FINITE TYPE INVARIANTS OF STRING LINKS AND THE HOMFLYPT POLYNOMIAL OF KNOTS

JEAN-BAPTISTE MEILHAN AND AKIRA YASUHARA

Abstract. A family of finite type invariants of string link is given by the HOMFLYPT polynomial of knots using various closure operations on (cabled) string links. In this talk we will show the following:

(1) These invariants, together with Milnor invariants of length ≤ 5, give classifications of n-string links up to C_{k}-equivalence for k ≤ 5, and give a complete set of finite type invariants of degree ≤ 4.

(2) Any Milnor invariant of length n + 1(> 2) of a C_{n}-trivial string link is expressed as a linear combination of such invariants.

1. String links and C_{k}-moves

The notion of string link was introduced by Le Dimet [3] and Habegger-Lin [5]. A string link is a kind of tangle without closed components in the cylinder, which generalizes pure braids.

Let D be the unit disk in the plane. Choose n points p_{1}, ..., p_{n} in the interior of D so that p_{1}, ..., p_{n} lie in order on the x-axis, see Figure 2.1. An n-string link L = K_{1} ∪ ⋯ ∪ K_{n} in D × [0, 1] is a disjoint union of oriented arcs K_{1}, ..., K_{n} such that each K_{i} runs from (p_{i}, 0) to (p_{i}, 1) (i = 1, ..., n). The string link K_{1} ∪ ⋯ ∪ K_{n} with K_{i} = {p_{i}} × [0, 1] (i = 1, ..., n) is called the trivial n-string link and denoted by 1_{n}.

The set SL(n) of isotopy classes of n-string links fixing the endpoints has a monoidal structure, with composition given by the stacking product and with the trivial n-string link 1_{n} as unit element.

Habiro [6] and Goussarov [4] introduced independently the notion of C_{k}-move as follows. (This notion can also be defined by using the theory of claspers, see Subsection 5.1.) A C_{k}-move is a local move on (string) links as illustrated in Figure 1.1, which can be regarded as a kind of ‘higher order crossing change’ (in particular, a C_{1}-move is a crossing change). The C_{k}-move generates an equivalence relation

Figure 1.1. A C_{k}-move involves k + 1 strands of a link, labelled here by integers between 0 and k.

on (string) links, called C_{k}-equivalence, which becomes finer as k increases. Thus we have a descending filtration

SL(n) = SL_{1}(n) ⊃ SL_{2}(n) ⊃ SL_{3}(n) ⊃ ...

The second author is partially supported by a Grant-in-Aid for Scientific Research (C)

(#20540065) of the Japan Society for the Promotion of Science.

where SL_{k}(n) denotes the set of C_{k}-trivial n-string links, i.e., string links which are C_{k}-equivalent to 1_{n}. For 1 ≤ k ≤ l, let SL_{k}(n)/C_{l} denote the set of C_{l}-equivalence classes of C_{k}-trivial n-string links. This is known to be a finitely generated nilpotent group. Furthermore, if l ≤ 2k, this group is abelian [6, Thm. 5.4].

2. Finite type invariants of string links

A singular n-string links is a proper immersion ⊔_{i=1}^{n}[0, 1]_{i} → D^{2} × [0, 1] of the disjoint union ⊔_{i=1}^{n}[0, 1]_{i} of n copies of [0, 1] in D^{2} × [0, 1] such that the image of [0, 1]_{i} runs from (p_{i}, 0) to (p_{i}, 1) (1 ≤ i ≤ n), and whose singularities are transverse double points (in finite number).

Denote by ZSL(n) the free abelian group generated by SL(n). A singular nstring link σ with k double points can be expressed as an element of ZSL(n) using the following skein formula.

(2.1)  =  −

Let A be an abelian group. An n-string link invariant f : SL(n) → A is a finite type invariant of degree ≤ k if its linear extension to ZSL(n) vanishes on every n-string-link with (at least) k + 1 double points. If f is of degree ≤ k but not of degree k − 1, then f is called a finite type invariant of degree k.

We recall several classical examples of such invariants in the next two subsections.

