NON-TRIVIAL LINKS AND PLATS

WITH TRIVIAL GASSNER MATRICES

Tim D. Cochran^{∗!}

Abstract. We define link-theoretic and plat-theoretic versions of the Gassner representation and show they are not injective for links of at least 6 components. In doing so we show how to construct non-trivial links satisfying all of Joan Birman’s conditions [Bi; p. 130] for a link which could be the closure of a braid in the kernel of the Gassner representation. We define a Gassner matrix for 2n-plats in such a way to extend the Gassner matrix for pure n-braids. We observe that the classical Gassner representation for pure braids has a kernel if and only if there is an element in the kernel which is homotopically essential.

§0.

Let B_{n} denote the Artin braid group on “n-strings” and PB_{n} its normal subgroup consisting of all the pure braids [Bi, Mo]. These groups have been considerably scrutinized by both topologists and algebraists [BL]. One question whose answer has so far eluded us is whether or not the Gassner representation G : PB_{n} ⟶ M_{n×n}(Λ), into the group of n-by-n matrices over Λ = Z[t_{1}^{±1}, …, t_{n}^{±1}], is faithful (see §1) [Bi; §3.3] [Ga]. Recently the less discriminating Burau representation B : PB_{n} ⟶ M_{n×n}(Z[t^{±1}]) was shown to have a non-trivial kernel for each n ≥ 6 [M, LP] but these techniques have not yet yielded an element of kernel(G). This paper is a partial step in that direction.

The connection between pure braids and links is strong. Thus Joan Birman in 1973 [Bi; pp. 130] states that if there exists a non-trivial element β ∈ PB_{n} such that G(β) = I, then the n-component link β̂ obtained by “closing” the braid would satisfy the following:

i) β̂ has at least 4 components, each of which is unknotted and which are pairwise unlinked.

ii) π_{1}(S^{3} − β̂) can be generated by n meridional elements.

iii) a certain specific Alexander matrix for β̂ is a matrix of zeros.

In particular the Alexander module of β̂ would be trivial, that is isomorphic to that of the trivial n-component link. Thus one can search for links which satisfy these criteria. Birman goes on to remark that “the only known class of links with zero Alexander polynomial, the so-called homology boundary links, cannot satisfy ii) unless the link is trivial (since

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*partially supported by the National Science Foundation

!This work was first presented at the regional meeting of the American Mathematical Society in Denton, Texas, October 1990

an n-component link is a homology boundary link if and only if its group G maps onto F_{n}, the free group of rank n, so that if G were already a quotient of F_{n}, it would follow that G ≅ F_{n}). In 1981 Jonathan Hillman exhibited ribbon links with trivial Alexander’s module [H1] (see also [La]). These also cannot satisfy ii) (see §4). In 1986, the author discovered new classes of links which have trivial Alexander’s modules and in this paper we show that these actually satisfy all of Birman’s conditions [C1, C2]. If any of these links is a “pure link”, that is if any of these n-component links is actually isotopic to β̂ for an n-string braid β, then the Gassner representation is far from faithful. We note that condition ii) above is the only (pre-Vaughn Jones) general condition on a link to be a pure link [Bi; p. 96, Appendix #11 p. 217] and so the links presented herein satisfy it. However one need only consider the difference between the class of 2-bridge links and the class of pure 2-braids to see that ii) is still far from characterizing a pure link. In fact preliminary calculations using the HOMFLY polynomial (truncated), done for me by Jim Hoste, have revealed that the examples of Figure 3.3 are not “pure links”.

In the present paper we formalize these results in the following way. Viewing ii) as essentially characterizing the class of n-component n-bridge links, we put a group structure (loosely speaking) on the set of n-component n-bridge links by considering the group of 2n plats P_{2n} in such a way that PB_{n} is a subgroup of P_{2n} [Bi; p. 192]. Then we extend the classical Gassner representation to a Gassner function G : P_{2n} ⟶ M_{n×n}(Λ). One then sees that a link satisfying Birman’s conditions will yield an element ρ ∈ P_{2n} such that G(ρ) = I. We then describe how to construct such examples for n ≥ 6.

In Section 4 we rephrase the question of the faithfulness of the Gassner representation in terms of Milnor’s μ̄-invariants. This allows us to see that our examples are, in a sense, the simplest possible examples. We close with several interesting open questions from this point of view.

§1. A Gassner Matrix for Plats.

