HODGE–TYPE STRUCTURES AS LINK INVARIANTS

MACIEJ BORODZIK AND ANDRÁS NÉMETHI

Abstract. Based on some analogies with the Hodge theory of isolated hypersurface singularities, we define Hodge–type numerical invariants of any, not necessarily algebraic, link in S^{3}. We call them H–numbers. They contain the same amount of information as the (normalized) real Seifert matrix. We study their basic properties, and we express the Tristram–Levine signatures and the higher order Alexander polynomial in terms of them. Motivated by singularity theory, we also introduce the spectrum (determined from these H–numbers), and we establish some semicontinuity properties for them. They can be related with skein–type relations, although they are not so precise as the classical skein relations.

1. Introduction

1.1. Although a Seifert matrix of a Seifert surface of a link is not a link invariant, it allows to define many knot invariants, which are on the one hand very deep, and easy to compute on the other. These invariants include the Alexander polynomial, the signature and the Tristram–Levine signatures. It might be quite surprising that the signature and the Alexander polynomial, although both come from Seifert matrix, have completely different properties. For example, the signature detects mirrors, and estimates the four-genus, while the Alexander polynomial estimates the three-genus and does not detect mirrors.

From a different point of view, if we consider a hypersurface singularity in (ℂ^{n+1}, 0), then we can associate many interesting objects with it, as the intersection form on the middle homology of the Milnor fiber, the monodromy matrix or the variation operator. These three objects, together with the space they act on, constitute a so–called variation structure. If the singularity is isolated, then its variation structure is determined by the variation operator, which is equivalent with the non–degenerate Seifert bilinear form associated with the germ of the singularity. Over real coefficients, each such variation structure is build up of some explicitly written indecomposable pieces. The number of times each such piece occurs in a concrete variation structure, is encoded in the so–called mod 2 equivariant (primitive) Hodge numbers associated with the singular germ. The name is motivated by the fact that they are, indeed, mod 2 reductions of the equivariant Hodge numbers associated with the mixed Hodge structure of the vanishing cohomology of the singularity, defined as in [Steen1]. In this way one also sees that the information codified in these numbers is equivalent with the real Seifert form [Nem2].

From the mod 2 equivariant Hodge numbers many other invariants can be reread, like the characteristic polynomial of monodromy, or different signature–type invariants. In fact, if n = 1, then from them one can even recover completely all the equivariant Hodge numbers of the mixed Hodge structure of the vanishing cohomology.

Date: April 30, 2010.

2010 Mathematics Subject Classification. primary: 57M25, secondary: 32S25, 14D07, 14H20.

Key words and phrases. Seifert matrix, Hodge numbers, Alexander polynomial, Tristram–Levine signature, variation structure, semicontinuity of the spectrum.

The first author is supported by Polish MNiSzW Grant No N N201 397937 and also by a Foundation for Polish Science FNP. The second author is partially supported by OTKA Grant K67928.

The algebraic links (case n = 1) form a bridge between the singularity theory and the knot theory (but the correspondences can be continued in higher dimensions too). The Alexander polynomial of an algebraic link is exactly the characteristic polynomial of the monodromy of the corresponding Milnor fibration. The Milnor fiber constitutes a natural Seifert surface of the algebraic link. The corresponding Seifert matrix is the transposed inverse of the variation operator. In other words, the variation structure of (plane curve) singularities is deeply related to link invariants of algebraic links.

Motivated strongly by the case of algebraic links, we associate a variation structure with an arbitrary link in S^{3}, and we define the analogs of mod 2 equivariant (primitive) Hodge numbers. The variation structure is built up from the real Seifert matrix of the link and determines the Seifert matrix up to real S-equivalence. In fact, the newly defined numbers codifies and determines this structure. Although the present work does not contain any Hodge theoretical discussion, motivated by the above correspondence we still call the introduced numbers H–numbers. Actually, the nilpotent part of the monodromy operator defines a weight filtration, and also one can define a mod 2–Hodge filtration similarly as in [Nem2], hence a ‘mod 2 Hodge structure’ exists (it would be interesting to extend it to a genuine Hodge structure).

