Finite type invariants of nullhomologous knots in 3-manifolds fibered over S 1 by counting graphs
aa r X i v : . [ m a t h . G T ] M a y Preprint (2015)
FINITE TYPE INVARIANTS OF NULLHOMOLOGOUS KNOTSIN 3-MANIFOLDS FIBERED OVER S BY COUNTING GRAPHS
TADAYUKI WATANABE
Abstract.
We study finite type invariants of nullhomologous knots in a closed3-manifold M defined in terms of certain descending filtration { K n ( M ) } n ≥ of the vector space K ( M ) spanned by isotopy classes of nullhomologous knotsin M . The filtration { K n ( M ) } n ≥ is defined by surgeries on special kinds ofclaspers in M . When M is fibered over S and H ( M ) = Z , we show thatthe natural surgery map from the space of Q [ t ± ]-colored Jacobi diagramson S of degree n to the graded quotient K n ( M ) / K n +1 ( M ) is injective for n ≤
2. This already shows that the classification of nullhomologous knotsin M by finite type invariants considered in this paper is rather fine. In theproof, we construct a finite type invariant of nullhomologous knots in M upto degree 2 that is an analogue of the invariant given in our previous paperarXiv:1503.08735, which is based on Lescop’s construction of Z -equivariantperturbative invariant of 3-manifolds evaluated with the explicit propagatorgiven by the author using parametrized Morse thoery. Introduction
There is a natural descending filtration K ( S ) ⊃ K ( S ) ⊃ K ( S ) ⊃ · · · onthe vector space (or Z -module) K ( S ) spanned by isotopy classes of knots in S ,called the Vassiliev filtration (after Vassiliev’s [Va1]), whose n -th term is spannedby alternating sums of possible resolutions of singular knots with n double points.In terms of the Vassiliev filtration, finite type (or Vassiliev) invariant of knots ofdegree n is defined as linear maps from K ( S ) / K n +1 ( S ) ([BL, BN2]). It is knownthat the natural “geometric realization” map gives an isomorphism from the vectorspace of certain trivalent graphs called the Jacobi diagrams ([BN1, BN2], see also[CDM]) to the graded quotients K n ( S ) / K n +1 ( S ). This very striking result hasbeen proved by Kontsevich in [Ko2] with the help of his diagram-valued universalfinite type invariant of knots in S , so called the Kontsevich integral of knots. Thereis another construction of finite type invariants of knots in S coming from Chern–Simons perturbation theory, which are given by integrations on configuration spaces([BN1, GMM, Koh, Ko1, BT]). It has been proved by Bott–Taubes [BT] (degree2) and Altschuler–Freidel [AF] (all degrees) that the configuration space integralinvariant give another universal finite type invariant of knots in S . Date : April 5, 2019.2000
Mathematics Subject Classification.
In this paper, we study finite type invariants of nullhomologous knots in an ori-ented closed 3-manifold M . As an analogoue of finite type invariants of knots in S defined by null-claspers (Garoufalidis–Rozansky [GR]), we introduce a descendingfiltration K ( M ) ⊃ K ( M ) ⊃ · · · of the space of isotopy classes of nullhomologousknots in M by using surgeries on certain claspers with nullhomologous leaves. Inthe case H ( M ) = Z and M is fibered over S , we show that the natural surgerymap from the space of Q [ t ± ]-colored Jacobi diagrams on S of degree n to thegraded quotient K n ( M ) / K n +1 ( M ) is injective for n ≤
2. The main idea is toconstruct a diagram-valued perturbative invariants of nullhomologous knots in afibered 3-manifold M in a similar method as [Wa3].We remark that finite type invariants of knots in general 3-manifolds have beendeveloped in [Ka, KL, Va2] by considering singular knots with double points. Al-though the definition of finite type invariants of [Ka, KL, Va2] is different fromthat studied in this paper, it can be considered as a noncommutative refinement ofours. We also remark that in [Ha], Habiro studied finite type invariants of links ingeneral 3-manifolds defined by using surgeries on graph claspers, which is differentfrom that studied in this paper too. Although Habiro’s definition of finite typeinvariant is natural from the point of view of clasper theory, we modify Habiro’sdefinition for our purpose. Other relevant works can be found in [Sch1, Sch2, Lie].We define the perturbative invariant as the trace of the generating functionof counts of certain graphs in M , which we call AL-graphs . The definition of ourinvariant is based on Lescop’s works on equivariant perturbative invariants of knotsand 3-manifolds [Les2, Les3, Les4] and on the explicit propagator of “AL-paths”in M given in [Wa2] using parametrized Morse theory. Our construction can beconsidered an analogue of that of the configuration space integral invariant.Throughout this paper, manifolds and maps between them are smooth unlessotherwise indicated. We use the inward-normal-first convention to orient bound-aries of manifolds. Homology groups are assumed to be with integer coefficientsunless otherwise specified.2. Finite type invariants of nullhomologous knots in M We study finite type invariants of nullhomologous knots in a closed 3-manifolddefined by using clasper surgeries. The theory of clasper surgery has been developedindependently by Goussarov and Habiro in [Gu, Ha]. We review some fundamentalproperties of claspers and prescribe the type of finite type invariants studied in thepresent paper. The main result of the present paper is stated in terms of claspersurgeries.2.1.
Null-claspers, filtration K n ( M ) and finite type invariant. We shall re-call definition of clasper surgery from [Ha]. Let M be a closed connected orientedRiemannian 3-manifold. A tree clasper for a knot K in M is a connected surfaceimmersed in M consisting of bands, nodes, leaves and disk-leaves (as in Figure 2). Aband is an embedded 2-dimensional 1-handle, a node is an embedded 2-dimensional0-handle on which three bands are attached. A leaf (resp. disk-leaf) is an embeddedannulus (resp. an embedded 0-handle) on which a band is attached. The union of INITE TYPE INVARIANTS OF NULLHOMOLOGOUS KNOTS 3
Figure 1. I -clasper and surgery on it (we use the blackboard-framing convention to represent ribbon structure of I -clasper) Figure 2. M -null tree clasper and the associated set of I -claspersbands, nodes and leaves is embedded in M \ K , whereas a disk-leaf may intersect K or leaves or bands transversally in its interior.A tree clasper without nodes is called an I -clasper . On an I -clasper, surgeryis defined as follows. In this paper, we only consider I -clasper with at least onedisk-leaves. Then surgery on such an I -clasper C is defined as the replacement asin Figure 1. One may also consider surgery on a tree clasper by replacing nodesand disk-leaves with collections of leaves as in Figure 2.We say that a tree clasper for a nullhomologous knot K in M is M -null if itsleaves consist of disk-leaves and at most one leaf that is nullhomologous in M . SeeFigure 2 for an example. A strict tree clasper is a tree clasper with only disk-leavesthat intersect only with the knot K . The degree of a tree clasper T is the numberof nodes in T plus 1. Let K T denote the knot in M obtained from K by surgeryalong the set of I -claspers associated to T . Lemma 2.1. If T is an M -null tree clasper for a nullhomologous knot K in M ,then K T is again a nullhomologous knot in M . Two nullhomologous knots in M are related by surgeries on finitely many strict I -claspers and at most one M -null I -clasper.Proof. By Habiro’s move 9 in [Ha, Proposition 2.7], surgery on an M -null treeclasper can be replaced with a sequence of surgeries on M -null I -claspers, namely, I -claspers with only disk-leaves or that with one disk-leaf and one nullhomologousleaf in M . TADAYUKI WATANABE
It is obvious that surgeries on such I -claspers do not change the homology class ofa knot in M .For the second assertion, let K , K be two nullhomologous knots in M . We mayassume without loss of generality that K and K are mutually disjoint. Take basepoints q , q on K , K respectively and a path γ in M from q to q . Let K bea knot in M given by a connected sum ( − K ) γ K taken along γ . Let C bean M -null I -clasper whose disk-leaf links K at q and the other leaf is a parallelof K disjoint from K ∪ K . Then K C is homotopic to K . Hence K is obtainedfrom K by surgery on C and several crossing changes. (cid:3) Lemma 2.1 motivates the definition of finite type invariants given below. Let K ( M ) be the vector space over Q spanned by isotopy classes of all nullhomologousknots in M . Let K n,k ( M ) (1 ≤ k ≤ n ) be the subspace of K ( M ) spanned by[ K ; G ] = X I ⊂{ , ,...,k } ( − k −| I | K G I , where • K is a nullhomologous knot in M , • G = { G , G , . . . , G k } is a disjoint collection of tree claspers with P ki =1 deg G i = n that consists of strict tree claspers and at most one M -null tree clasper. • G I = S i ∈ I G i .The alternating sum [ K ; G ] as above is called an M -null forest scheme of degree n ,size k . When all G i are strict, then [ M ; G ] is called a strict forest scheme . We put K ( M ) = K ( M ) , K n ( M ) = n X k =1 K n,k ( M ) (for n ≥ . Lemma 2.2 ([Ha, page 48]) . (1) If ≤ k ≤ k ′ ≤ n , then K n,k ( M ) ⊂ K n,k ′ ( M ) .In particular, K n ( M ) = K n,n ( M ) . (2) [ K ; G , G , . . . , G k ] = [ K G ; G , . . . , G k ] − [ K ; G , . . . , G k ] . (3) [ K ; S ∪ T, G , . . . , G k ] = [ K ; S, G , . . . , G k ] + [ K S ; T, G , . . . , G k ] . Definition 2.3.