Let B_{n} denote the Artin braid group on “n strings” and PB_{n} its normal subgroup consisting of all braids whose corresponding permutation is the identity [Bi, Mo]. Recall that the braid group has a faithful representation as a group of right automorphisms of a free group F = F ⟨x_{1}, …, x_{n}⟩ where if σ̂ denotes the automorphism corresponding to σ ∈ PB_{n} then σ̂(x_{i}) = λ_{i}x_{i}λ_{i}^{−1} for some λ_{i} ∈ F for i = 1, …, n [Bi; Corollary 1.83]. We may assume that the exponent sum of x_{i} in λ_{i} is zero. Let Λ denote Z[t_{1}^{±1}, …, t_{n}^{±1}], that is, the integral group ring of the abelianization of F. Recall the Gassner representation G : PB_{n} ⟶ M_{n×n}(Λ) which is given by G(σ) = π (\frac{∂σ̂(x_{i})}{∂x_{j}}) where ∂/∂x_{j} is Fox’s free derivative and π is induced by the homomorphism π : ZF ⟶ Λ with π(x_{i}) = t_{i} [Bi; 3.1–3.2]. For the reader’s convenience, we recall that the Fox derivative D_{i} = ∂/∂x_{i} is determined by the rules: D_{i}(zw) = D_{i}(z) + zD_{i}(w), D_{i}(1) = 0, D_{i}(x_{j}) = δ_{ij} and D_{i}(x_{i}^{−1}) = −x_{i}^{−1}.

Figure 1

We define a pure 2n-plat ρ to be merely a pure 2n-braid and denote the group of such as P_{2n}. Thus the group P_{2n} is “identical” to PB_{2n} and there is the standard embedding PB_{n} ↪^{i} P_{2n} = PB_{2n}. However a plat differs from a braid in the definition of its closure. The closure L(β) of a pure n-braid β can be defined to be the oriented, ordered n-component link obtained by closing the braid in the standard fashion and considering all strings to be oriented upwards. The closure L(ρ) of a pure 2n-braid ρ we shall define as the oriented, ordered n-component link obtained by closing the 2n-braid as shown in Figure 1, and considering the orientation induced by allowing the first n strands of the 2n-braid to go “up” and the second n strands to go “down”. The reader must be warned that the “usual” definition of closing a plat is as in Figure 2 [Bi]. Our definition is more convenient for our purposes but is by no means substantively different. The following are trivial consequences of the definitions.

Figure 2

Proposition 1.1. If β is a pure n-braid then L(β) is isotopic to L(i(β)) as oriented links.

Proposition 1.2. An n-component oriented link L is isotopic to the closure of a pure plat ρ ∈ P_{2n} if and only if L is an n-component n-bridge link.

Thus we would like to think of P_{2n} as one way of imposing a group structure on the set of n-component n-bridge links in such a way that P B_{n} is a natural subgroup.

We shall now extend the notion of Gassner Matrix to elements of P_{2n} in such a way as to retain the strong relationship between G(ρ) and an Alexander matrix for L(ρ) (see §2). Given ρ ∈ P_{2n}, consider ρ as a 2n-braid to which is associated an automorphism ρ̂(x_{i}) = λ_{i}x_{i}λ_{i}^{−1} i = 1, …, 2n. Recall that ρ̂ determines the set {λ_{i}x_{i}^{n} | n ∈ Z} but not λ_{i} itself. To clarify the definition of λ_{i}, we make the following conventions. Choose λ_{i} such that the exponent sum of x_{i} in λ_{i} equals the exponent sum of x_{i} in λ_{2n−i+1} (the latter exponent is determined by ρ̂), and the exponent sum of x_{2n−i+1} in λ_{2n−i+1} is the exponent sum of x_{2n−i+1} in λ_{i}. This ensures that, in λ_{2n−i+1}^{−1}λ_{i}, the exponent sums of x_{i} and x_{2n−i+1} are zero. Let ϕ : F ⟨x_{1}, …, x_{2n}⟩ ⟶ F ⟨x_{1}, …, x_{n}⟩ be given by ϕ(x_{j}) = x_{j} and ϕ(x_{2n−j+1}) = x_{j}^{−1} for 1 ≤ j ≤ n.

Definition 1.3: The Gassner Matrix G(ρ) of the pure 2n-plat ρ is

G_{ij} = π (\frac{∂ϕ(λ_{2n−i+1}^{−1}λ_{i}x_{i}λ_{i}^{−1}λ_{2n−i+1})}{∂x_{j}})

for 1 ≤ i, j ≤ n.

Proposition 1.4. The function G : P_{2n} ⟶ M_{n×n}(Λ) extends the Gassner representation G : P B_{n} ⟶ M_{n×n}(Λ).