Both the (higher) Alexander polynomial and the Tristram–Levine signatures can very easily expressed from the H–numbers of the link. Their symmetries and their behaviour under taking mirrors, allow us to explain e.g. why the Alexander polynomial does not distinguish mirrors, and while the signatures do.

In the Hodge theory of hypersurface singularities, the (numerical part) of the mixed Hodge structure was codified by Steenbrink and Varchenko in the so–called spectrum. This codification was motivated by the extremely powerful and mysterious semicontinuity behavior of it under the deformation.

In our present context we also introduce the mod 2 spectrum of a link in S^{3}, and we relate with the classical link invariants as the higher Alexander polynomials and Tristram–Levine signatures. This also emphasizes the unifying power of the newly introduced invariants, which gather together more conceptually all the classical properties and invariants associated with Seifert matrices. On the other hand, since we know only the classification of the variation structures over reals, and not over integers, in this discussion we loose some information regarding the integer Seifert matrices, like the determinant.

After the foundations, we try to alloy the two main strategies used in the two theories: the technique of skein relations of classical link theory with the semicontinuity of the spectrum (known in singularity theory). Although at the origin and substance of both sit in a surgery, for the second case one needs a special surgery with intrinsic monotonicity structure (this, in the singularity theory, is guaranteed by the presence of the deformation). In our results we will assume the monotonicity of the rank of the Alexander polynomial.

The basic motivation for studying semicontinuity of our structures is the following. First, we believe that the newly introduced H–numbers do not admit so precise skein relations similarly as some of the classical link invariants; or, their form should be packed in a more intelligent way. We believe that this ‘packing’ goes through the spectrum, and the corresponding semicontinuity relations will guide the corresponding surgery formulas. Similarly as in the case of analytic singularities, where the semicontinuity had remarkable applications (see e.g. [Var]), we expect in the future similar consequences for the newly introduced spectrum too. For more comments see (6.2).

1.2. The structure of the paper is the following. We begin with a definition and examples of variation structures in Section 2. Then we recall the classification theorem of [Nem2] regarding real (simple) variation structures, and define the H–numbers for links. In Section 4 we relate the classical link invariants (higher Alexander polynomials, rational Nakanishi index, Tristram–Levine signatures) to the H–numbers. In the next section we show some examples. In the last section we gather results about the skein relation, the proofs mostly go through skein relation for the Alexander polynomial or the signatures. Also, we establish some semicontinuity results for the spectrum. By them we wish to draw the attention of the readers to this new phenomenon with the hope that this will bring some deep and powerful instrument in the near future.

Acknowledgements. The authors wish to express their thanks to Rényi Institute for hospitality, to L. Kaufmann, A. Stoimenow, P. Traczyk, H. Trotter and H. Zoladek for many fruitful discussions on the subject, and to S. Friedl for pointing out the relation of the newly discussed invariants with the Nakanishi index.

2. Variations structures. Definitions and examples

2.1. Definitions. Here we recall some definitions from [Nem3, Section 2]. We begin with fixing some standard notation.

For a finite dimensional complex vector space U, we denote its dual by U^{∗}. The natural isomorphism θ: U → U^{∗∗} is given by θ(u)(ϕ) = ϕ(u). The complex conjugation is denoted by a bar ¯·; for any ϕ : U → V its dual map is denoted by ϕ^{∗} : V^{∗} → U^{∗}. Let us also recall that, if ϕ is represented by a matrix S in some basis, then ϕ^{∗} is represented by a transpose S^{T} in the dual basis. For a matrix S, S_{kl} or S_{k,l} denotes the coefficient of S in k-th row and l-th column.

It is convenient to regard hermitian forms in the following way.

Definition 2.1.1. A ℂ–linear endomorphism b: U → U^{∗} with \overline{b^{∗} ∘ θ} = εb (where ε = ±1) is called ε–hermitian form on U.

We do not assume here that b is an automorphism. The automorphisms of b consists of isomorphisms h: U → U preserving b, i.e. with h̄^{∗} ∘ b ∘ h = b.

Definition 2.1.2. An ε–hermitian variation structure (abbreviated by HVS) over ℂ is a quadruple (U; b, h, V), where

(1) U is finite dimensional vector space over ℂ;

(2) b: U → U^{∗} is an ε–hermitian form on U;

(3) h: U → U is a b–orthogonal automorphism of U;

(4) V : U^{∗} → U is a ℂ–linear endomorphism such that

\overline{θ^{−1} ∘ V^{∗}} = −εV ∘ \overline{h}^{∗}
V ∘ b = h − I.