Let V be a vector space over Q . We say that a linear map f : K ( M ) → V is a finite type invariant of M -null type n if f ( K n +1 ( M )) = 0.A fundamental problem in the thoery of finite type invariant is to determine thestructure of the quotient space K n ( M ) / K n +1 ( M ). The restriction of tree claspersin the definition of K n,k ( M ) may not look natural. However, this definition is niceto relate the graded quotient with the space of Λ-colored Jacobi diagrams definedbelow.2.2. Λ -colored Jacobi diagrams. A Jacobi diagram on S is a connected triva-lent graph with oriented edges and with a distinguished simple oriented cycle, calledthe Wilson loop ([BN1, BN2], see also [CDM, Ch.5]). A labeled Jacobi diagram is a Jacobi diagram Γ equipped with bijections α : { , , . . . , n } → V (Γ) and β : { , , . . . , n } → E (Γ), where V (Γ) (resp. E (Γ)) is the set of vertices (resp.edges) of Γ (including the edges in the Wilson loop). Let E W (Γ) be the subsetof E (Γ) consisting of edges in the Wilson loop and let E nW (Γ) = E (Γ) \ E W (Γ). INITE TYPE INVARIANTS OF NULLHOMOLOGOUS KNOTS 5
Figure 3.
The relations AS, IHX, STU and FI.Let V W (Γ) be the subset of V (Γ) consisting of vertices on the Wilson loop and let V nW (Γ) = V (Γ) \ V W (Γ). Let P (Γ) be the set of components in the subgraph of Γformed by E nW (Γ). A Jacobi diagram Γ on S with V nW (Γ) = ∅ is called a chorddiagram . A vertex-orientation of a trivalent vertex v in a Jacobi diagram is a cyclicordering of the edges incident to v . A Jacobi diagram all of whose trivalent verticesare equipped with vertex-orientations is said to be vertex-oriented.Let R be a commutative ring with 1. For a Jacobi diagram Γ on S , an R -coloring of Γ is an assignment of an element of R to every edge of Γ. An R -coloringis represented by a map φ : E (Γ) → R . The degree of a Jacobi diagram is definedas half the number of vertices. A vertex of Γ that is on the Wilson loop is called a univalent vertex and otherwise a trivalent vertex . Definition 2.4.
Let Λ M = Q [ H ( M )] and let b Λ M be the total ring of fractions Q (Λ M ) = Q ( H ( M )). Let A n ( S ; Λ M ) (resp. A n ( S ; b Λ M )) be the vector space over Q spanned by pairs (Γ , φ ), where Γ is a Jacobi diagram of degree n with vertex-orientation and φ is a Λ M -coloring (resp. b Λ M -coloring) of Γ, quotiented by therelations AS, IHX, STU, FI, Orientation reversal, Linearity, Holonomy (Figure 3and 4) and automorphisms of oriented graphs.There is a canonical way to fix an equivalence class of vertex-orientation modulothe AS relation from labelings α, β on a Jacobi diagram (see e.g., [CV]).We denote a pair (Γ , φ ) by Γ( φ ) or by Γ( φ ( e ) , φ ( e ) , . . . , φ ( e ℓ )). We say thata Λ M -colored graph Γ( φ ) is a monomial if for each edge e of Γ, φ ( e ) is an el-ement of H ( M ). In this case, we may consider φ as a map E (Γ) → H ( M ).There is a bijective correspondence between the equivalence class of a labeledmonomial Jacobi diagram Γ( φ ) modulo the Holonomy relation and the homo-topy class of a continuous map c : Γ → K ( H ( M ) , c ] ∈ H (Γ; H ( M )) = [Γ , K ( H ( M ) , c isnullhomotopic. Definition 2.5.
Let A NH n ( S ; Λ M ) be the subspace of A n ( S ; Λ M ) spanned by theset G NH n ( S ; Λ M ) of all monomial Jacobi diagrams such that ∗ Q e ∈ E W (Γ) φ ( e ) = 1.By definition, the natural map A NH n ( S ; Λ M ) → A n ( S ; b Λ M ) is an embedding. ∗ We consider the group structure of H ( M ) as multiplication. TADAYUKI WATANABE
Figure 4.
The relations Orientation reversal, Linearity and Ho-lonomy. p, q, r ∈ Λ M (or p, q, r ∈ b Λ M ), α ∈ Q , t ± ∈ H ( M ). Theexponent ε i is 1 if the i -th edge is oriented toward v and otherwise −
1. The edge in the Linearity relation is either of E W (Γ) or of E nW (Γ).2.3. Surgery map ψ n . Let ι : M → K ( H ( M ) ,
1) be a map that representsthe class in H ( M ; H ( M )) = [ M, K ( H ( M ) , H ( M ) , H ( M )) = H ( M ; H ( M )). Consider a monomial Λ M -colored Jacobidiagram Γ ∈ G NH n ( S ; Λ M ) and a piecewise smooth embedding ρ : Γ → M thatpreserves its vertex-orientation and such that the homotopy class of the compositionΓ ρ → M ι → K ( H ( M ) ,
1) represents the Λ M -coloring of Γ. We assign a collectionof tree claspers G = { G , . . . , G k } with only disk-leaves to ρ as follows. We replacevertices in the embedded graph ρ (Γ) with leaves or nodes of claspers as follows:so that each component in P (Γ) is mapped to a (connected) tree clasper. Here isan example.One may see that the graph scheme [ K ; G ] represents an element of K n,n ( M ) byapplying Habiro’s move 9 of [Ha, Proposition 2.7] several times. Proposition 2.6.
The assignment Γ [ K ; G ] induces a well-defined linear map ψ n : A NH n ( S ; Λ M ) → K n ( M ) / K n +1 ( M ) . If π ( M ) is abelian, then ψ n is surjective † . † In [Ha], Habiro has obtained similar result, showing that the natural surgery map gives asurjection from the space of Jacobi diagrams with H ( M )-colored univalent vertices to the graded INITE TYPE INVARIANTS OF NULLHOMOLOGOUS KNOTS 7
Proof.
First, we prove that the assignment Γ [ K ; G ] gives a well-defined map ψ n : G NH n ( S ; Λ M ) → K n ( M ) / K n +1 ( M ) . Let [ K ; G ] and [ K ′ ; G ′ ] be two forest schemes of degree n , size k that correspondto a monomial Λ M -colored graph Γ ∈ G NH n ( S ; Λ M ). Then [ K ; G ] and [ K ′ ; G ′ ] arerelated by a sequence of the following moves:(1) A crossing change of knot.(2) A crossing change between edges of tree claspers.(3) A crossing change between an edge of a tree clasper and knot.(4) A bordism change of knot.(5) A bordism change of an edge of a tree clasper.(6) A swapping of a node and a disk-leaf that correspond to trivalent verteces.In each case, the difference of a change is given by an element of K n +1 ,k +1 ( M ).Indeed, the case (1) is obvious. In the cases (2) and (3), the difference is of the form[ K ; G ∪ C, G , . . . , G k ] − [ K ; G , . . . , G k ], where C is an I -clasper whose leaves maylink with edges of tree claspers as follows.The right hand side is obtained by Habiro’s move 12 of [Ha, Proposition 2.7]. Thetwo boxes can be moved by Habiro’s move 11 along the tree claspers toward theunivalent ends, so that the component T including the node in the right hand sideof the above picture does not have boxes.Here if the two leaves of C links with only G , then the strand labeled l may bedoubled after the slides of the boxes. Hence by Lemma 2.2 (3) we have[ K ; G ∪ C, G , . . . , G k ] − [ K ; G , . . . , G k ]= [ K ; G ′ ∪ T, G , . . . , G k ] − [ K ; G , . . . , G k ]= [ K ; G ′ , G , . . . , G k ] + [ K G ′ ; T, G , . . . , G k ] − [ K ; G , . . . , G k ]= [ K G ′ ; T, G , . . . , G k ] = 0 ∈ K n ( M ) / K n +1 ( M ) . In the case (4), the difference is of the form [ K C ; G , . . . , G k ] − [ K ; G , . . . , G k ] =[ K ; C, G , . . . , G k ] ∈ K n +1 ,k +1 ( M ), where C is an I -clasper for K with one disk-leaf quotient in his filtration, without any assumption on π ( M ). The techniques used in Proposi-tion 2.6 are almost the same as those used in Habiro’s result. TADAYUKI WATANABE and one leaf that is nullhomologous in M . In the case (5), the difference is of theform [ K ; G ∪ C, G , . . . , G k ] − [ K ; G , . . . , G k ], where C is an I -clasper as follows.The rest is similar to the case (2) except that a disk-leaf of C is replaced with anullhomologous leaf. For the case (6), see e.g., [Oh, Appendix E].Next, we shall see that the images of the relations for A NH n ( S ; Λ M ) under ψ n iszero in K n ( M ) / K n +1 ( M ). The proofs that the AS, IHX relations are mapped by ψ n to K n +1 ( M ) are similar as in [Ha, GGP]. The STU relation holds by a result of[Ha, § ρ , ρ : Γ → M are edge-bordant if and only if ι ◦ ρ and ι ◦ ρ arehomotopic, and ψ n is invariant under bordism changes of ρ , where we say that twoembedded graphs are edge-bordant if they are related by homotopies and relativebordism changes of edges.Finally, we shall check the surjectivity of ψ n when π ( M ) is abelian. Let [ K ; G ]be any M -null forest scheme in K n ( M ). By the STU relations, we may assume that[ K ; G ] ∈ K n,n ( M ), where G consists of n disjoint collection of I -claspers. If G hasonly disk-leaves, then by Lemma 2.2 (3) it can be modified modulo K n +1 ( M ) to asum of forest schemes with only simple disk-leaves (a simple disk-leaf is a disk-leafthat intersects K in a single point transversally. See Figure 2). Each such forestscheme is clearly obtained by the construction ψ n . If G has an I -clasper C with anullhomologous leaf, then the leaf is nullhomotopic since π ( M ) is abelian. Thusthe surgery on C can be replaced with a sequence of surgeries on strict I -claspersand [ K ; G ] can be rewritten as a sum of forest schemes with only strict I -claspers.The rest is the same as above. (cid:3) The main theorem of the present paper is the following.