Proof of 1.4: If β ∈ P B_{n} with β̂(x_{i}) = α_{i}x_{i}α_{i}^{−1} and i(β) = ρ, then clearly ρ̂(x_{i}) = x_{i} for i > n, and λ_{i} for i ≤ n is merely α_{i}. Thus, with reference to 1.3, λ_{2n−i+1} = e and ϕ(λ_{i}) = α_{i}, implying the result. □

Unfortunately G does not appear to be a homomorphism and several attempts at modifying it to be such have failed.

It is not reasonable to expect that G is injective. In fact if β is any n-braid and ρ is the 2n-plat of Figure 3, then G(ρ) = I. Note that the closure of ρ is a trivial link. The appro-

Figure 3

priate question is “Is there a non-trivial n-component n-bridge link L such that for some 2n-plat representation ρ, G(ρ) = I?” This link-theoretic version of the question of the injectivity of the Gassner representation was essentially formulated by Birman [Bi; page 130]. In §2 and §3 we will see that there are many such links, answering Birman’s question in the negative. The simplest of such can be shown to not be the closures of n-braids so we have not, as of this writing, been able to decide whether or not the classical Gassner representation is faithful.

§2. Relationship of Gassner matrix to the Alexander matrix.

It is well known that the Gassner matrix of a pure braid β is equal to the sum of the identity matrix and a certain matrix presenting the Alexander module A(L(β)) of the closure of β [Bi 3.10, Mo 13.7]. We shall extend these results to our more general situation and use these in §3 to establish the existence of non-trivial n-component n-bridge links and pure 2n-plats with trivial Gassner matrix.

Proposition 2.1. Let ρ ∈ P_{2n} and suppose that the action of ρ on the free group F ⟨x_{1}, …, x_{2n}⟩ is given by ρ̂(x_{i}) = λ_{i}x_{i}λ_{i}^{−1} 1 ≤ i ≤ 2n. Then the fundamental group π_{1}(S^{3} − L(ρ)) admits the presentation:

⟨x_{1}, …, x_{n} | ϕ (λ_{2n−i+1}^{−1}λ_{i}x_{i}λ_{i}^{−1}λ_{2n−i+1}x_{2n−i+1}) , 1 ≤ i ≤ n⟩

where ϕ is as in §1.

Proof of 2.1: Decompose S^{3} via a 2-sphere which slices through the level \frac{1}{2} of the braid. Let A, B denote the resulting 3-balls. Clearly π_{1}(A − L(β)) has a presentation ⟨x_{i} 1 ≤ i ≤ 2n | x_{i} = x_{2n−i+1}^{−1} 1 ≤ i ≤ n⟩ and π_{1}(B − L(ρ)) has a presentation ⟨z_{i} 1 ≤ i ≤ 2n | z_{i} = z_{2n−i+1}^{−1} 1 ≤ i ≤ n⟩. An application of the Seifert Van-Kampen theorem yields the presentation ⟨x_{i}, z_{i} 1 ≤ i ≤ 2n | x_{i} = x_{2n−i+1}^{−1}, z_{i} = z_{2n−i+1}^{−1}, z_{i} = ρ̂(x_{i})⟩ for π_{1}(S^{3} − L(ρ)). Eliminating z_{i} yields the relations {x_{i} = x_{2n−i+1}^{−1}, λ_{i}x_{i}λ_{i}^{−1} = λ_{2n−i+1}x_{2n−i+1}^{−1}λ_{2n−i+1}^{−1} | 1 ≤ i ≤ n}. Eliminating {x_{2n−i+1} | 1 ≤ i ≤ n} yields the desired presentation. □

Recall that the Alexander matrix associated to the presentation ⟨x_{i} | r_{i}⟩ is defined by A_{ij} = π (\frac{∂r_{i}}{∂x_{j}} ) [F; p. 100] and, in the case that ⟨x_{i} | r_{i}⟩ ≅ π_{1}(S^{3} − L(ρ)), presents the Alexander’s module A(ρ) of L(ρ) [Cr; §3].

Proposition 2.2. If ρ ∈ P_{2n} then G(ρ) − I is the Alexander matrix of L(ρ) associated to the presentation of 2.1.