2.1. Finite type knot invariants. Recall that the Conway polynomial of a knot K has the form

∇_{K}(z) = 1 + ∑_{k≥1} a_{2k}(K)z^{2k}.

It is not hard to show that the z^{2k}-coefficient a_{2k} in the Conway polynomial is a finite type invariant of degree 2k [1].

Recall also that the HOMFLYPT polynomial of a knot K is of the form

P (K; t, z) = ∑_{k=0}^{N} P_{2k}(K; t)z^{2k},

where P_{2k}(K; t) ∈ ℤ[t^{±1}] is called the 2kth coefficient polynomial of K. Denote by P_{2k}^{(l)}(K) the lth derivative of P_{2k}(K; t) evaluated at t = 1. It was proved by Kanenobu and Miyazawa that P_{2k}^{(l)} is a finite type invariant of degree 2k + l [9].

Note that both the Conway and HOMFLYPT polynomials of knots are invariant under orientation reversal, and that both are multiplicative under the connected sum [12].

2.2. Milnor invariants of string links. Given an n-component oriented, ordered link L in S^{3}, Milnor invariants \overline{μ}_{L}(I) of L are defined for each multi-index I = i_{1}i_{2}...i_{m} (i.e., any sequence of possibly repeating indices) among {1, ..., n} [18, 19]. The number m is called the length of Milnor invariant \overline{μ}(I), and is denoted by |I|. Unfortunately, the definition of these \overline{μ}(I) contains a rather intricate self-recurrent indeterminacy.

Habegger and Lin showed that Milnor invariants are actually well defined integervalued invariants of string links [5], and that the indeterminacy in Milnor invariants of a link is equivalent to the indeterminacy in regarding it as the closure of a string link.

FINITE TYPE INVARIANTS OF STRING LINKS AND THE HOMFLYPT POLYNOMIAL 3

In the unit disk D^{2}, we chose a point e ∈ ∂D and loops α_{1}, ..., α_{n} as illustrated in Figure 2.1. For an n–component string link L = K_{1} ∪ ⋯ ∪ K_{n} in D^{2} × [0, 1] with ∂K_{j} = {(p_{j}, 0), (p_{j}, 1)} (j = 1, ..., n), set Y = (D^{2} × [0, 1]) \ L, Y_{0} = (D^{2} × {0}) \ L, and Y_{1} = (D^{2} × {1}) \ L. We may assume that each π_{1}(Y_{t}) (t ∈ {0, 1}) with base point (e, t) is the free group F (n) on generators α_{1}, ..., α_{n}. We denote the image of α_{j} in the lower central series quotient F (n)/F (n)_{q} again by α_{j}. By Stallings’ theorem [23], the inclusions i_{t} : Y_{t} ⟶ Y induce isomorphisms (i_{t})_{∗} : π_{1}(Y_{t})/π_{1}(Y_{t})_{q} ⟶ π_{1}(Y)/π_{1}(Y)_{q} for any positive integer q. Hence the induced map (i_{1})_{∗}^{−1} ∘ (i_{0})_{∗} is an automorphism of F (n)/F (n)_{q} and sends each α_{j} to a conjugate l_{j}α_{j}l_{j}^{−1} of α_{j}, where l_{j} is the longitude of K_{j} defined as follows. Let γ_{j} be a zero framed parallel of K_{j} such that the endpoints (c_{j}, t) ∈ D^{2}×{t} (t = 0, 1) lie on the x–axis in ℝ^{2} ×{t}. The longitude l_{j} ∈ F (n)/F (n)_{q} is an element represented by the union of the arc γ_{j} and the segments e×[0, 1], c_{j}e×{0, 1} under (i_{1})_{∗}^{−1}. The coefficient μ_{L}(i_{1}i_{2}...i_{k−1}j) (k ≤ q) of X_{i_{1}} ⋯ X_{i_{k−1}} in the Magnus expansion E(l_{j}) is well-defined invariant of L, and it is called a Milnor μ–invariant of length k. It is known that Milnor μ–invariants of length k are finite type invariants of degree k − 1 for string links [2, 13].