Theorem 2.7. If H ( M ) = Z and M is fibered over S , then for n ≤ , the map ψ n is injective. Theorem 2.7 will be proved in § § M -null type invariants is rather fine. As a corollary to Proposition 2.6 andTheorem 2.7, we have the following. Corollary 2.8. If M = S × S , then for n ≤ , the map ψ n is an isomorphism. Conjecture 2.9.
Theorem 2.7 holds for all n ≥ and for all closed connectedoriented 3-manifold M . INITE TYPE INVARIANTS OF NULLHOMOLOGOUS KNOTS 9
Degree 1 part.
The following remark is an expansion from a comment ofK. Habiro. Since any nullhomologous knot can be unknotted by a surgery onan M -null I -clasper, we have K ( M ) / K ( M ) ∼ = Q . The degree 1 part is morecomplicated. Let ̟ ′ be the preimage of 0 of the natural map π ( LM ) → H ( M )where LM is the free loop space of M and let Q ̟ ′ be the vector space over Q spanned by the set ̟ ′ . Let η : K ( M ) / K ( M ) → Q ̟ ′ be the linear map that assigns to each forest scheme [ K ; G ] = K G − K its class[ K G ] − [ K ]. This is well-defined because any forest scheme of degree 2, size 2contains a strict I -clasper and because surgery on a strict I -clasper does not changethe free homotopy class of knot. Now we assume that H ( M ) = Z and that M isfibered over S . Then by Theorem 2.7 we have the following chain complex, whichmay be non-exact only at K ( M ) / K ( M ).(2.1) 0 −→ A NH1 ( S ; Λ) ψ −→ K ( M ) / K ( M ) η −→ Q ̟ ′ ε −→ Q −→ , where ε is the augmentation map. To make the sequence exact, it may be neces-sary to consider a noncommutative refinement of the map ψ n as in [Ka, KL, Va2]by the resolutions of singular knots (or allowing only strict I -claspers) and to re-strict underlying knots in the definition of the filtration K n ( M ) to (base-pointed)nullhomotopic knots in M .We will see in Appendix A that there is a natural isomorphism A NH1 ( S ; Λ) ∼ = Q [ t ] / Q . 3. Perturbative invariants of nullhomologous knots in M Let Λ = Q [ t ± ] and b Λ = Q ( t ). In this section, we define a map Z n : K ( M ) → A n ( S ; b Λ) for n = 1 , Z n is defined by intersectionsof certain fundamental chains in Lescop’s equivariant version of the configurationspaces ( § § § § Z n can be interpreted as the trace of a generating function of counts of certaingraphs in M , which we call AL-graphs ( § Equivariant configuration space.
From now on we consider an orientedsurface bundle κ : M → S with a fiberwise gradient ξ of an oriented fiberwiseMorse function for κ . Let K : S → M be a nullhomologous knot. We shall definea configuration space that is suitable to our purpose, based on Lescop’s equivariantconfiguration space [Les2, Les3, Les4].Let Conf q ( X ) denote the Fulton–MacPherson–Kontsevich compactification ofthe configuration space Conf q ( X ) of q distinct points on a compact real orientedmanifold X (see [Ko1, FM, BT, Les1] etc. for detail). Let Conf q ( S ) < denote thecomponent of Conf q ( S ) for a fixed ordering of the points on S . Let Conf t,q ( M, K ) be the pullback in the following commutative diagram:Conf t,q ( M, K ) / / (cid:15) (cid:15) Conf t + q ( M ) π t,q (cid:15) (cid:15) Conf q ( S ) < C Kq / / Conf q ( M )where π t,q is the natural map associated to the projection Conf t + q ( M ) → Conf q ( M )and C Kq is the extension of K × · · · × K : Conf q ( S ) < → Conf q ( M ).Let Γ be a labeled Jacobi diagram with q univalent and t trivalent vertices. Bythe labeling α : { , , . . . , t + q } → V (Γ), we identify E (Γ) with the set of orderedpairs ( i, j ), i, j ∈ { , , . . . , t + q } . Let M Γ denote the space of tuples( x , x , . . . , x n ; { γ ij } ( i,j ) ∈ E (Γ) ) , where x i ∈ M and γ ij is the homotopy class of continuous maps c ij : [0 , → S relative to the endpoints such that c ij (0) = κ ( x i ) and c ij (1) = κ ( x j ). We consider M Γ as a topological space as follows. Let C (Γ , S ) be the space of continuousmaps Γ → S equipped with the C -topology and let C Γ be the space that is thepullback in the following commutative diagram. C Γ / / (cid:15) (cid:15) C (Γ , S ) (cid:15) (cid:15) M t + q κ ×···× κ / / ( S ) t + q The fiberwise quotient map C Γ → M Γ by the homotopy relation of edges gives M Γ the quotient topology.Let Conf Γ ( M ) denote the space obtained from M Γ by blowing-up along all thelifts of the diagonals in M t + q . Let Conf Γ ( M, K ) denote the pullback in the followingcommutative diagram: Conf Γ ( M, K ) / / (cid:15) (cid:15) Conf Γ ( M ) (cid:15) (cid:15) Conf q ( S ) < C KO / / Conf O ( M )where O is the subgraph of Γ given by the Wilson loop and C KO is the natural mapinduced by K × · · ·× K , i.e., C Kq together with the relative homotopy classes of arcsrepresented by those in K . The forgetful map π : Conf Γ ( M, K ) → Conf t,q ( M, K )is a Z n − q -covering. Since Conf Γ ( M, K ) is naturally a Z n − q -space by the coveringtranslation, the twisted homology H i (Conf Γ ( M, K )) ⊗ Λ Γ b Λ Γ , where Λ Γ = Q [ { t ± ij } ( i,j ) ∈ E nW (Γ) ] and b Λ Γ = N ( i,j ) ∈ E nW (Γ) Q ( t ij ) (tensor prod-uct of Q -modules), is defined. Here, Λ Γ acts on b Λ Γ by ( Q ( i,j ) t k ij ij )( N ( i,j ) f ij ) = N ( i,j ) t k ij ij f ij . INITE TYPE INVARIANTS OF NULLHOMOLOGOUS KNOTS 11
Fiberwise Morse functions and their concordances.
Let κ : M → S be a smooth fiber bundle with fiber diffeomorphic to a closed connected oriented2-manifold Σ. We equip M with a Riemannian metric. We fix a fiberwise Morsefunction f : M → R and its gradient ξ : M → Ker dκ along the fibers that satisfiesthe parametrized Morse–Smale condition, i.e., the descending manifold loci and theascending manifold loci are mutually transversal in M . We consider only fiberwiseMorse functions that are oriented , i.e., the bundles of negative eigenspaces of theHessians along the fibers on the critical loci are oriented. There always exists anoriented fiberwise Morse function on M (e.g., [Wa2]). Definition 3.1. A generalized Morse function (GMF) is a C ∞ function on a mani-fold with only Morse or birth-death singularities ([Ig, Appendix]). A fiberwise GMF is a C ∞ function f : M → R whose restriction f c = f | κ − ( c ) : κ − ( c ) → R is a GMFfor all c ∈ S . A critical locus of a fiberwise GMF is the subset of M consisting ofcritical points of f c , c ∈ S . A fiberwise GMF is oriented if it is oriented outsidebirth-death loci and if birth-death pairs near a birth-death locus have incidencenumber 1.It is known that for a pair of fiberwise Morse functions f , f : M → R , thereexists a homotopy e f = { f s } s ∈ [0 , between f and f in the space of orientedGMF’s on M , which gives an oriented fiberwise GMF on the surface bundle κ × id : M × [0 , → S × [0 , Definition 3.2.