Proof: Note that r_{i} = γ_{i}x_{i}^{−1} where γ_{i} = ϕ (λ_{2n−i+1}^{−1}λ_{i}x_{i}λ_{i}^{−1}λ_{2n−i+1}). Then \frac{∂r_{i}}{∂x_{j}} =

\frac{∂γ_{i}}{∂x_{j}} + γ_{i} \frac{∂x_{i}^{−1}}{∂x_{j}} = \frac{∂γ_{i}}{∂x_{j}} − γ_{i}x_{i}^{−1}δ_{ij} where δ_{ij} = 1 only when i = j. Since π(γ_{i}) = 1 · e = 1 in

Λ, A_{ij} = π (\frac{∂γ_{i}}{∂x_{j}} ) − δ_{ij} = G_{ij} − δ_{ij}, as desired. □

Since the triviality of the Gassner matrix is now seen to entail the Alexander matrix of a specific presentation being the zero matrix, we investigate when the latter may occur. First we observe that this is only a property of the link, not of the particular n-bridge presentation.

Proposition 2.3. The Alexander matrix for ⟨x_{1}, …, x_{n} | r_{1}, …, r_{n}⟩ is a zero matrix if and only if any n × n Alexander matrix presenting the Alexander module of the associated group is a zero matrix.

Proof of 2.3: Let π ≅ ⟨x_{1}, …, x_{n} | r_{1}, …, r_{n}⟩. By definition there is a short exact se-

quence 0 ⟶ image(Â) ↪^{i} Λ^{n} ⟶^{f} A ⟶ 0 where Â: Λ^{n} ⟶ Λ^{n} is given by multiplication by the Alexander matrix and A is the Alexander module [Tr; §3]. If for one presentation Â = 0 then clearly A ≅ Λ^{n}. But the isomorphism class of the Alexander module is an invariant of the isomorphism class of the group π so for any presentation of the group, A ≅ Λ^{n}. But f is then an epimorphism between finitely-generated free modules of identical rank over a Noetherian ring, hence an isomorphism. Thus image (Â) = 0 and A_{ij} is always a matrix of zeros. □.

Proposition 2.4. Suppose the exponent sum of x_{i} in w_{i} is zero for all 1 ≤ i ≤ n. Then the Alexander matrix of ⟨x_{i} | [w_{i}, x_{i}], 1 ≤ i ≤ n⟩ is zero if and only if w_{i} ∈ F′′ (the second derived) for each i.

Proof of 2.4: The proof of Theorem 3.16 of [Bi] suffices. □

The following extends [Bi; Thm. 3.16] and [CL; Thm. 3.10] to our present situation.

Theorem 2.5. Suppose L is an n-component link which admits an n-bridge presentation or suppose that L = L(ρ) is the closure of a pure 2n-plat ρ. Then the Gassner matrix of L (equivalently of ρ) is the identity if and only if the longitudes of L lie in the second derived subgroup of π_{1}(S^{3} − L) (equivalently if and only if w_{i} = ϕ(λ_{2n−i+1}^{−1}λ_{i}) lies in F′′ for all 1 ≤ i ≤ n).

Proof of 2.5: Suppose the Gassner matrix of ρ is the identity. Then by 2.2 the associated Alexander matrix for L(ρ) is the zero matrix. By [C2, Thm. 3.1], π_{1}(S^{3} − L(ρ)) ≡ π has a presentation of the form ⟨x_{i} | [v_{i}, x_{i}]⟩ where v_{i} represents an i^{th} longitude l_{i} of the link. By 2.3 the Alexander matrix associated to this presentation is the zero matrix and by 2.4 it follows that v_{i} ∈ F′′ for each i (since it is well known that if l_{i} ∉ F′ then some linking numbers are non-zero and the Alexander matrix cannot be zero [Bi; Cor. 3.14.1].) It follows immediately that l_{i} ∈ π′′. On the other hand, by 2.1, π has a presentation ⟨x_{i} | [w_{i}, x_{i}]⟩ where w_{i} = ϕ(λ_{2n−i+1}^{−1}λ_{i}) so by 2.4 w_{i} ∈ F′′. For the other implication, suppose that the longitudes of l_{i} lie in π′′. It follows that F/F′′ ≊ π/π′′ and hence that the v_{i} ∈ F′′ (see presentation above) [C2, see proof of 3.2 from 3.1]. Apply 2.4 to see that the Alexander matrix associated to ⟨x_{i} | [v_{i}, x_{i}]⟩ is the zero matrix then apply 2.3 and 2.2 to yield the desired “triviality” of the Gassner matrix. □

§3. n-component n-bridge links with trivial Gassner matrix.

We review a construction of [C2, §1].