Figure 2.1

Convention 2.1. As said above, each Milnor invariant μ(I) for n-string links is indexed by a sequence I of possibly repeating integers in {1, ..., n}. In the following, when denoting indices of Milnor invariants, we will always let distinct letters denote distinct integers, unless otherwise specified. For example, μ(iijk) (1 ≤ i, j, k ≤ n) stands for all Milnor invariants μ(iijk) with i, j, k ∈ {1, ..., n} pairwise distincts.

2.3. Closure invariants. Given an n-string link L and a sequence I = i_{1}i_{2} ⋯ i_{m} of m elements in \{1, ..., n, \overline{1}, ..., \overline{n}\}, we will construct in the next subsection an oriented knot K(L; I) in S^{3} as a closure of L with respect to I. Roughly speaking, we build the knot in S^{3} by connecting the endpoints of the i_{j}th components of L (j = 1, ..., m) so that, when running along the knot following the orientation, we meet these components in the order i_{1}, i_{2}, ⋯, i_{m}. Indices contained in {1, ..., n}, resp. in \{\overline{1}, ..., \overline{n}\}, correspond to components whose orientation agree, resp. disagree, with the orientation of the knot. If some index appears more than one in I, then we properly take parallels of the corresponding component of L.

2.3.1. Definition of the knot K(L; I) for a sequence I without repetition. Let I = i_{1}i_{2} ⋯ i_{m} be a sequence of m elements in \{1, ..., n, \overline{1}, ..., \overline{n}\} without repeated number, i.e., for each i = 1, ..., n, the number of times that i or \overline{i} appears in I is at most one. Let L = K_{1} ∪ ... ∪ K_{n} be an n-string link in D^{2} × [0, 1] ⊂ S^{3}.

Suppose that ∂K_{i} = p_{i} × {0, 1} ⊂ D^{2} × {0, 1}. For each I, we choose a tangle T_{I} in S^{3} \ (D^{2} × [0, 1]) as follows:

• If i_{k} and i_{k+1} are in {1, ..., n} then connect p_{i_{k}} × {1} and p_{i_{k+1}} × {0} in

S^{3} \ (D^{2} × [0, 1]).

• If i_{k} is in \{1, ..., n\} and i_{k+1} is in \{\overline{1}, ..., \overline{n}\} then connect p_{i_{k}} × \{1\} and

p\overline{i_{k+1}} × \{1\} in S^{3} \ (D^{2} × [0, 1]).

• If i_{k} and i_{k+1} are in \{\overline{1}, ..., \overline{n}\} then connect p\overline{i_{k}} × \{0\} and p\overline{i_{k+1}} × \{1\} in

S^{3} \ (D^{2} × [0, 1]).

• If i_{k} is in \{\overline{1}, ..., \overline{n}\} and i_{k+1} is in \{1, ..., n\} then connect p\overline{i_{k}} × \{0\} and

p_{i_{k+1}} × {0} in S^{3} \ (D^{2} × [0, 1]).

Here we implicitely mean that \overline{\overline{i}} = i and i_{m+1} = i_{1} in our notation. Let L_{I} be the m-string link obtained from L by deleting all components K_{j} of L such that neither j nor \overline{j} appears in I. Then we have a knot K(L; I) := L_{I} ∪ T_{I} in S^{3}. See Figure 2.2 for an example. For each I, we choose T_{I} so that K(1_{n}; I) is the trivial knot. While there are several choices of T_{I} tangles for each I, we choose one and fix it.

Figure 2.2

2.3.2. Definition of the knot K(L; I) for an arbitrary sequence I. Let L = K_{1} ∪ ⋯ ∪ K_{n} be an n-string link. Let I = i_{1}i_{2} ⋯ i_{m} be a sequence of m elements of \{1, ..., n, \overline{1}, ..., \overline{n}\}, where for each number i(= 1, ..., n), the number of times that i or \overline{i} appears in I is r_{i}. Let m = ∑_{i} r_{i}. Denote by D_{I}(L) the m-string link obtained from L as follows:

• Replace each string K_{i} by r_{i} zero-framed parallel copies of it, labeled from K_{(i,1)} to K_{(i,r_{i})} according to the natural order induced by the orientation of the diametral axis in D^{2}. If r_{i} = 0 for some index i, simply delete K_{i}.