We say that the homotopy e f is a concordance if each birth-deathlocus of e f in M × [0 ,
1] projects to a closed curve that is not nullhomotopic in S × [0 , AL-paths.
Let π : f M → M be the Z -covering associated to κ . Let κ : f M → R be the lift of κ , f : f M → R denote the Z -invariant lift of f , and ξ denote the liftof ξ . We say that a piecewise smooth embedding σ : [ µ, ν ] → M is descending if κ ( σ ( µ )) ≥ κ ( σ ( ν )) and f ( σ ( µ )) ≥ f ( σ ( ν )), where σ : [ µ, ν ] → f M is a lift of σ . Wesay that σ is horizontal if Im σ is included in a single fiber of κ and say that σ is vertical if Im σ is included in a critical locus of f . Definition 3.3.
Let x, y be two points of M . An AL-path from x to y is a sequence γ = ( σ , σ , . . . , σ n ), where(1) for each i , σ i is a descending embedding [ µ i , ν i ] → M for some real numbers µ i , ν i such that µ i < ν i ,(2) for each i , σ i is either horizontal or vertical with respect to f ,(3) if σ i is horizontal, then σ i is a flow line of − ξ , possibly broken at criticalloci,(4) σ ( µ ) = x , σ n ( ν n ) = y ,(5) σ i ( ν i ) = σ i +1 ( µ i +1 ) for 1 ≤ i < n ,(6) if σ i is horizontal (resp. vertical) and if i < n , then σ i +1 is vertical (resp.horizontal).We say that two AL-paths are equivalent if they differ only by reparametrizationson segments. For generic ξ , there is special horizontal flow-line between critical loci, called1 / -intersection , which is the transversal intersection of the descending manifoldlocus of a critical locus of ξ of index 1 and the ascending manifold locus of anothercritical locus of index 1. There are finitely many 1 / ξ . Most ofthe horizontal segments in AL-paths are 1 / Equivariant propagator.
Let ξ be the fiberwise gradient of an oriented fiber-wise Morse function on M . We say that an AL-path γ in M with positive lengthis a closed AL-path if the endpoints of γ coincide. A closed AL-path γ gives apiecewise smooth map ¯ γ : S → M , which can be considered as a “closed orbit”in M . We will also call ¯ γ a closed AL-path. A closed AL-path has an orientationthat is determined by the orientations of descending and ascending manifolds lociof e ξ . See [Wa3, § ε ( γ ) ∈ {− , } and theperiod p ( γ ) of γ by p ( γ ) = |h [ dκ ] , [¯ γ ] i| , ε ( γ ) = h [ dκ ] , [¯ γ ] i|h [ dκ ] , [¯ γ ] i| . Let S ( T M ) be the subbundle of
T M of unit tangent vectors. Let S ( T γ ) be thepullback ¯ γ ∗ S ( T M ), which can be considered as a piecewise smooth 3-dimensionalchain in ∂ Conf K ( M ). We say that two closed AL-paths γ and γ are equivalent if there is a degree 1 homeomorphism g : S → S such that ¯ γ ◦ g = ¯ γ . Theindices of vertical segments in a closed AL-path must be all equal since an AL-pathis descending. We define the index ind γ of a closed AL-path γ to be the index ofa vertical segment (critical locus) in γ , namely, the index of the critical point of f | κ − ( c ) for any c ∈ S that is the intersection of γ with κ − ( c ).Let M = M \ S γ : critical locus γ and let s ξ : M → S ( T M ) be the normalization − ξ/ k ξ k of the section − ξ . The closure s ξ ( M ) in S ( T M ) is a smooth manifold withboundary whose boundary is the disjoint union of circle bundles over the criticalloci γ of ξ , for a similar reason as [Sh, Lemma 4.3]. The fibers of the circle bundlesare equators of the fibers of S ( T γ ). Let E − γ be the total space of the 2-disk bundleover γ whose fibers are the lower hemispheres of the fibers of S ( T γ ) which lie belowthe tangent spaces of the level surfaces of κ . Then ∂s ξ ( M ) = S γ ∂E − γ as sets. Let s ∗ ξ ( M ) = s ξ ( M ) ∪ [ γ E − γ ⊂ S ( T M ) . This is a 3-dimensional piecewise smooth manifold. We orient s ∗ ξ ( M ) by extendingthe natural orientation ( s − ξ ) ∗ o ( M ) on s ξ ( M ) induced from the orientation o ( M )of M . The piecewise smooth projection s ∗ ξ ( M ) → M is a homotopy equivalenceand s ∗ ξ ( M ) is homotopic to s ˆ ξ .Let C be a knot in M such that h [ dκ ] , [ C ] i = −
1. Let M AL K ( ξ ) be the set of allAL-paths in M . There is a natural structure of non-compact manifold with cornerson M AL K ( ξ ). For a closed AL-path γ , we denote by γ irr the minimal closed AL-pathsuch that γ is equivalent to the iteration ( γ irr ) k for a positive integer k and we call γ irr the irreducible factor of γ . This is unique up to equivalence. If γ = γ irr , wesay that γ is irreducible. We orient S ( T γ irr ) so that [ S ( T γ irr )] = p ( γ irr )[ S ( T C )].
INITE TYPE INVARIANTS OF NULLHOMOLOGOUS KNOTS 13
Note that this may not be the one naturally induced from the orientation of γ irr but from ε ( γ irr ) γ irr . Theorem 3.4 ([Wa2]) . Let M be the mapping torus of an orientation preservingdiffeomorphism ϕ : Σ → Σ of closed, connected, oriented surface Σ . Let ξ be thefiberwise gradient of an oriented fiberwise Morse function f : M → R . (1) There is a natural closure M AL K ( ξ ) of M AL K ( ξ ) that has the structure of acountable union of smooth compact manifolds with corners whose codimen-sion 0 strata are disjoint from each other. (2) Let ¯ b : M AL K ( ξ ) → M K be the evaluation map, which assigns the pair of theendpoints of an AL-path γ together with the homotopy class of κ ◦ γ relativeto the endpoints. Let Bℓ ¯ b − ( e ∆ M ) ( M AL K ( ξ )) denote the blow-up of M AL K ( ξ ) along ¯ b − ( e ∆ M ) . Then ¯ b induces a map Bℓ ¯ b − ( e ∆ M ) ( M AL K ( ξ )) → Conf K ( M ) and it represents a 4-dimensional b Λ -chain Q ( ξ ) in Conf K ( M ) that satisfiesthe identity ∂Q ( ξ ) = s ∗ ξ ( M ) + X γ ( − ind γ ε ( γ ) t p ( γ ) S ( T γ irr ) , where the sum is taken over equivalence classes of closed AL-paths in M .Moreover, (1 − t ) ∆( t ) Q ( ξ ) is a Λ -chain, where ∆( t ) is the Alexander poly-nomial of the fibration κ : M → S . Equivariant intersection form.