Figure 4

Let A be the set of letters {x_{i} | i = 1, …, n}. A 1-bracket is an element of A. An mbracket is a symbol (β_{1}, β_{2}) where β_{1} is a k-bracket k ≥ 1 and β_{2} is an m − k bracket (m − k ≥ 1). We say that the weight of an m-bracket is m. To each pair of brackets {β_{1}, β_{2}} we shall associate an unoriented link with n_{1} + n_{2} components (where w(β_{1}) = weight (β_{1}) = n_{1} and w(β_{2}) = n_{2}). Begin with the Hopf link with the components labelled by β_{1}, β_{2} respectively. If w(β_{1}) ≠ 1 then β_{1} = (β_{11}, β_{12}) and we describe a procedure for replacing that component by two components labelled by β_{11} and β_{12} respectively. Continuing this procedure (applied to β_{2} also) until all labels have weight 1 yields a link with w(β_{1})+w(β_{2}) components. The procedure is shown in Figure 4 and is usually called “Bing-Doubling”. There is a choice of clasp which is not important to our present considerations. Any such link L(β) = L({β_{1}, β_{2}}) is clearly an n-component n-bridge link where n = w(β_{1})+w(β_{2}).

Define a generalized simple bracket, GSB, to be any bracket of the form (x_{i}, α) or (α, x_{i}) where α is a GSB of lesser weight, and a GSB of weight 1 is merely any 1-bracket. Thus all m-brackets are GSB for m = 1, 2, 3 but ((x_{1}, x_{2}), (x_{3}, x_{4})) is not a GSB. In [C2] the following criterion was derived for when the longitudes of L({β_{1}, β_{2}}) lie in G′′.

Theorem 3.2. (2.2 of [C2]) Suppose (β_{1}, β_{2}) contains no repeated letters, w(β_{i}) > 1 for i = 1, 2, and at least one of β_{1}, β_{2} is not a generalized simple bracket. Then the ncomponent n-bridge link L({β_{1}, β_{2}}) described above has the property that its longitudes lie in π′′.

Examples: The examples of minimal weight are several 6-component links corresponding to β_{1} = (x_{1}, x_{2}) β_{2} = ((x_{3}, x_{4}), (x_{5}, x_{6})) which are isotopic to those obtained from the Borromean Rings by “Bing-Doubling” each component and assigning orientations arbitrarily. Several distinct ordered, oriented links arise in such a way. One is shown in Figure 5.

More generally let β_{1} = (x_{1}, x_{2}) and β_{2} = ((x_{3}, x_{4}), σ) where σ is any bracket of weight (n − 4), n ≥ 6, whose underlying sequence is x_{5}x_{6} … x_{n} or any permutation of that sequence. This link is shown in Figure 6.

All of the links which arise as in Theorem 3.2 are non-trivial, in fact none is link homotopic to a trivial link because the Bing-Double of any homotopically essential link is homotopically essential [C1; Corollary 8.2]. This reference also shows the equivalent fact that Milnor’s invariant \overline{μ}(i_{1}i_{2} … i_{n}), for some permutation of \{1, …, n\}, will be ±1 for the n-component links of Theorem 3.2. In summary we have shown:

Theorem 3.5. Suppose ρ is a pure 2n-plat arising from the Brunnian n-component nbridge link L({β_{1}, β_{2}}) as in Theorem 3.2. Then ρ and L({β_{1}, β_{2}}) are non-trivial (even homotopically essential) but have trivial Gassner matrices. If any such L({β_{1}, β_{2}}) were the closure of a pure n-braid then the Gassner representation on pure n-braids would have a non-trivial kernel. In particular each of these links satisfies all the conditions i)–v) of [Bi, p. 130].

§4. Relations with Milnor’s μ̄-invariants and Massey products.

In this section we translate the question of the faithfulness of the classical Gassner representation to the language of some well-studied link invariants, Milnor’s μ̄-invariants. This allows us to see that examples of the sort suggested by Theorem 3.5 would yield (if they could be realized by pure links) the simplest possible (from this other point of view) examples of braids in the kernel of the classical Gassner representation, and are the simplest examples for our extended Gassner representation. Many aspects of this translation to μ̄-invariants were known to Jon Hillman [Hi2] and perhaps others, but they are certainly not as generally known as they deserve to be. Also, Birman’s question 20 [Bi; p. 218] translates the faithfulness question to a purely algebraic question about elements of the free group which is intimately related to the present discussion.

Let us briefly recall the ideas and notation behind Milnor’s μ̄-invariants for an ordered, oriented, n-component link L in S^{3}. Given a positive integer k, called the weight, and a sequence I = i_{1} … i_{k} of elements of the set {1, …, n} Milnor’s invariant μ̄(i_{1} … i_{k}) for L is an integer (well defined modulo the ideal generated by certain μ̄-invariants of lesser weight). These invariants of weight k detect precisely if the longitudes of L lie in the k^{th} term of the lower central series G_{k} of the link group G = π_{1}(S^{3} − L).