Here and afterwards I denotes the identity map. The name of the structure is inherited from the operator V, which usually is a ‘variation map’, cf. (2.4). V, respectively h will be called variation, respectively monodromy operator.

Observe that from (4) it follows immediately that

(2.1.3)  b ∘ V = \overline{h}^{∗}^{−1} − I, and h ∘ V ∘ h̄^{∗} = V.

Definition 2.1.4. The HVS (U; b, h, V) will be called non–degenerate (respectively simple) if b (respectively V) is an isomorphism.

We will need following simple lemmas from [Nem2]:

Lemma 2.1.5. For a triple (U; b, h) satisfying points (1)–(3) from Definition 2.1.2, if b is non–degenerate, then there exists a unique V, namely V = (h − I)b^{−1}, such that (U; b, h, V) constitutes a HVS.

The non–degenerate triplets (U; b, h) are classified by Milnor [Mil2], see also [Neu1].

Lemma 2.1.6. For a pair (U; V) with V an isomorphism, there exist unique b and h such that (U; b, h, V) is a HVS. Indeed, h = −εV V̄^{∗−1} and b = −V^{−1} −εV̄^{∗−1} satisfy the axioms.

From the last lemma it follows that the classification of simple HVS is equivalent to the classification of ℂ–linear isomorphisms V : U^{∗} → U.

There is a natural notion of an isomorphism of a HVS:

Definition 2.1.7. Two HVS (U; b, h, V) and (U′; b′, h′, V′) are isomorphic, denoted by ≃, if there exists an isomorphism ϕ : U → U′ such that b = ϕ̄^{∗}b′ϕ, h = ϕ^{−1}h′ϕ, and V = ϕ^{−1}V′(ϕ̄^{∗})^{−1}.

There is an obvious construction of a direct sum of two or more HVS with agreeing signs ε. For sum of m copies of V we write m · V.

The conjugate of V = (U; b, h, V) is defined as V̄ = (U; b̄, h̄, V̄).

2.2. Examples and classification of HVS. Here we shall follow closely [Nem2]. For k ≥ 1, J_{k} denotes the (k × k)–Jordan block with eigenvalue 1.

Example 2.2.1. For λ ∈ ℂ^{∗} \ S^{1} and k ≥ 1, the quadruple

V_{λ}^{2k} = (ℂ^{2k}; (\begin{matrix}0 & I \\ εI & 0 \end{matrix}) , (\begin{matrix}λJ_{k} & 0 \\ 0 & \frac{1}{λ̄}J_{k}^{∗}^{−1} \end{matrix}) , (\begin{matrix}0 & ε(λJ_{k} − I) \\ \frac{1}{λ̄}J_{k}^{∗}^{−1} − I & 0 \end{matrix}))

defines a HVS. Moreover, V_{λ}^{2k} and V_{1/λ̄}^{2k} are isomorphic.

Before we show the next example we need a computational lemma; here one needs to consider the two square roots of ε. The two canonical sign choices for them are motivated by Hodge theoretical sign–conventions (cf. [Nem2], Sections 5 and 6) of the variation structures associated with isolated hypersurface singularities (ℂ^{n+1}, 0) → (ℂ, 0), where ε = (−1)^{n}.

Lemma 2.2.2. For any k > 1 there are precisely two non-degenerate ϵ–hermitian forms (up to a real positive scaling), denoted by b_{±}^{k}, such that

b̄^{∗} = εb and J_{k}^{∗}bJ_{k} = b.

By convention, the signs are fixed by (b_{±}^{k})_{1,k} = ±i^{−n^{2}−k+1}, where ϵ = (−1)^{n}. The entries of b also satisfy: b_{i,j} = 0 for i + j ≤ k and b_{i,k+1−i} = (−1)^{i+1}b_{1,k}.

(Maybe it is worth to mention that in matrix notations, of B is a bilinear form on U, and b : U → U^{∗} is defines by b(v) := B(v, ·), then the matrix of b is the transposed of the matrix of B.)