• Let D_{I}(L) = K′_{1} ∪ ⋯ ∪ K′_{m} be the m-string link ⋃_{i,j} K_{(i,j)} with the order induced by the lexicographic order of the index (i, j). This ordering defines a bijection

φ : {(i, j) | 1 ≤ i ≤ n, 1 ≤ j ≤ r_{i}} → {1, ..., m}.

We also define a sequence D(I) of elements of {1, ..., m} without repeated number as follows. First, consider a sequence of elements of {(i, j); 1 ≤ i ≤ n, 1 ≤ j ≤ r_{i}} by replacing each number i in I with (i, 1), ..., (i, r_{i}) in this order. For example if I = 12\overline{2}31, we obtain the sequence (1, 1), (2, 1), \overline{(2, 2)}, (3, 1), (1, 2). Next replace each term (i, j) of this sequence with φ((i, j)). Hence we have D(12\overline{2}31) = 13\overline{4}52. Since D(I) does not contain repeated number, we have a closure K(D_{I}(L); D(I)) of L with respect to the sequence D(I). We call the knot K(D_{I}(L); D(I)) the closure knot with respect to I.

It is not hard to show the following proposition.

Proposition 2.2. Let I a sequence of elements in \{1, ..., n, \overline{1}, ..., \overline{n}\}, and let v_{m} be a finite type invariant of degree m for knots. Then the assignement L ↦ v_{m}(K(D_{I}(L); D(I))) defines a finite type invariant of degree m for n-string links.

Convention 2.3. Let v_{m} be a finite type invariant of degree m(≥ 2) for knots. For an n-string link L and a sequence I = i_{1}i_{2} ⋯ i_{m} of m elements of \{1, ..., n, \overline{1}, ..., \overline{n}\}, we denote v_{m}(K(D_{I}(L); D(I))) by v_{m}(L; I) or v_{m}(D_{I}(L); D(I)). For example, we denote P_{0}^{(m)}(K(D_{I}(L); D(I))) and a_{m}(K(D_{I}(L); D(I))) by P_{0}^{(m)}(L; I) and a_{m}(L; I) respectively. We call P_{0}^{(m)}(L; I) and a_{m}(L; I) the P_{0}^{(m)}-closure invariant and the a_{m}-closure invariant respectively.

3. C_{k}-moves and finite type invariants

3.1. The Goussarov-Habiro Conjecture. Goussarov and Habiro showed independently the following.

Theorem 3.1 ([4, 6]). Two knots (1-string links) cannot be distinguished by any finite type invariant of degree ≤ k if and only if they are C_{k}-equivalent.

It is known that the ‘if’ part of the statement holds for links as well, but explicit examples show that the ‘only if’ part of Theorem 3.1 does not hold for links in general, see [6, §7.2].

However, Theorem 3.1 may generalize to string links.

Conjecture (Goussarov-Habiro ; [4, 6]). Two string links of the same number of components share all finite type invariant of degree ≤ k − 1 if and only if they are C_{k}-equivalent.

As in the link case, the ‘if’ part of the conjecture is always true. The ‘only if’ part is also true for k = 1 (in which case the statement is vacuous) and k = 2; the only finite type string link invariant of degree 1 is the linking number, which is known to classify string links up to C_{2}-equivalence [21]. (Note that this actually also applies to links). The Goussarov-Habiro conjecture was (essentially) proved for k = 3 by the first author in [15]. Massuyeau gave a proof for k = 4, but it is mostly based on algebraic arguments and thus does not provide any information about the corresponding finite type invariants [14]. In [16], we classify n-string links up to C_{k}-move for k ≤ 5, by explicitly giving a complete set of low degree finite type invariants. In addition to Milnor invariants, these include several closure invariants of string links. In the next subsection, we give the statements of these results. As a consequence, we show that the Goussarov-Habiro Conjecture is true for k ≤ 5.