Our first result says that for the class of n-bridge n-component links, the property of being trivial is equivalent to the vanishing of all the μ̄-invariants. This would certainly be false for more general links since, for example, the μ̄-invariants of any boundary link are zero.

Theorem 4.1. Suppose ρ ∈ P_{2n} (or σ ∈ PB_{n}). Then the following are equivalent. a) L(ρ) (or L(σ)) is a trivial n-component link

b) G = π_{1}(S^{3} − L(ρ)) (or L(σ)) is free of rank n

c) the longitudes of L(ρ) (or L(σ)) are trivial in G

d) the longitudes of L(ρ) (or L(σ)) lie in G_{k} for each k ∈ Z_{+}

e) Milnor’s invariants μ̄(i_{1} … i_{k}) vanish for L for any I = i_{1} … i_{k}

f) all Massey products in S^{3} − L are trivial

g) σ is the trivial braid

Proof of 4.1: The following implications are trivial: g ⇒ a ⇒ c ⇒ d ⇔ e. The following are well-known a ⇔ b ⇔ c, e ⇔ f [Po, Tu]. It suffices therefore to show d ⇒ b and, in the case of braids, b ⇒ g. Suppose d holds for L(ρ) (or L(σ)). By [C2, Thm. 3.1], G has a presentation of the form < x_{i} | [x_{i}, w_{i}] i = 1, …, n > where w_{i} represents an i^{th} longitude of L. We shall establish, by induction, that w_{i} ∈ ⋂_{k=1}^{∞} F_{k}, the intersection of the terms of the lower-central series of F < x_{1}, …, x_{n} >. Since this intersection is well-known to be trivial, b would follow. Suppose, by induction that w_{i} ∈ F_{k} for each i. Since l_{i}, the i^{th} longitude, lies in G_{k+1}, w_{i} must lie in the normal subgroup generated by F_{k+1} and the relations [x_{i}, w_{i}]. But since [x_{i}, w_{i}] ∈ F_{k+1}, it follows that w_{i} ∈ F_{k+1}. This completes the inductive step establishing d ⇒ b. Now suppose σ is a braid such that G is free. Since G has a presentation < x_{i} | [x_{i}, λ_{i}] i = 1, …, n > where σ̂(x_{i}) = λ_{i}x_{i}λ_{i}^{−1} and the exponent sum of x_{i} in λ_{i} is zero (a result of Artin [Bi; 2.2]), it follows that [x_{i}, λ_{i}] = e for each i (free groups are Hopfian). The restriction on the exponent sum then implies λ_{i} = e for all i which implies σ̂ is the identity endomorphism. It follows immediately that σ is trivial [Bi; 1.8.3]. □

This is to be contrasted with our 2.5 and Birman’s 3.16 [Bi] which say that the Gassner Matrix is trivial if and only if the longitudes of the associated link lie in G′′ = [G_{2}, G_{2}]. Thus the translation of the question of the faithfulness of the Gassner representation to links is “If the Alexander module of a link is trivial, does it have vanishing μ̄-invariants?”. It is known that the Alexander module controls some μ̄-invariants. For example, for a 2-component link, if the Alexander polynomial is trivial then each μ̄(111 … 12 … 2) is zero [Sm]. In fact Lorenzo Traldi has worked out a presentation matrix for the Alexander module in terms of the μ̄-invariants [Tr]. Each entry of this matrix involves a certain “symmetric sum” of μ̄-invariants so that if the Alexander matrix is the zero matrix, one gets systems of equations. Linear dependencies among the μ̄-invariants provide further equations. Using these facts Traldi achieved some limited but interesting results about the vanishing of μ̄-invariants. Nonetheless the author settled the long-standing question in [C2, C1] by exhibiting many links with trivial Alexander module and non-vanishing μ̄-invariants. Thus many μ̄-invariants are simply not controlled by the Alexander module. Therefore the answer to the Gassner question translated to arbitrary links is decidedly “No”. Links are much more non-abelian than the Alexander module can capture. However these examples are in the category of arbitrary links. Therefore the question remaining is

Question 4.2. If L is an n-bridge n-component link (respectively the closure of a pure 2n-plat, respectively the closure of a pure n-braid) what restrictions are placed on the μ̄-invariants?