Example 2.2.3. Let λ ∈ S^{1}. Up to isomorphism there are two non–degenerate HVS such that h = λJ_{k}. These are

V_{λ}^{k}(±1) = (ℂ^{k}; b_{±}^{k}, λJ_{k}, (λJ_{k} − I)(b_{±}^{k})^{−1}) .

Notice that these structures are simple unless λ = 1. In fact, if λ ≠ 1 then any HVS with h = λJ_{k} is both non-degenerate and simple. The case with eigenvalue 1 admits also a pair of degenerate HVS.

Lemma 2.2.4. For k ≥ 2 there are two degenerate HVS with h = J_{k}. They are

Ṽ_{1}^{k}(±1) = (ℂ^{k}; b̃_{±}, J_{k}, Ṽ_{±}^{k}) ,

where

b̃_{±}^{k} = (\begin{matrix}0 & 0 \\ 0 & b_{±}^{k−1} \end{matrix})

and Ṽ_{±}^{k} is uniquely determined by b and h (up to an isomorphism). Moreover, Ṽ_{1}^{k}(±1) is simple. In fact, the entries of V^{−1} satisfy: (V^{−1})_{i,j} = 0 for i + j ≥ k + 2, (V^{−1})_{i,k+1−i} = ±(−1)^{i+1}i^{−n^{2}−k}. In order to recognize the isomorphism type, we have to recognize these entries up to a real positive re-scaling.

For k = 1 (i.e. U = ℂ), and h = I, the structures can be written down more explicitly; there are the following five ε–HVS’s with ε = (−1)^{n}:

V_{1}^{1}(±1) = (ℂ, ±i^{−n^{2}}, I, 0)
Ṽ_{1}^{1}(±1) = (ℂ, 0, I, ±i^{n^{2}+1})
T = (ℂ, 0, I, 0).

From all these examples the structures V_{1}^{k}(±1) and T are non–simple, and Ṽ_{1}^{1}(±1) are simple. In other words, for any λ ∈ S^{1}, there are precisely two non-equivalent simple variation structures with h = λJ_{k} in each dimension. We use the following uniform notation for them:

(2.2.5)  W_{λ}^{k}(±1) = \begin{cases}V_{λ}^{k}(±1) & if λ ≠ 1 \\ Ṽ_{1}^{k}(±1) & if λ = 1. \end{cases}

Proposition 2.2.6. A simple HVS is uniquely expressible as a sum of indecomposable ones up to ordering of summands and up to an isomorphism. The indecomposable pieces are

W_{λ}^{k}(±1) for k ≥ 1, λ ∈ S^{1}
V_{λ}^{2k}  for k ≥ 1, 0 < |λ| < 1.

Convention 2.2.7. From now on, all HVS we shall discuss, are assumed to be simple.

The above proposition allows us to define some invariants of HVS.

Definition 2.2.8. Let us be given a simple HVS V and express this structure, according to Proposition 2.2.6, as

(2.2.9)  V = ⨁_{\begin{matrix}0<|λ|<1 \\ k≥1 \end{matrix}} p_{λ}^{k} · V_{λ}^{2k} ⊕ ⨁_{\begin{matrix}|λ|=1 \\ k≥1, u=±1 \end{matrix}} p_{λ}^{k}(u) · W_{λ}^{k}(u),

where p_{λ}^{k}·V_{λ}^{2k} is a shorthand for a sum V_{λ}^{2k} ⊕⋯⊕V_{λ}^{2k} (p_{λ}^{k} times), cf. (2.1.7). Then {p_{λ}^{k}|}_{|λ|<1} and {p_{λ}^{k}(±1)}_{λ∈S^{1}} are called the H–numbers of V.

Remark 2.2.10. The above classification result is over ℂ or, equivalently, over ℝ. One can consider the HVS over ℤ, but then the classification is unknown.

If V is defined over the real numbers, then the above decomposition has some symmetries. Let s be 1 if λ = 1 and 0 if λ ∈ S^{1} \ {1}. Then, with ε = (−1)^{n},

(2.2.11)  \overline{V_{λ}^{2k}} = V2k ; λ̄ for λ ∉ S^{1}, and
(2.2.12)  \overline{W_{λ}^{k}(±1)} = Wk ; λ̄(±(−1)^{n+k+1+s}) for λ ∈ S^{1}.