3.2. Invariants of degree ≤ 4. In this subsection, we give a C_{k}-classification of string links for k ≤ 5. While the statemants here look different from the statements in [16], they are essentially the same (we just use a different notation for closure invariants).

Recall that there is essentially only one finite type knot invariant of degree 2, namely a_{2}, and that there is essentially only one finite type knot invariant of degree 3, namely P_{0}^{(3)}. There are essentially two linearly independent finite type knot invariants of degree 4, namely a_{4} and P_{0}^{(4)}. We will use these knot invariants to define a number of finite type string links invariants of degree ≤ 4 by using some closure. These various invariants, together with Milnor invariants of length ≤ 5, give the following classification of n-string links up to C_{k}-equivalence for k ≤ 5.

Theorem 3.2 ([15]). Let L, L′ ∈ SL(n). Then the following assertions are mutually equivalent:

(1) L and L′ are C_{3}-equivalent,

(2) L and L′ share all finite type invariants of degree ≤ 2,

(3) a_{2}(L; i) = a_{2}(L′; i) (1 ≤ i ≤ n), a_{2}(L; i\overline{j}) = a_{2}(L′; i\overline{j}) (1 ≤ i < j ≤ n), μ_{L}(ij) = μ_{L′}(ij) (1 ≤ i < j ≤ n) and μ_{L}(ijk) = μ_{L′}(ijk) (1 ≤ i < j < k ≤ n).

Theorem 3.3 ([16]). Let L, L′ ∈ SL(n). Then the following assertions are mutually equivalent:

(1) L and L′ are C_{4}-equivalent,

(2) L and L′ share all finite type invariants of degree ≤ 3, (3) L and L′ share all finite type invariants of degree ≤ 2, and

P_{0}^{(3)}(L; i) = P_{0}^{(3)}(L′; i) (1 ≤ i ≤ n), P_{0}^{(3)}(L; i\overline{j}) = P_{0}^{(3)}(L′; i\overline{j}) (1 ≤ i < j ≤ n) P_{0}^{(3)}(L; ik\overline{j}) = P_{0}^{(3)}(L′; ik\overline{j}) (1 ≤ i < j < k ≤ n), μ_{L}(iijj) = μ_{L′}(iijj) (1 ≤ i < j ≤ n), μ_{L}(ijkl) = μ_{L′}(ijkl) (1 ≤ i, j < k < l ≤ n) and μ_{L}(ijkk) = μ_{L′}(ijkk) (1 ≤ i, j, k ≤ n, i < j).

Theorem 3.4 ([16]). Let L, L′ ∈ SL(n). Then the following assertions are equivalent:

(1) L and L′ are C_{5}-equivalent,

(2) L and L′ share all finite type invariants of degree ≤ 4,

(3) L and L′ share all finite type invariants of degree ≤ 3, and a_{4}(L; i) = a_{4}(L′; i), P_{0}^{(4)}(L; i) = P_{0}^{(4)}(L′; i) (1 ≤ i ≤ n), a_{4}(L; i\overline{j}) = a_{4}(L′; i\overline{j}), P_{0}^{(4)}(L; i\overline{j}) = P_{0}^{(4)}(L′; i\overline{j}), a_{4}(L; ii\overline{j}) = a_{4}(L′; ii\overline{j}), P_{0}^{(4)}(L; ii\overline{j}) = P_{0}^{(4)}(L′; ii\overline{j}), P_{0}^{(4)}(K(L; i\overline{j}\overline{j})) = P_{0}^{(4)}(K(L′; i\overline{j}\overline{j})) (1 ≤ i < j ≤ n), a_{4}(L; i\overline{j}\overline{k}) = a_{4}(L′; i\overline{j}\overline{k}), P_{0}^{(4)}(L; i\overline{j}\overline{k}) = P_{0}^{(4)}(L′; i\overline{j}\overline{k}), a_{4}(L; i\overline{k}j) = a_{4}(L′; i\overline{k}j), P_{0}^{(4)}(L; i\overline{k}j) = P_{0}^{(4)}(L′; i\overline{k}j), a_{4}(L; ik\overline{j}) = a_{4}(L′; ik\overline{j}), P_{0}^{(4)}(L; ik\overline{j}) = P_{0}^{(4)}(L′; ik\overline{j}), P_{0}^{(4)}(L; i\overline{j}k) = P_{0}^{(4)}(L′; i\overline{j}k) (1 ≤ i < j < k ≤ n), P_{0}^{(4)}(L; i\overline{j}k\overline{l}) = P_{0}^{(4)}(L′; i\overline{j}k\overline{l}), P_{0}^{(4)}(L; i\overline{j}l\overline{k}) = P_{0}^{(4)}(L′; i\overline{j}l\overline{k}), P_{0}^{(4)}(L; i\overline{k}j\overline{l}) = P_{0}^{(4)}(L′; i\overline{k}j\overline{l}) (1 ≤ i < j < k < l ≤ n), μ_{L}(ijklm) = μ_{L′}(ijklm) (1 ≤ i, j, k < l < m ≤ n), μ_{L}(iiijk) = μ_{L′}(iiijk), μ_{L}(ijjkk) = μ_{L′}(ijjkk), μ_{L}(jikll) = μ_{L′}(jikll) (1 ≤ i, j, k, l ≤ n, j < k).