Theorem 3.5 shows that in the category of 2n-plats, μ̄(123 … n) (or any permutation) is not controlled by the Alexander module so the analogue of the Gassner representation here has a “kernel”. Indeed these are the simplest possible examples within this class as the following shows. Recall that a Brunnian link (or braid) is one for which any proper sublink (proper subbraid) is trivial.

Proposition 4.3. Suppose L is a non-trivial n-component link which is the closure of the plat ρ ∈ P_{2n} whose Gassner matrix is trivial, and suppose n is minimal with respect to these properties. Then L is a Brunnian link (if L = L(β) then β is a Brunnian braid [Bi; Theorem 3.18]) and so all μ̄-invariants of weight less than n vanish as do all μ̄-invariants μ̄(I) where I fails to contain each element of {1, …, n} at least once. Therefore the μ̄- invariants of least weight which might fail to vanish are μ̄(I) where I is a permutation of the sequence 12 … n.

Proof of 4.3: Suppose G(ρ) = I. By 2.5 the longitudes of L lie in the second derived G′′. Suppose J is a proper sublink of L. It is easy to see that the longitudes of J lie in its second derived group so, by 2.5 and minimality, J is a trivial link. But Milnor himself showed that μ̄(I) depends only on the sublink J consisting of those components of L whose corresponding index appears in I. Therefore if some index is missing from I, μ̄(I) is zero since J is trivial. □

Moreover, Traldi has shown that these first-possible non-vanishing invariants {μ̄(i_{1} … i_{n}) | i_{1} … i_{n} a permutation of 1 … n} vanish when the Alexander Module is trivial in the cases of fewer than 6 components [Tr; p. 423 last paragraph]. Thus the example of Figure 5 is the simplest possible example of this phenomenon.

The reader will recall that these μ̄-invariants (with no repeated indices) vanish precisely when the link is link-homotopic to the trivial link (see for example [HL]). The connection to link homotopy is fascinating. For example we can argue that these homotopy invariants are the only ones we need concern ourselves with.

Proposition 4.4. Suppose that the Gassner representation for pure braids has a nontrivial kernel. Then there exists a pure braid in the kernel of the Gassner representation which is homotopically essential.

Proof of 4.4: Let β be a non-trivial pure braid such that G(β) = I and such that β is of minimal braid index n. By 4.1 there is some sequence I = i_{1}i_{2} … i_{k} of letters from {1, …, n} such that μ̄(I) is non-zero for L(β) and such that all μ̄-invariants of weight less than k vanish. Suppose that there are n_{j} occurences of the letter j in I. Form a new braid β′ by replacing the j^{th} strand by n_{j} parallel strands. Since G(β) = I, all linking numbers are zero, so that the obvious choice of parallel is 0-framed. Milnor showed that the μ̄- invariants of L(β′) are closely related to those of L(β) [Mi; Thm. 7]. In particular if I′ is the sequence obtained from I by replacing the occurences of the letter j by {j_{1}, …, j_{n_{j}}} then μ̄(I′) for L(β′) equals μ̄(I) for L(β). If J is any sequence with no repeated indices for the link L(β′) whose weight is less than that of I′, application of the same theorem reveals that μ̄(J) is equal to a μ̄-invariant of weight less than k evaluated on L(β), and is thus zero. It follows that L(β′) is homotopically non-trivial [Mi; §5].

Moreover, by 2.5, to show G(β′) = I, it suffices to show the longitudes of L(β′) lie in the second derived of its link group. But it is an easy exercise to see that if the longitudes of L(β) lie in the second derived of its group then the same will be true of any parallel link. □

Note that the braid β′ of 4.4 is certainly not Brunnian since β is a proper sub-braid. The question of which μ̄-invariants may be realized as the first non-vanishing μ̄-invariants

of a pure link, is fascinating, even without considering the Alexander module. My first intuition was that only the homotopy invariants could be realized but this is only true for n = 2. In fact Deborah Goldsmith long ago characterized what braids were null-homotopic (see [G]). In fact I do not know any restrictions on the μ̄-invariants of a pure 3-component link.

§5. Questions.

1. Which μ̄-invariants can be realized (as the first non-vanishing μ̄-invariant) by pure links, that is links which are closures of pure braids? More generally what are restrictions on the μ̄-invariants of a pure link?

2. The construction of §3 uses a link-theoretic construction, Bing-Doubling, to create links with desired μ̄-invariants. The set of plats is closed under this operation but not the set of pure links. Is there a braid-theoretic operation which can be used to create braids with desired μ̄-invariants? For example it follows from recent work of T. Stanford, X.S. Lin and D. Bar-Natan, that a braid lying in the k^{th} term of the lower central series of the pure braid group will have vanishing μ̄-invariants up to a high weight depending on k [BN, Li2, St]. It should not be so difficult to make this more explicit.