Therefore:

Lemma 2.2.13. If in the HVS V, the matrix V is defined over reals, then

p_{λ}^{k} = pk ; λ̄ (for |λ| < 1) and p_{λ}^{k}(±1) = pk ; λ̄(±(−1)^{n+k+1+s}) (for |λ| = 1).

Moreover, by an easy check (of the coefficient b_{k,1}) one has:

Lemma 2.2.14. Let V be the variation operator of the simple structure V. Let −V be the structure corresponding to the variation operator −V (see Lemma 2.1.6). Then −W_{λ}^{k}(±1) ≃ W_{λ}^{k}(∓1), and −V_{λ}^{2k} ≃ V_{λ}^{2k}.

One needs slightly more computations to verify:

Lemma 2.2.15. Let V be the variation operator of the simple structure V. Let V^{T} be the structure determined by the variation operator V^{T}. Then V^{T} and V are isomorphic.

Proof. The statement is clear for V = V_{λ}^{2k}. Hence, assume that V = (U; b, h, V) is W_{λ}^{k}(±1). Consider the new structure with variation operator W = V^{T}. Since for ϕ := V^{T} one has ϕ^{−1}V^{T} \overline{ϕ}^{∗,−1} = \overline{V}^{−1}, the variation structures associated with W and \overline{V}^{−1} are isomorphic. The monodormy operator of \overline{V}^{−1} is −ε\overline{V}^{−1}V^{∗} = h^{∗} = λJ_{k}^{T}. Next, consider the anti-diagonal matrix A with A_{ij} = 1 if i + j = k + 1 and zero otherwise; it satisfies A = A^{−1} = A^{T} = \overline{A}. Base change by A has the effect A(λJ_{k}^{T})A = λJ_{k}, hence the monodromy operator is twisted to a ‘normal form’, which agrees with h. In particular, it is enough to compare the two variation operators V and A\overline{V}^{−1}A. If λ = 1, by the last sentence of (2.2.4), it is enough to compare the anti-diagonals of these two operators, which clearly agree. If λ ≠ 1 use (2.2.2) and the same type of argument. □

2.3. Spectrum and the extended spectrum. One can extract from a variation structure a weaker invariant, whose motivation will be explained in the next subsection when we discuss the spectrum of an isolated hypersurface singularity.

Definition 2.3.1. Let V be a HVS. The mod 2–spectrum (or, shortly, the spectrum) of V is a finite set Sp of real numbers from (0, 2] such that any real non-integer number α ∈ (0, 2] occurs in Sp precisely s(α) times, where

s(α) = ∑_{\begin{matrix}k odd \\ u=±1 \end{matrix}} \frac{k − uv}{2} · p_{λ}^{k}(u) + ∑_{\begin{matrix}k even \\ u=±1 \end{matrix}} \frac{k}{2} · p_{λ}^{k}(u),

where

e^{2πiα} = λ and (−1)^{⌊α⌋} = v.

The H–numbers p_{1}^{k}(±1) correspond to elements 1 and 2 in the spectrum, appearing precisely k/2 times each if k is even, and (k ± 1)/2 and (k ∓ 1)/2 times if k is odd.

A consequence of Lemma 2.2.13 is the following symmetry property

Corollary 2.3.2. If V is a real matrix, and ϵ = −1, then Sp \ ℤ is symmetric with respect to 1.

Notice that Sp contains no information regarding the blocks with eigenvalues λ ∉ S^{1}. To enclose the information regarding {p_{λ}^{k}}_{|λ|<1} we define the extended spectrum. For a different codification see [Nem1, (3.9)]. (This construction has no counterpart in the classical Hodge theory.)