Remark 3.5. A complete set of finite type link invariant of degree ≤ 3 has been computed in [10] using weight systems and chord diagrams. For 2-component links, this has been done for degree ≤ 4 invariants in [11]. All invariants are given by coefficients of the Conway and HOMFLYPT polynomials of sublinks.

4. Milnor invariants and P_{0}^{(m)}-closure invariants

We start by expressing Milnor’s link homotopy invariants, i.e., Milnor invariants μ(I) with I a sequence without repeated number, in terms of the closure invariants defined in Subsection 2.3.

Theorem 4.1 ([17]). Let m ≥ 2. Let L be a C_{m}-trivial n-string link (m + 1 ≤ n). Let I be a sequence of m + 1 elements of {1, ..., n} without repeated number. Then

μ_{L}(I) = \frac{±1}{m!2^{m}} ∑_{J⊂I,J≠∅}(−1)^{m−|J|}P_{0}^{(m)}(L; J),

where the sum runs over all nonempty subsequences J of I.

Remark 4.2. (1) By [6], the fact that L is C_{m}-trivial implies that μ_{L}(I) = 0 for any sequence I of length |I| ≤ m.

(2) Any link-homotopically trivial Brunnian n-string link is C_{n}-trivial [8, 20], and any Brunnian n-string link whose Milnor invariants of length ≤ n+1 vanish is C_{n+1}- trivial [16]. Since a Brunnian n-string link whose Milnor invariants with length ≤ n vanish is link-homotopically trivial [18], for m = n + 1 or n, a Brunnian n-string link whose Milnor invariants with length ≤ m vanish is C_{m}-trivial. Moreover, any Brunnian n-string link is C_{n−1}-trivial [7, 20] and has vanishing Milnor invariants with length ≤ n − 1, so this holds for m = n − 1 as well.

(3) Since there exists no degree one invariant of knots, such a formula does not hold for the linking number, hence the assumption m ≥ 2. In order to give such a formula one should consider ‘closure links’, that is more general closure operations on string links that can produce links of several components.

By combining [19, Thm. 7] and Theorem 4.1, we have the following theorem.

Theorem 4.3 ([17]). Let m ≥ 2. Let L be a C_{m}-trivial n-string link. Let I be a sequence of m + 1 elements of {1, ..., n}. Then

μ_{L}(I) = μ_{D_{I}(L)}(D(I)) = \frac{±1}{m!2^{m}} ∑_{J⊂D(I),J≠∅}(−1)^{m−|J|}P_{0}^{(m)}(D_{I}(L); J),

where the sum runs over all nonempty subsequences J of D(I).