Let κ : M → S be a fibration. Fix acompact connected oriented 2-submanifold Σ of M without boundary such that theoriented bordism class of Σ in M corresponds to [ κ ] via the canonical isomorphismΩ ( M ) = H ( M ) ∼ = H ( M ). Let Γ be a labeled oriented Jacobi diagram of degree n and let E I (Γ) (resp. E ρ (Γ)) be the subset of E nW (Γ) consisting of non self-loopedges (resp. self-loop edges).(1) If e ∈ E I (Γ), then let ψ e : Conf Γ ( M, K ) → Conf K ( M ) denote the projec-tion that gives the endpoints of e together with the associated path pro-jected by κ in S . Take a compact oriented 4-submanifold F e in Conf K ( M )with corners.(2) If e ∈ E ρ (Γ), then let ψ e : Conf Γ ( M, K ) → M denote the projectionthat gives the unique endpoint of e together with the associated path pathprojected by κ in S . Take a compact oriented 1-submanifold F e in M withboundary.Note that in both cases F i is of codimension 2. Now we put E nW (Γ) = { i , . . . , i n − q } , E W (Γ) = { j , . . . , j q } ,V nW (Γ) = { k , . . . , k n − q } , V W (Γ) = { ℓ , . . . , ℓ q } . Then we define h F i , F i , . . . , F i n − q i Γ = n − q \ e =1 ψ − i e ( F i e ) , which gives a compact 0-dimensional submanifold in Conf Γ ( M, K ) if the intersec-tion is transversal. We equip each point ( u ℓ , . . . , u ℓ q , v k , . . . , v k n − q ; γ , γ , . . . , γ n ) of h F i , F i , . . . , F i n − q i Γ with a coorientation (a sign) in Conf Γ ( M, K ) by ^ e ∈ E (Γ) ψ ∗ e o ∗ Conf K ( M ) ( F e ) ( x e ,y e ,γ e ) ∈ V • T ( u ℓ ,...,u ℓq ,v k ,...,v k n − q ) Conf t,q ( M, K ) . Here, we identify a neighborhood of a point in Conf Γ ( M, K ) with its image of theprojection in Conf t,q ( M, K ). The coorientation gives a sign as the sign of µ inthe equation V e ψ ∗ e o ∗ Conf K ( M ) ( F e ) = µ o ( S ) u ℓ ∧ · · · ∧ o ( S ) u ℓq ∧ o ( M ) v k ∧ · · · ∧ o ( M ) v k n − q , where the order of the product is determined by the vertex label-ing, namely, so that it agrees with the exterior product of o ( S ) and o ( M ) in theorder of the vertex labeling. By this, h F i , F i , . . . , F i n − q i Γ represents a 0-chainin Conf Γ ( M, K ). This can be extended to tuples of codimension 2 Q -chains bymultilinearity. We will denote the homology class of h F i , F i , . . . , F i n − q i Γ (inte-ger) by the same notation. Note that a point of Conf Γ ( M, K ) possesses canon-ical homotopy classes of edges of E W (Γ) in S determined by the embedding C KO : Conf q ( S ) < → Conf O ( M ).We extend the form h· , . . . , ·i Γ to tuples of codimemsion 2 Λ-chains F ′ k in Conf K ( M )or M as follows. When k ∈ E I (Γ), suppose that F ′ k is of the form P N k λ k =1 µ ( k ) λ k σ ( k ) λ k ,where µ ( k ) λ k ∈ Λ and σ ( k ) λ k is a compact oriented smooth 4-submanifold in Conf K ( M )[0],where Conf K ( M )[0] is the subspace of Conf K ( M ) consisting of ( x , x , γ ) suchthat γ has a lift σ : [0 , → M connecting x and x with [ κ ◦ σ ] = γ relativeto the boundary whose interior has algebraic intersection number 0 with Σ. When k ∈ E ρ (Γ), suppose that F ′ k is of the form P N k λ k =1 µ ( k ) λ k σ ( k ) λ k , where σ ( k ) λ k is a piecewisesmooth path in M transversal to Σ and µ ( k ) λ k ( t ) = α ( k ) λ k t σ ( k ) λk · Σ , α ( k ) λ k ∈ Q . Then wedefine h F ′ i , F ′ i , . . . , F ′ i n − q i Γ = X λ ,λ ,...,λ n − q µ (1) λ ( t i ) µ (2) λ ( t i ) · · · µ (3 n − q ) λ n − q ( t i n − q ) (cid:10) σ (1) λ , σ (2) λ , . . . , σ (3 n − q ) λ n − q (cid:11) Γ ∈ C (Conf Γ ( M, K )) ⊗ Q , which can be considered as a 0-chain in Conf t,q ( M, K ) with coefficients in Λ Γ . Thisis multilinear by definition.Next, we extend the form h· , . . . , ·i Γ to tuples of codimension 2 b Λ-chains inConf K ( M ) or M as follows. Let Q i , Q i , . . . , Q i n − q be codimension 2 b Λ-chains inConf K ( M ) or M depending on whether the corresponding edge is not a self-loopor a self-loop. Then there exist p , p , . . . , p n − q ∈ Λ such that p k Q i k is a Λ-chainfor each k . Then we define h Q i , Q i , . . . , Q i n − q i Γ = h p Q i , p Q i , . . . , p n − q Q i n − q i Γ p ( t i ) − p ( t i ) − · · · p n − q ( t i n − q ) − ∈ C (Conf Γ ( M, K )) ⊗ Λ Γ b Λ Γ , which can be considered as a 0-chain in Conf t,q ( M, K ) with coefficients in b Λ Γ . Thisdoes not depend on the choices of p , . . . , p n − q and this is multilinear by definition.Note that the multilinear form h· , . . . , ·i Γ depends on the choice of Σ. INITE TYPE INVARIANTS OF NULLHOMOLOGOUS KNOTS 15
Figure 5.
Jacobi diagrams of degree 1 and 2 (edge-orientations omitted).Let ̟ O : C (Conf Γ ( M, K )) ⊗ Λ Γ b Λ Γ → C (Conf Γ ( M )) ⊗ Λ ⊗ n b Λ ⊗ n be the naturalmap induced by C KO and we define a linear mapTr Γ : C (Conf Γ ( M )) ⊗ Λ ⊗ n b Λ ⊗ n → A n ( S ; b Λ)by settingTr Γ (cid:16) q ( t ) p ( t ) ⊗ q ( t ) p ( t ) ⊗ · · · ⊗ q n ( t n ) p n ( t n ) (cid:17) = h Γ (cid:16) q ( t ) p ( t ) , q ( t ) p ( t ) , . . . , q n ( t ) p n ( t ) (cid:17)i . Definition of Z n . Let κ , κ , . . . , κ n : M → S be fibrations isotopic to κ .Let f i : M → R , i = 1 , , . . . , n , be oriented fiberwise Morse functions for κ i such that ( κ i , f i ) is concordant to a pair isotopic to ( κ, f ). Let ξ i be the fiberwisegradient of f i . Let Q ( ξ i ) be the equivariant propagator in Theorem 3.4 for ξ i . Wedefine the 1-cycle Q ′ ( ξ i ) = X γ γ ∈ C ( M ; b Λ)in M , where the sum is over equivalence classes of all closed AL-paths γ for ξ i considered as oriented 1-cycles. This is an infinite sum but is well-defined as a b Λ-chain. The orientation of a closed AL-path γ is given by ε ( γ ) times the downwardorientation on γ . Choosing the closed oriented surface Σ ⊂ M and κ i , ξ i generically,we may define I Γ ( K ) = Tr Γ ̟ O h Q ◦ ( ξ i ) , Q ◦ ( ξ i ) , . . . , Q ◦ ( ξ i n − q ) i Γ ∈ A n ( S ; b Λ) , where Q ◦ ( ξ i ) is Q ( ξ i ) or Q ′ ( ξ i ) depending on whether the corresponding edge in Γis not a self-loop or a self-loop, as in [Wa3]. Theorem 3.5.
For n ≥ , we define Z n ( K ) = 1(2 n )!(3 n )! X Γ I Γ ( K ) ∈ A n ( S ; b Λ) , where the sum is over all labeled Jacobi diagrams on S of degree n for all possibleedge orientations (Jacobi diagrams of degree 1 and 2 are shown in Figure 5.). For n = 1 , , Z n ( K ) is invariant under isotopy of K and concordance of f . Let E ( κ ) be the set of concordance classes of oriented fiberwise Morse functionsfor a fibration κ : M → S . Theorem 3.5 says that for n = 1 , Z n gives a familyof knot invariants parametrized by E ( κ ). Figure 6.
An AL-graph for a knot3.7. Z n and the generating function of counts of AL-graphs. Let f , f , . . . , f n and ξ , ξ , . . . , ξ n be as in § K be a nullhomologous knot in M . Definition 3.6.
Let Σ = κ − (0). Suppose that no 1 / ξ i is onΣ. For a labeled Jacobi diagram Γ of degree n , we define M ALΓ( ~k ) (Σ; ξ , ξ , . . . , ξ n ), ~k = ( k , k , . . . , k n ), as the set of piecewise smooth maps I : Γ → M such that(1) if i ∈ E nW (Γ), the restriction of I to the i -th edge is an AL-path of ξ i ,(2) if i ∈ E W (Γ), the restriction of I to the i -th edge is embedded to a segmentin K in an orientation preserving way,(3) the algebraic intersection of the restriction of I to the i -th edge e i with Σis k i .We call such maps AL-graphs for (Σ; ξ , ξ , . . . , ξ n ) of type ~k . We define a topologyon M ALΓ( ~k ) (Σ; ξ , ξ , . . . , ξ n ) as the transversal intersection of smooth submanifoldsof Conf Γ ( M ), as in § M ALΓ( ~k ) (Σ; ξ , ξ , . . . , ξ n ) for generic choice of ( κ i , f i ).)We may identify a point of M ALΓ( ~k ) (Σ; ξ , . . . , ξ n ) with an oriented 0-manifoldin Conf Γ ( M, K ). Hence the moduli space M ALΓ( ~k ) (Σ; ξ , . . . , ξ n ) can be countedwith signs. The sum of signs agrees with the sum of coefficients of the terms of t k t k · · · t k n n in the power series expansion of ̟ O h Q ◦ ( ξ i ) , Q ◦ ( ξ i ) , . . . , Q ◦ ( ξ i n − q ) i Γ .We denote the sum of signs by M ALΓ( ~k ) (Σ; ξ , . . . , ξ n ).For generic choices of Σ , κ , . . . , κ n , ξ , . . . , ξ n , an AL-graph I ∈ M ALΓ( ~k ) (Σ; ξ , . . . , ξ n )consists of finitely many horizontal components and some AL-paths that connecttwo univalent vertices of horizontal components or one univalent vertex of a hori-zontal component and the knot K (see Figure 6). If an AL-path that starts at (resp.end at) a point of K has a vertical segment, then its first (resp. last) horizontalsegment σ is the transversal intersection of K and the ascending (resp. descending)manifold locus of a vertical segment next to σ (resp. previous to σ ). The followingproposition follows from Theorem 3.4 and from definition of Z n . Proposition 3.7.