Definition 2.3.3. The extended spectrum ESp of a HVS V is a finite subset of complex numbers from (0, 2] × iℝ of the form ESp = Sp ∪ ISp, where ISp ∩ ℝ = ∅ and any non-real number z ∈ (0, 2] × iℝ, z = x + iy, occurs in ISp precisely s(z) times, where

s(z) = \begin{cases}∑ k · p_{λ}^{k} & if x ≤ 1, y > 0 and e^{2πiz} = λ \\ ∑ k · p_{λ}^{k} & if x > 1, y < 0 and e^{2πiz} = 1/λ̄ \\ 0 & if x ≤ 1 and y < 0, or x > 1 and y > 0. \end{cases}

In other words, a block V_{λ}^{2k} (where |λ| < 1) contributes k times to both (x, y) and (1 + x, −y), if e^{2π(−y+ix)} = λ and x ∈ (0, 1].

We have the following two important properties of ESp.

Lemma 2.3.4. For any u ∈ (0, 1), let H_{u} = (u, u + 1) × iℝ. Then, if ISp ∩ ∂H_{u} = ∅, we have

#ISp ∩ H_{u} = #ISp \ H_{u}.

Proof. This follows directly from a simple observation that from the two numbers (x, y) and (1 + x, −y), one of them lies in H_{u}, and one of them does not. □

Lemma 2.3.5. If V is real and ε = −1, then ESp \ ℤ is symmetric (via point–reflection) with respect to 1.

Proof. By Corollary 2.3.2, it is enough to prove that ISp is symmetric. But this follows from the fact that λ = e^{2π(−y+ix)} provides the points (x, y) and (1 + x, −y) in ISp, while λ̄ = e^{2π(−y−ix)} the points (1 − x, y) and (2 − x, −y). □

2.4. Variations structures of Milnor fibers. The motivation of the definition of HVS comes from the topological invariants of complex isolated hypersurface singularities and their relationship with the mixed Hodge structure on the vanishing cohomology.

Let f : (ℂ^{n+1}, 0) → (ℂ, 0) (n ≥ 0) be an analytic germ such that f^{−1}(0) has an isolated singularity at the origin.

Let S^{2n+1} be a small sphere around 0, K = S^{2n−1} ∩ {f = 0} the link, and

ϕ: S^{2n+1} \ K → S^{1}, ϕ(z) = f(z)/|f(z)|

the Milnor fibration (see [Mil1]) with fiber F = ϕ^{−1}(1). Set F_{t} = ϕ^{−1}(e^{2πit}) for t ∈ [0, 1) (with F = F_{0}). Then the trivialization of the bundle over [0, 1) gives diffeomorphisms (defined up to isotopy) γ_{t}: F → F_{t} for t ∈ [0, 1), and extended to t = 1, the geometric monodromy γ_{1} : F → F. They give rise to a well-defined map

Γ_{t}: H̃_{n}(F_{1}) → H̃_{n}(F_{t})

and the monodromy map

h = Γ_{1} : H̃_{n}(F) → H̃_{n}(F).

One also defines the intersection form on b : H̃_{n}(F)^{⊗2} → ℝ which is (−1)^{n} symmetric. Since γ_{1} is chosen such that it is identity on ∂F, one also defines a variation map V : H̃_{n}(F, ∂F) → H̃_{n}(F) (see [Zo, Chapter 4.2] or [AVG, Chapter 1.2]). Here, by Lefschetz duality one has the identification H̃_{n}(F, ∂F) ≃ Hom(H̃_{n}(F), ℝ). The next fact is well–known (see e.g. [Nem2]):

Proposition 2.4.1. The quadruple U = H̃_{n}(F, ℂ), b, h, V (extended from ℤ, or ℝ to ℂ) form a HVS with ε = (−1)^{n}.

Definition 2.4.2. The variation structure defined above is called the variation structure of the singularity f and it is denoted by V_{f}.

Notice that V_{f} is defined over ℝ. Additionally, it has some other particular properties as well. First of all, by the Monodromy Theorem (see e.g. [AVG, Theorems 3.11 and 3.12] or [Zo, Chapter 7, §4]), all the eigenvalues of h are roots of unity. Moreover, the block–decomposition of V_{f} is closely related with the mixed Hodge structure of U.