3. Is there a non-trivial 2n-plat with trivial Gassner matrix for n = 4 and n = 5?

4. What is the inverse image of the identity for the Gassner function on P_{2n}?

References

[BN] Bar-Natan, D., Vassiliev homotopy string link invariants, preprint, Harvard University, 1993.

[Bi] Birman, Joan S., Braids, Links and Mapping Class Groups, Annals of Math. Studies 82, Princeton Univ. Press (1975), Princeton, New Jersey.

[BL] Birman, J.S. and A. Libgober eds., Braids: Proceedings of a Summer Research Conference held July 13–26, 1986, Contemporary Mathematics volume 78, American Mathematical Society, Providence, R.I., 1988.

[C1] Cochran, Tim D., Links with trivial Alexander’s Module but non-vanishing Massey Products, Topology 29 (1990), no. 2, 189–204.

[C2] , Derivatives of Links: Milnor’s invariants and Massey products, Memoirs of Amer. Math. Soc. #427, Volume 84, American Mathematical Society (1990), Providence, Rhode Island.

[CL] Cooper, D. and D.D. Long, Derivative varieties and the pure braid group, American Journal of Math. 115 (1993), 137–160.

[Cr] Crowell, R.H., The derived module of a homomorphism, Adv. in Math. 6 (1971), 210–238.

[F] Fox, R.H., Quick trip through knot theory, Topology of 3-manifolds and related topics (M.K. Fort, ed.), Prentice Hall, 1962.

[Ga] Gassner, B.J., On braid groups, Abh. Math. Sem., Hamburg Univ., 25, pp. 19–22.

[G] Goldsmith, D.L., Homotopy of Braids: in answer to a question of Artin, Topology Conference VPI 1973, Lecture Notes in Mathematics #375, Springer Verlag, 1974, Berlin-Heidelberg.

[HL] Habegger, N. and Xiao-Song Lin, The classification of links up to link-homotopy, Journal of American Math. Soc. 3 (1990), 389–419.

[H1] Hillman, J.A., A link with Alexander module free which is not an homology boundary link, J. Pure Appl. Alg. 20 (1981), 1–5.

[H2] , Alexander Ideals of Links, Lecture Notes in Math. 895, Springer-Verlag, Berlin-Heidelberg, 1981.

[La] Lambert, H.W., A 1-linked link whose longitudes lie in the second commutator subgroup, Trans. Amer. Math. Soc. 147 (1970), 261–269.

[Li] Lin, Xiao-Song, Artin-type representation theorems and Milnor’s μ̄-invariants, Ph.D. thesis, Univ. of California at San Diego, 1988.

[Li2] , Milnor link invariants are all of finite type, preprint, Columbia Univ., 1993.

[Mi1] Milnor, J., Link groups, Annals of Math. 59 (1954), 177-195.

[Mi2] , Isotopy of links, Algebraic geometry and topology: A symposium in honor of S. Lefshetz, Princeton Univ. Press, Princeton, N.J. (1957), 280–306.

[M] Moody, J., The Burau representation is unfaithful for large n, Bull. Amer. Math. Soc. (1991), 379–384.

[Mo] Moran, Siegfried, The Mathematical Theory of Knots and Braids: An Introduction, North Holland Mathematics Studies 82, Elsevier Science Publishers B.V., 1983, Amsterdam, Netherlands.

[Po] Porter, Richard, Milnor’s μ̄-invariants and Massey products, Trans. Amer. Math. Soc. 257 (1980), 39–71.

[Sm] Smythe, N., Isotopy invariants of links and the Alexander matrix, Amer. Journal of Math. 89 (1967), 693–704.

[St] Stanford, T., Braid commutators and finite type invariants of knots and links, preprint, University of Geneva, 1993.

[Tr] Traldi, Lorenzo, Milnor’s invariants and the completions of link modules, Trans. Amer. Math. Soc. 284 (1984), 401–423.

[Tu] Turaev, V.G., The Milnor invariants and Massey products, in Studies in Topology-II, Zap. Nanc. Sem. Lenin. Ot. Mat. Kogo oi Stet. Acad. Nauk USSR 66 (1976) in translation Journal of Soviet Mathematics.

Tim D. Cochran cochran@rice.edu

Mathematics Department Rice University, P.O. Box 1892

Houston, Texas 77251