K. Habiro has pointed out the following remark.

Remark 4.4. It is not hard to see that the 6-string link L illustrated in Figure 4.1 is C_{5}-trivial and satisfies μ_{L}(123456) = ±1. By Theorem 4.1, μ_{L}(123456) can be expressed as a linear combination of P_{0}^{(5)}-closure invariants of L. (By applying the theorem, we have μ_{L}(123456) = (±1/5!2^{5})P_{0}^{(5)}(L; 123456).) In contrast, since a_{5} of knots always vanish, it is impossible to express μ_{L}(123456) by any linear combination of a_{5}-closure invariants of L. Moreover we notice that L is equivalent to 1_{6} up to doubled-delta move, which is a local move on links defined by Naik and Stanford [22]. Hence any closure knot, and more generally any closure link (see Remark 4.2(3)) obtained from L is equivalent to a trivial knot or link up to doubleddelta moves. Since the doubled-delta move preserves the Alexander invariant, the Conway polynomial of any closure link obtained from L vanishes.

Figure 4.1

5. Claspers and P_{0}^{(m)}-closure invariants

5.1. Claspers. For a general definition of claspers, we refer the reader to [6]. Let L be a (string) link. A surface G embedded in D^{2} × [0, 1) is called a graph clasper for L if it satisfies the following three conditions:

(1) G is decomposed into disks and bands, called edges, each of which connects two distinct disks.

(2) The disks have either 1 or 3 incident edges, and are called leaves or nodes respectively.

(3) G intersects L transversely, and the intersections are contained in the union

of the interiors of the leaves. In particular, if a connected graph clasper G is simply connected, we call it a tree clasper.

A graph clasper for a (string) link L is simple if each of its leaves intersects L at one point. The degree of a connected graph clasper G is defined as half of the number of nodes and leaves. We call a degree k connected graph clasper a C_{k}-graph. A tree clasper of degree k is called a C_{k}-tree.

Given a graph clasper G for a (string) link L, there is a procedure to construct a framed link, in a regular neighbourhood of G. There is thus a notion of surgery along G, which is defined as surgery along the corresponding framed link. In particular, surgery along a simple C_{k}-tree is a local move as illustrated in Figure 5.1, which is equivalent to a C_{k}-move as defined in Section 1 (Figure 1.1).

Figure 5.1. Surgery along a simple C_{5}-tree.

The C_{k}-equivalence (as defined in Section 1) coincides with the equivalence relation on string links generated by surgeries along C_{k}-graphs and isotopies. In particular, it is known that two links are C_{k}-equivalent if and only if they are related by surgery along simple C_{k}-trees [6, Thm. 3.17].

For k ≥ 3, a C_{k}-tree G having the shape of the tree clasper in Figure 5.1 is called a linear C_{k}-tree. The left-most and right-most leaves of G in Figure 5.1 are called the ends of G, and the remaining (k − 1) leaves are called the internal leaves of G.

Suppose that the two ends of a linear C_{k}-tree are denoted by f and f′. Let S be a nonempty subset of the set of all internal leaves of T. We have a labeling from 1 to |S| of the leaves in S by travelling along the boundary of the disk^{1} T from f to f′ so that all leaves are visited. We call this labeling the linear labeling of S, from f to f′.

5.2. Generators of SL_{m}(n)/C_{m+1}. Let m ≥ 3 be an integer. In this section we find generators for the abelian group SL_{m}(n)/C_{m+1} and show that for each of these generators, there is a P_{0}^{(m)}-closure invariant which detects it.

For a simple tree clasper Γ for a string link, let r_{i}(Γ) denote the number of leaves intersecting the ith component of the string link.

Let L ∈ SL_{m}(n) be a C_{m}-trivial n-string link. By Calculus of Claspers [16, Lem.3.2] and the AS and IHX relations [16, Lem.3.3], L is C_{m+1}-equivalent to a product ∏ T_{i} of n-string links T_{1}, ..., T_{l}, where each T_{k} is obtained from 1_{n} by surgery along a simple linear C_{m}-tree Γ_{k}. Actually, by the IHX relation we may assume that each Γ_{k} satisfies one of the following;

(1) all leaves of Γ_{k} intersect a single component of 1_{n},

(2) |{i | r_{i}(Γ_{k}) = 1}| ≥ 2, and the ends intersect the pth and qth components of 1_{n}, where p = min{i | r_{i}(Γ_{k}) = 1} and q = min{i | r_{i}(Γ_{k}) = 1, i ≠ p},