Recall (see e.g. [Nem2] for the facts below) that U carries a mixed Hodge structure compatible with the monodromy action. Let us denote the corresponding equivariant Hodge numbers by h_{λ}^{a,b}. The nilpotent part of the monodromy defines a morphism of Hodge structures of type (−1, −1), let us denote by p_{λ}^{a,b} the dimensions of the corresponding primitive λ–generalized eigenspaces (which are, in general, non–trivial for a + b ≥ n + s). Then

p_{λ}^{a,b} = h_{λ}^{a,b} − h_{λ}^{a+1,b+1} and h_{λ}^{a,b} = ∑_{l≥0} p_{λ}^{a+l,b+l}

for any a + b ≥ n + s. Moreover, since h_{λ}^{a,b} = hn+s−a,n+s−b ; λ̄, the system of Hodge numbers {h_{λ}^{a,b}}_{a,b} is equivalent with the system of primitive Hodge numbers {p_{λ}^{a,b}}_{a+b≥n+s}.

The point is that by [Nem2, Theorem 6.1] one has the following isomorphism of variation structures:

V_{f} ≃ ⨁_{λ} ⨁_{2n≥a+b≥n+s} p_{λ}^{a,b} · W_{λ}^{a+b+1−n−s}((−1)^{b}).

In particular, for any k ≥ 1 and u = ±1 one has

(2.4.3)  p_{λ}^{k}(u) = ∑_{\begin{matrix}a+b=k+n+s−1 \\ (−1)^{b}=u \end{matrix}} p_{λ}^{a,b}.

This fact motivates to call the numbers p_{λ}^{k}(u) the mod–2 primitive Hodge numbers of f, or, of the corresponding variation structure.

The relation with Hodge theory can be continued. Recall that for any f as above one extract from the equivariant Hodge numbers the spectrum. Now, if V is a variation structure associated to an isolated hypersurface singularity, then Sp (defined in (2.3.1)) is the spectrum of the singularity reduced modulo 2, i.e. if α belongs to the spectrum, then α ∈ (0, 2] (mod 2) belongs to Sp. In the case of isolated curve singularities Sp is just the spectrum of the singular germ.

Obviously, in general, Sp does not always contain enough information to recover V. However, if all the monodormy eigenvalues are different, then the dimension of all Jordan blocks is one, and V is determined by Sp. This simple case contains for example all spectra of cuspidal plane curve singularities.

At the end of this subsection we recall the connection of the variation structures with the Seifert form.

Definition 2.4.4. Let us be given two cycles α, β ∈ H̃_{n}(F). The Seifert form of the Milnor fibration is defined to be

(2.4.5)  S(α, β) = L(α, Γ_{1/2}β),

where L is the linking number of two n-dimensional cycles in S^{2n+1}.

There is a standard fact that S(α, β) = ⟨V^{−1}(α), β⟩, where ⟨·, ·⟩ is the Lefschetz pairing H̃_{n}(F, ∂F) × H̃_{n}(F) → ℝ . In particular, in matrix notations, the variation operator is the transposed inverse of the Seifert form.

3. H–numbers for links

3.1. Definitions and first properties. Let us consider S^{3} with its standard orientation, and let L ∈ S^{3} be an oriented link. Let S be its (integral) Seifert form. By our convention, the Seifert form is S(α, β) = L(α, β^{+}), where α, β are cycles on the Seifert surface and β^{+} is the push-forward of β in the positive direction. This is the convention adopted by e.g. [BZ, Liv1, Mur2]. Some authors like [Kau, Kaw] define S(α, β) as L(α^{+}, β). This amounts to transposition of S.

We say that the matrices S and ASA^{T} are ‘congruent’ if A is an invertible integral matrix.

The next results were proved in [Keef] as a generalization of Theorem 12.2.9 of [Kaw] (we would like to thank to H. Trotter for drawing our attention to Keef’s paper):

Proposition 3.1.1. (a) [Keef, Proposition 3.1] Let S : V × V → ℝ be a Seifert form of a link. Then, either S is real S–equivalent to the empty matrix or is real S-equivalent to

(\begin{matrix}S_{0} & 0 \\ 0 & S_{ndeg} \end{matrix}), where det S_{ndeg} ≠ 0 and S_{0} is a zero matrix.

(b) [Keef, Theorem 3.5] Let us be given two matrices S and T = T_{0} ⊕ T_{ndeg}, which are S

equivalent. Assume that they are of the form S = (\begin{matrix}S_{0} & 0 \\ 0 & S_{ndeg} \end{matrix}), T = (\begin{matrix}T_{0} & 0 \\ 0 & T_{ndeg} \end{matrix}). Then

S_{ndeg} and T_{ndeg} are congruent.