For generic choices of ξ , ξ , . . . , ξ n and for a labeled trivalentgraph Γ , let F Γ (Σ; ξ , ξ , . . . , ξ n ) be the generating function X ~k =( k ,...,k n ) ∈ Z n M ALΓ( ~k ) (Σ; ξ , ξ , . . . , ξ n ) t k t k · · · t k n n , INITE TYPE INVARIANTS OF NULLHOMOLOGOUS KNOTS 17 where M ALΓ( ~k ) (Σ; ξ , ξ , . . . , ξ n ) is the count of AL-graphs of type ~k of generic type.Then there exist Laurent polynomials P ( µ ) i ( t ) ∈ Λ , i = 1 , , . . . , n , µ = 1 , , . . . , N ,such that F Γ (Σ; ξ , ξ , . . . , ξ n ) = N X µ =1 3 n Y i =1 P ( µ ) i ( t i ) n − q Y k =1 { (1 − t i k ) ∆( t i k ) } − . Considering this as an element of Q ( t ) ⊗ Q Q ( t ) ⊗ Q · · · ⊗ Q Q ( t n ) , we have Z n ( K ) = 1(2 n )!(3 n )! X Γ Tr Γ F Γ (Σ; ξ , ξ , . . . , ξ n ) . Well-definedness of Z n (proof of Theorem 3.5) Let I = [0 , K , K ⊂ M be two nullhomologous knots that are isotopic. Let e K : S × I → M × I bean isotopy between K and K and for s ∈ I let K s : S → M be the embedding K s ( x ) = e K ( x, s ). Let Conf t,q ( M × I, e K ) be the Conf t,q -bundle over I given by thepullback in the following commutative diagramConf t,q ( M × I, e K ) / / (cid:15) (cid:15) Conf t + q ( M ) (cid:15) (cid:15) Conf q ( S ) < × I C f Kq / / Conf q ( M )where C e Kq is the natural map induced by e K . Let Γ be a labeled Jacobi diagramwith t trivalent and q univalent vertices. Let Conf Γ ( M, K ) be the pullback in thefollowing commutative diagramConf Γ ( M × I, e K ) / / (cid:15) (cid:15) Conf Γ ( M ) (cid:15) (cid:15) Conf q ( S ) < × I C f KO / / Conf O ( M )where C e KO is the natural map induced by e K . Then Conf Γ ( M × I, e K ) forms a fiberbundle over I with fiber diffeomorphic to Conf Γ ( M, K ).4.2. Bifurcations of AL-graphs in 1-parameter family, ξ i fixed. By replac-ing Q ◦ ( ξ i ) with e Q ◦ ( ξ i ) = Q ◦ ( ξ i ) × I in the definition of I Γ ( K ), we may define I Γ ( e K ) = Tr Γ ̟ O h e Q ◦ ( ξ i ) , . . . , e Q ◦ ( ξ i n − q ) i Γ . If e K is generic, this gives a piecewise smooth 1-chain in C (Conf t,q ( M × I, e K )) ⊗ Q A n ( S ; b Λ). A bordism change of Σ may change the value of ̟ O h Q ◦ ( ξ i ) , . . . , Q ◦ ( ξ i n − q ) i Γ . However, one may see that its trace is invariant under a bordism of Σ, by exactlythe same argument as [Wa3, Lemma 4.1]. We have Z n ( K ) − Z n ( K ) = ± n )!(3 n )! X Γ ∂I Γ ( e K ) . We may arrange that the possible contributions in the boundary of I Γ ( e K ) in the1-parameter family are of the following forms:(1) AL-graph with an edge in Γ collapsed to a point.(2) AL-graph with a nonseparated full subgraph Γ ′ of Γ with at least 3 verticescollapsed to a point, where we say that Γ ′ is full if every edge ( i, j ) ∈ E nW (Γ) with i, j ∈ V (Γ ′ ) belongs to E nW (Γ ′ ) and if the vertices in V W (Γ ′ )are successive in Γ, and we say that Γ ′ is nonseparated if Γ is not of theform Closure(Γ ′ ) ′′ for any Jacobi diagram Γ ′′ .(3) (Anomalous face) A separated component in P (Γ) collapsed to a point ona knot, where we say that a component Γ in P (Γ) is separated if Γ can bewritten in the form Γ ′ ′′ for Jacobi diagrams Γ ′ and Γ ′′ such that Γ ′ isnullhomotopic ‡ and P (Γ ′ ) = { Γ } .These correspond to the intersection of I Γ ( e K ) with ∂ Conf t,q ( M × I, e K ). Note thatit is not necessary to consider an AL-graph with a non self-loop edge forming aclosed AL-path in M since such an AL-graph and an AL-graph with one 4-valentvertex do not occur simultaneously in a generic 1-parameter family. Lemma 4.1.
For n ≥ , Z n is invariant under bifurcations (1) and (2).Proof. The invariance under a bifurcation of type (1) follows by the IHX and STUrelations. Roughly, the AL-graph with an edge labeled k collapsed to a point hasone 4-valent vertex whose contribution in F Γ (Σ; ξ , . . . , ξ n ) is a rational functionwith no terms of nonzero exponents of t k . The sum of contributions of such AL-graphs come from terms in the IHX or the STU relations and they are set to bezero by the relations. For the detail, see [AF], where the boundary contribution isgiven by integrations in place of the counts.The invariance under a bifurcation of type (2) follows by an analogue of Kont-sevich’s lemma ([Ko1]) as in [Wa3] and by dimensional reasons. See the proof of[Wa3, Lemma 4.4] for the detail. Here, we must take care of the orientations ofthe faces of the moduli spaces at the boundary of Conf Γ ( M × I, e K ). However, theproblem is local and the orientations of the faces can be treated in almost the sameway as the case of knots in S given in [BT, AF]. (cid:3) We shall consider the invariance under a bifurcation of type (3). Let s ∈ I be aparameter at which a bifurcation of type (3) occurs and put J = [ s − ε, s + ε ] for ε > e K , we may assume that there are finitelymany such bifurcation time s in I . We may assume without loss of generality thatonly one bifurcation occurs in J . The change of the value of Z n for the bifurcationis given by sum of contributions of the anomalous face S of ∂ Conf Γ ( M × J, e K ), ‡ If one of two Jacobi diagrams is nullhomotopic, then the connected sum of the two is well-defined (i.e., independent of the choice of arcs for the connected sum).
INITE TYPE INVARIANTS OF NULLHOMOLOGOUS KNOTS 19 which can be described as follows, following [BT, BC]. Let Γ be a labeled Jacobidiagram on S with q univalent and t trivalent vertices. We consider Γ that is anullhomotopic Λ-colored Jacobi diagram. Suppose for simplicity that the univalentvertices of Γ are labeled by 1 , , . . . , q and they are cyclically ordered in this order.Let P → M be the orthonormal frame bundle associated to T M . Then the unittangent bundle S ( T M ) is identified with P × SO (3) S . Let ¯ β : S ( β ∗ T M ) → S ( T M )be the natural bundle map that covers β = proj ◦ e K : S × J → M . Let B t,q bethe space of points ( a, u , . . . , u q , v , . . . , v t ) in S × ( R ) q × ( R ) t such that • u < · · · < u q , or its cyclic permutations, • v i = v j if i = j , • u i a = v j for all i ∈ { , . . . , q } , j ∈ { , . . . , t } , • q X i =1 u i + t X j =1 k v j k = 1 and q X i =1 u i + t X j =1 h v j , a i = 0.The forgetful map B t,q → S is a fiber bundle. Let ν : S × J → S ( β ∗ T M ) be thesection given by the unit tangent vectors of knots e K | S ×{ s } : S × { s } → M × { s } for each s ∈ J and let S be the pullback in the following commutative diagram. S E ( ¯ β ◦ ν ) / / (cid:15) (cid:15) P × SO (3) B t,q (cid:15) (cid:15) S × J ¯ β ◦ ν / / S ( T M )Then S can be naturally identified with the (interior of the) anomalous face of ∂ Conf t,q ( M × J, e K ).Let π ij : P × SO (3) B t,q → S ( T M ) denote the fiberwise Gauss map, namely, itgives the direction of the straight line that connects the i -th and the j -th vertex in T x M . Let θ ℓ = E ( ¯ β ◦ ν ) − π − ij ( s ∗ ξ ℓ ( M )) ⊂ S , where the edge labeled ℓ is ( i, j ). This is a piecewise smooth submanifold of S andhas a natural coorientation induced from that of s ∗ ξ ℓ ( M ) in S ( T M ). Let Γ be aseparated component in P (Γ) and let h θ , . . . , θ k i Γ be the count of T kℓ =1 θ ℓ withsigns and put I Γ ( e K ) = h θ , . . . , θ k i Γ [Γ ] ∈ A n ( S ; Q ) , where Γ is the union of Γ and the Wilson loop, and n = deg Γ . If ε issmall enough, then I Γ ( e K ) for the bifurcation of type (3) over J is of the form I Γ ( e K ) I Γ ( K s − ε ) for Γ such that Γ = Γ . Lemma 4.2.