(3) r_{i}(Γ_{k}) = 2 for some i, |{i | r_{i}(Γ_{k}) = 1}| < 2, and the ends intersect the pth component of 1_{n}, where p = min{i | r_{i}(Γ_{k}) = 2},

(4) Γ_{k} is not of type (1), r_{i}(Γ_{k}) ≠ 2 for any i, |{i | r_{i}(Γ_{k}) = 1}| < 2, and the ends intersect the pth component of 1_{n}, where (r_{p}(Γ_{k}), p) is the minimum among {(r_{i}(Γ_{k}), i) | i = 1, ..., n, r_{i}(Γ_{k}) ≥ 3} with respect to the lexicographic order.

This implies that SL_{m}(n)/C_{m+1} is generated by all string links obtained from 1_{n} by surgery along a C_{m}-tree of one of the 4 types above.

1Recall that a clasper is an embedded surface: in particular, since T is a tree clasper, the underlying surface is isotopic to a disk.

Let us reduce the number of generators of type (4). Let T_{p} be the set of linear C_{m}-trees of type (4) with ends intersecting the pth component of 1_{n}. Each tree in T_{n} has a unique leaf not intersecting the nth component of 1_{n}. By [16, Lem.3.6], the case reduces to trees of type (3). Hence we may assume that p ≠ n. By the IHX relation, we may assume that the two ends are the ‘top’, resp. ‘bottom’, leaves on the pth component of 1_{n}, which are defined as the last, resp. first, leaf we meet while traveling along this component from the initial point to the terminal point. For a C_{m}-tree Γ ∈ T_{p} with top end f and bottom end f′, we consider the linear labeling (from 1 to m − 1) of the set of all internal leaves of Γ, from f′ to f (see Section 5.1). Suppose that while traveling along the pth component from f′ to f, we meet s leaves labeled by i_{1}, ..., i_{s} ∈ {1, ..., m − 1} in this order. We say that Γ is flat (on the pth component of 1_{n}) if i_{1} < i_{2} < ⋯ < i_{s}. Let F_{p} be the set of flat trees in T_{p}.

Define F_{p}^{0} as set of C_{k}-trees in F_{p} which do not contain a fork. Here we say that a tree clasper T for 1_{n} contains a fork if there exists a 3-ball that intersects 1_{n} ∪ T as represented in Figure 5.2

Figure 5.2

Proposition 5.1 ([17]). For an integer m ≥ 3, SL_{m}(n)/C_{m+1} is generated by string links obtained from 1_{n} by surgery along linear trees of type (1), (2), (3) or in F_{p}^{0} (p = 1, ..., n − 1).

The abelian group SL_{m}(n)/C_{m+1} can be decomposed into a direct sum G_{1} ⊕G_{2}, where G_{1} (resp. G_{2}) is the subgroup generated by string links obtained from 1_{n} surgery along a linear C_{m}-tree of type (1) (resp. of type (2), (3) or in F_{p}^{0} (p = 1, ..., n − 1)). By the Goussarov-Habiro Theorem [4, 6], G_{1} is classified by finite type invariants. For the group G_{2}, we have the following

Theorem 5.2 ([17]). Let m ≥ 3 be an integer. For any simple linear C_{m}-tree Γ for 1_{n} of type (2), (3) or in F_{p}^{0} (p = 1, ..., n − 1), there is a sequence I of elements of {1, ..., n} such that P_{0}^{(m)}((1_{n})_{Γ}; I) = ±m!2^{m}. Hence (1_{n})_{Γ} has infinite order in SL_{m}(n)/C_{m+1}.

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Institut Fourier, Université Grenoble 1, 100 rue des Maths - BP 74, 38402 St Martin d’Hères , France

   E-mail address: jean-baptiste.meilhan@ujf-grenoble.fr

Tokyo Gakugei University, Department of Mathematics, Koganeishi, Tokyo 184-8501, Japan

   E-mail address: yasuhara@u-gakugei.ac.jp