manifold is genus-minimizing in its homology class. The symplectic fillings of lens spaces have beeninvestigated by Lisca [11], McDuff [14] and recently by Etnyre and Roy [4]. Supposethat W p,q is a symplectic filling of the lens space L p,q , equipped with the standardcontact structure. Define the smooth 4-genus of a knot K in L p,q to be the minimumgenus of a smooth surface embedded in W p,q , whose boundary is K . Before statingthis as a definition, of course, one should check whether this minimum genus dependson the choice of the symplectic filling W p,q . If it does not, then we could explorefurther to see whether the symplectic Thom conjecture implies results about thesmooth 4-genus of algebraic knots in L p,q . Acknowledgements
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