For n = 1 , , Z n is invariant under bifurcation (3).Proof. By the same reason as [BT, Theorem 1.6], the sum of I Γ ( e K ) for all labeledJacobi diagram Γ and for all possible edge orientations vanishes if the degree of Γ is even. Note that in [BT], a cancellation by a symmetry of a (unlabeled) graph isconsidered, whereas we consider a cancellation between two graphs with differentlabellings. If the degree of Γ is 1, the contributions of the graphs that are relevant to theanomalous faces vanish by the FI relation. This completes the proof. (cid:3) Bifurcations of AL-graphs in 1-parameter family, ξ i perturbed.Lemma 4.3. For n = 1 , , Z n is invariant under a concordance of ξ i .Proof. Let e ξ i be the 1-parameter family for a concordance of ξ i . By [Wa3, § § I Γ ( e K ) = Tr Γ ̟ O h e Q ◦ ( e ξ i ) , . . . , e Q ◦ ( e ξ i n − q ) i Γ in C (Conf t,q ( M × J, e K )) ⊗ Q A n ( S ; b Λ). For the rest of the proof, one may see byan argument similar to [Wa3] that the boundaries of I Γ ( e K ) that may contribute areof the same kinds as (1)–(3) listed above. Proof of the invariance under bifurcationsof types (1) and (2) is the same as Lemma 4.1.To prove (3), assume that n ≤
2. The anomalous face f S over a closed interval J ⊂ I is the pullback in the following commutative diagram. f S E ( e ν ) / / (cid:15) (cid:15) ( P × SO (3) B t,q ) × J (cid:15) (cid:15) S × J e ν / / S ( T M ) × J where e ν : S × J → S ( T M ) × J is the section given by the unit tangent vectors ofknots e K | S ×{ s } : S ×{ s } → M ×{ s } . Let ( M × J ) = ( M × J ) \ S σ : critical locus of e ξ ℓ σ and let s e ξ ℓ : ( M × J ) → S ( T M ) × J be the normalization − e ξ ℓ / k e ξ ℓ k of the section − e ξ ℓ . The closure s e ξ ℓ (( M × J ) ) in S ( T M ) × J is a smooth manifold with boundarywhose boundary in S ( T M ) × Int J is the disjoint union of circle bundles over thecritical loci σ of e ξ ℓ (including the birth-death locus γ ). The fibers of the circlebundles are equators of the fibers of S ( T σ ). Let E − σ be the total space of the 2-diskbundle over σ whose fibers are the lower hemispheres of the fibers of S ( T σ ) which liebelow the tangent spaces of the level surfaces of κ . Then ∂s e ξ ℓ (( M × J ) ) = S σ ∂E − σ as sets. Let s ∗ e ξ ℓ ( M × J ) = s e ξ ℓ (( M × J ) ) ∪ [ σ E − σ ⊂ S ( T M ) × J. This is a 4-dimensional piecewise smooth manifold. We orient s ∗ e ξ ℓ ( M × J ) byextending the natural orientation ( s − e ξ ℓ ) ∗ o ( M × J ) on s e ξ ℓ (( M × J ) ). Let e θ ℓ = E ( e ν ) − e π − ij ( s ∗ e ξ ℓ ( M × J )) ⊂ f S , where e π ij : ( P × SO (3) B t,q ) × J → S ( T M ) × J denote the fiberwise Gauss map. Weput e I Γ ( e K ) = h e θ , . . . , e θ k i Γ [Γ ] ∈ A n ( S ; Q ) , where Γ is the union of Γ and the Wilson loop, and n = deg Γ . If a bifurcationof type (3) occurs at s and J = [ s − ε, s + ε ] small enough, then I Γ ( e K ) for INITE TYPE INVARIANTS OF NULLHOMOLOGOUS KNOTS 21 the bifurcation of type (3) is of the form e I Γ ( e K ) I Γ ( K s − ε ) for Γ such thatΓ = Γ . The reason of the vanishing of the sum of I Γ ( e K ) is the same asLemma 4.2. (cid:3) Theorem 3.5 now follows as a corollary of Lemmas 4.1, 4.2 and 4.3.5.
Injectivity of ψ n when H ( M ) = Z (proof of Theorem 2.7) Surgery formula.Theorem 5.1.
Let κ : M → S be a smooth bundle with fiber diffeomorphic to anoriented connected closed surface. For n ≤ , Z n satisfies the following properties. (1) Z n is a finite type invariant of M -null type n . Hence Z n induces a map Z n : K n ( M ) / K n +1 ( M ) → A n ( S ; b Λ) . (2) If H ( M ) = Z , then Z n ( ψ n (Γ( φ ))) = 2 n [Γ( φ )] for a monomial Λ -coloredJacobi diagram Γ( φ ) ∈ G NH n ( S ; Λ) . For general H ( M ) , the followingdiagram is commutative: A NH n ( S ; Λ M ) ψ n / / κ ∗ (cid:15) (cid:15) K n ( M ) / K n +1 ( M ) Z n v v ❧❧❧❧❧❧❧❧❧❧❧❧❧ A n ( S ; b Λ) where κ ∗ is the natural map induced by κ ∗ : K ( H ( M ) , → K ( H ( S ) , .Proof. We shall prove the theorem only for n = 2. The case n ≤ S given in [BT, AF].(1) It suffices to check Z ( K , ( M )) = 0. Let [ K ; G ] ∈ K , ( M ), G = { G , G , G } ,be an M -null forest scheme consisting of three I -claspers and let R i be a regu-lar neighborhood of G i in M . (If G i is a strict I -clasper, then R i is an openball. If G i has one nullhomologous leaf, then R i is an open solid torus.) Byshrinking strict I -claspers by isotopy, we may assume that R i ’s are mutually dis-joint and that R i is a small ball in M if G i is a strict I -clasper. We show that I Γ ([ K ; G ]) = P I ⊂{ , , } ( − −| I | I Γ ( K G I ) vanishes for such a forest scheme [ K ; G ]and for any Jacobi diagram Γ of degree 2.Recall from Proposition 3.7 that I Γ is given by counts of AL-graphs. By Lemma 2.2(2), AL-graphs X of degree 2 that may contribute to the alternating sum I Γ ([ K ; G ])should be such that for every i , R i is occupied , i.e., on each strand of R i ∩ K G I there is at least one univalent vertex of X . Note that if G i is a strict I -clasper, then R i ∩ K G i consists of two strands, and if G i is an I -clasper with a nullhomologousleaf, then R i ∩ K G i consists of one strand. Hence X should have at least 5 univalentvertices. But this is impossible if X is of degree 2. This completes the proof of Z ([ K ; G ]) = 0 and of (1).(2) It suffices to check the assertion for forest schemes in K , ( M ) that has onlydisk-leaves since any Λ-colored Jacobi diagram on S can be written as a sum ofchord diagrams by the STU relation. Let [ K ; G ] ∈ K , ( M ), G = { G , G } , bea forest scheme corresponding to a chord diagram Γ( φ ) such that G and G arestrict I -claspers. Moreover, we may assume that the I -claspers are shrunk into small balls. By the argument of (1), the AL-graphs X that may survive in thealternating sum Z ([ K ; G ]) should be such that for i = 1 , R i is occupied. Suchan AL-graph corresponds to a chord diagram. Furthermore, the two univalentvertices in each R i should be connected by an AL-path included in R i , i.e., anAL-path without vertical segments, because if not, the small crossing change in R i does not change the value of I Γ .For the monomial Λ-colored chord diagram Γ( φ ), there are
4! 6! | Aut Γ( φ ) | differentways of labelings and edge-orientations on Γ( φ ), where | Aut Γ( φ ) | is the order ofthe group of automorphisms of Γ( φ ) which preserves the homotopy class of Λ-coloring. Each term of these labeled oriented chord diagrams in the alternatingsum Z ([ K ; G ]) contribute as | Aut Γ( φ ) | [Γ( φ )]. The terms for other graphs vanish.Hence we have Z ([ K ; G ]) = 14! 6! 2
4! 6! | Aut Γ( φ ) | | Aut Γ( φ ) | [Γ( φ )] = 2 [Γ( φ )] . This completes the proof of (2). (cid:3)
Proof of Theorem 2.7.
By Theorem 5.1, the image of Z n : ψ n ( A NH n ( S ; Λ)) → A n ( S ; b Λ) is A NH n ( S ; Λ). Hence Z n ◦ ψ n = 2 n id and ψ n is injective. (cid:3) Example 5.2.
Let M = S × S and K = { p } × S ( p ∈ S ). Let O be theunknot in a small ball in M . We consider the Whitehead double Wh ( K ) withrespect to the product framing on K . By Theorem 5.1, we see that Z (Wh ( K )) − Z ( O ) = 2[Θ(0 , , where Θ( p, q ) is the Λ-colored Jacobi diagram defined in Appendix A below. It iseasy to check that Z ( O ) = 0. Hence we have Z (Wh ( K )) = 2[Θ(0 , . It follows from the result of Appendix A that Z (Wh ( K )) is nontrivial. Appendix A. The structure of A NH1 ( S ; Λ)For p, q ∈ Z , we putΘ( p, q ) = , Ω( p ) = . Lemma A.1. (1) [Ω( p )] = 0 in A NH1 ( S ; Λ) . (2) A NH1 ( S ; Λ) is spanned by { [Θ( p, q )] } p,q ∈ Z .Proof. The claim (1) follows immediately by the STU relation. Since any Jacobidiagram in A NH1 ( S ; Λ) of degree 1 with a self-loop is equivalent to Ω( p ) for some p modulo the Holonomy relation and since [Ω( p )] = 0 by (1), A NH1 ( S ; Λ) is generatedby chord diagrams of the form Θ( p, q ). This completes the proof of (2). (cid:3) INITE TYPE INVARIANTS OF NULLHOMOLOGOUS KNOTS 23
Consider Q as the linear subspace of Q [ t ] of constants. Let W : A NH1 ( S ; Λ) → Q [ t ] / Q be the linear map defined by W ([Θ( p, q )]) = t p + q (mod Q ) . This is well-defined since W respects all the relations for A NH1 ( S ; Λ). Proposition A.2.
The map W is a linear isomorphism.Proof. Let L : Q [ t ] / Q → A NH1 ( S ; Λ) be the linear map defined by L ( t p ) = [Θ(0 , p )]for p ≥
0, which is well-defined. We have L ( W ([Θ( p, q )])) = L ( t p + q ) = [Θ(0 , p + q )] = [Θ( p, q )] by the Holonomy relation and we have W ( L ( t p )) = W ([Θ(0 , p )]) = t p (mod Q ). This completes the proof. (cid:3) Acknowledgments.
The author is supported by JSPS Grant-in-Aid for Young Scientists (B) 26800041.He is grateful to Professor Kazuo Habiro for pointing out mistakes in an earlier ver-sion of this paper and for helpful comments on clasper theory. He also would liketo thank Professor Efstratia Kalfagianni for giving him information on finite typeinvariants of knots in 3-manifolds.
References [AF] D. Altschuler, L. Freidel,
Vassiliev knot invariants and Chern-Simons perturbation theoryto all orders , Comm. Math. Phys. (1997), 261–287.[BL] J. Birman, X. S. Lin,
Knot polynomials and Vassiliev’s invariants , Invent. Math. (1993), 225–270.[BN1] D. Bar-Natan,
Perturbative Aspects of the Chern-Simons Topological Quantum FieldTheory , Ph.D. thesis, Princeton Univ. (1991).[BN2] D. Bar-Natan,
On the Vassiliev knot invariants , Topology (1995), 423–472.[BC] R. Bott, A. Cattaneo, Integral invariants of 3-manifolds , J. Differential Geom. (1998),no. 1, 91–133.[BT] R. Bott, C. Taubes, On the self-linking of knots , J. Math. Phys. , (1994), 5247–5287.[CDM] S. Chmutov, S. Duzhin, J. Mostovoy, Introduction to Vassiliev Knot Invariants , Cam-bridge Univ. Press (2012).[CV] J. Conant, K. Vogtmann,
On a theorem of Kontsevich , Algebr. Geom. Topol. (2003),1167–1224.[FM] W. Fulton, R. MacPherson, A compactification of configuration spaces , Ann. of Math. (1994), 183–225.[GGP] S. Garoufalidis, M. Goussarov, M. Polyak,
Calculus of clovers and finite type invariantsof 3–manifolds , Geom. Topol. (2001), 75–108.[GR] S. Garoufalidis, L. Rozansky, The loop expansion of the Kontsevich integral, the null-move and S -equivalence , Topology (2004), 1183–1210.[GMM] E. Guadagnini, M. Martellini, M. Mintchev, Wilson lines in Chern-Simons theory andlink invariants , Nuclear Phys. B (1990), 575–607.[Gu] M. Gusarov,
Variations of knotted graphs, geometric technique of n-equivalence , St. Pe-tersburg Math. J. (4) (2001).[Ha] K. Habiro, Claspers and finite type invariants of links , Geom. Topol. (2000), 1–83.[Ig] K. Igusa, The space of framed functions , Trans. Amer. Math. Soc. , no. 2 (1987),431–477. [Ka] E. Kalfagianni,
Finite type invariants for knots in 3-manifolds , Topology , no. 3 (1998),637–707.[KL] P. Kirk, C. Livingston, Type 1 knot invariants in 3-manifolds , Pacific J. Math. , no.3 (1998), 305–331.[Koh] T. Kohno,
Vassiliev invariants and de Rham complex on the space of knots , Symplecticgeometry and quantization (Sanda and Yokohama, 1993), Contemp. Math. , Amer.Math. Soc., Providence, RI (1994), 123–138.[Ko1] M. Kontsevich,
Feynman diagrams and low-dimensional topology , First European Con-gress of Mathematics, Vol. II (Paris, 1992), Progr. Math. (Birkhauser, Basel, 1994),97–121.[Ko2] M. Kontsevich,
Vassiliev’s knot invariants , Adv. Soviet. Math. (2) (1993), 137–150.[Les1] C. Lescop, On the Kontsevich–Kuperberg–Thurston construction of a configuration-space invariant for rational homology 3-spheres , math.GT/0411088, Pr´epublication del’Institut Fourier (2004).[Les2] C. Lescop,
On the cube of the equivariant linking pairing for knots and 3-manifolds ofrank one , arXiv:1008.5026.[Les3] C. Lescop,
Invariants of knots and 3-manifolds derived from the equivariant linking pair-ing , Chern–Simons gauge theory: 20 years after, AMS/IP Stud. Adv. Math. (2011),Amer. Math. Soc., Providence, RI, 217–242.[Les4] C. Lescop, A universal equivariant finite type knot invariant defined from configurationspace integrals , arXiv:1306.1705.[Lie] J. Lieberum,
Universal Vassiliev invariants of links in coverings of 3-manifolds , J. KnotTheory Remifications no. 4 (2004), 515–555.[Oh] T. Ohtsuki, Quantum Invariants: A Study of Knots, 3-Manifolds, and Their Sets , Serieson Knots and Everything , World Scientific Publishing Co., 2002.[Sh] T. Shimizu, An invariant of rational homology 3-spheres via vector fields ,arXiv:1311.1863.[Sch1] R. Schneiderman,
Algebraic linking numbers of knots in 3-manifolds , Algebr. Geom.Topol. (2003), 921–968.[Sch2] R. Schneiderman, Stable concordance of knots in 3-manifolds , Algebr. Geom. Topol. no. 1 (2010), 373–432.[Va1] V. A. Vassiliev, Cohomology of knot spaces , Theory of Singularities and Its Applications(V. I. Arnold ed.). Amer. Math. Soc., Providence, RI (1990), 23–69.[Va2] V. A. Vassiliev,
On invariants and homology of spaces of knots in arbitrary manifolds ,Topics in quantum groups and finite-type invariants, Amer. Math. Soc. Transl. Ser. 2, (1998), Amer. Math. Soc. Providence, RI, 155–182.[Wa1] T. Watanabe,
Higher order generalization of Fukaya’s Morse homotopy invariant of 3-manifolds I. Invariants of homology 3-spheres , arXiv:1202.5754.[Wa2] T. Watanabe,
Morse theory and Lescop’s equivariant propagator for 3-manifolds with b = 1 fibered over S , arXiv:1403.8030.[Wa3] T. Watanabe, An invariant of fiberwise Morse functions on surface bundle over S bycounting graphs , arXiv:1503.08735. Department of Mathematics, Shimane University, 1060 Nishikawatsu-cho, Matsue-shi,Shimane 690-8504, Japan
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