Let us define

V := (S_{ndeg}^{T})^{−1}.

and take the associated HVS with ε = −1. Its parts are the following: U = ℂ^{m}, where m := rank(V), b = S_{ndeg} − S_{ndeg}^{T}, and h = (S_{ndeg}^{T})^{−1} · S_{ndeg}.

Observe that taking a conjugate of the Seifert matrix results in an isomorphism of HVS. Hence, the whole structure is independent (up to an isomorphism) of the specific choice of the Seifert matrix. Hence it is a link invariant.

Definition 3.1.2. The variation structure (U; b, h, V) defined above is called the variation structure of the link L and is denoted by V_{L}.

According to Definition 2.2.8, we can define the numbers {p_{λ}^{2k}}_{|λ|<1} and {p_{λ}^{k}(±1)}_{λ∈S^{1}} of the corresponding HVS.

Definition 3.1.3. The numbers {p_{λ}^{2k}}_{|λ|<1} and {p_{λ}^{k}(±1)}_{λ∈S^{1}} will be called the H–numbers of the link L.

From basic properties of Seifert matrices we get

Lemma 3.1.4. (a) The H–numbers are symmetric in the sense that for 0 < |λ| < 1 one

has p_{λ}^{k} = pk ; λ̄, and

p_{λ}^{k}(±1) = pk ; λ̄(±(−1)^{k+s}) for λ ∈ S^{1}.

(b) H–numbers are additive with respect to the connected sum of links.

(c) If L^{or} is the link L with all its components with opposite orientation, then the H– numbers of L and L^{or} are the same.

(d) If L^{mir} is the mirror of L, then the H–numbers are changed as follows: p_{λ}^{2k}(L^{mir}) = p_{λ}^{2k}(L) for any |λ| < 1 and

p_{λ}^{k}(±1)(L^{mir}) = p_{λ}^{k}(∓1)(L) for λ ∈ S^{1}.

Proof. (a) follows immediately from Lemma 2.2.13. As for (b), observe that the Seifert matrix of the connected sum is the direct sum of the Seifert matrices of the summands. (c) and (d) follows from the classical facts that the S(L^{or}) = S(L)^{T} and S(L^{mir}) = −S(L)^{T}, cf.

[Mur2] Propositions (5.4.6) and (5.4.7), combined with (2.2.13), (2.2.14) and (2.2.15). □

If L is an algebraic link, i.e. a link of a plane curve singularity, it has two HVS’s: the variation structure of the singularity V_{f} (see Definition 2.4.2) and the variation structure of the oriented link V_{L}. Obviously, they agree V_{f} ≃ V_{L} (thanks to the relation V = (S^{T})^{−1}, cf. the discussion after (2.4.4)).

One has very strong restrictions for primitive Hodge numbers of algebraic links: from the classical monodromy theorem (see e.g. [AVG] or [Zo, Chapter 7, §4]) one reads:

Corollary 3.1.5. If L is an algebraic link then p_{λ}^{k}(±1) = 0 if at least one of the following conditions is satisfied

• λ is not a root of unity;

• λ ≠ 1 and k > 2;

• λ = 1 and k > 1.

Corollary 3.1.5 admits further improvements, see e.g. [Nem2, Proposition 6.14].

Lemma 3.1.6. If L is algebraic link and λ ∈ S^{1} \ {1}, then p_{λ}^{2}(−1) = 0 (p_{λ}^{2}(+1) can be positive) and p_{1}^{1}(−1) = 0.

4. Classical link invariants and H–numbers

Having defined the H–numbers, we wish to study their relationship with classical invariants of the link L. Recall that we have the decomposition S = S_{ndeg} ⊕ S_{0} and the newly defined numbers are associated with S_{ndeg}, see (3.1.3).

4.1. Alexander polynomial. Define the polynomial P (t) ∈ ℝ[t] by

P (t) := ∏_{|λ|=1}(t − λ)^{∑_{k,u} kp_{λ}^{k}(u)} ∏_{0<|λ|<1} ((t − λ)(t − 1/λ̄))^{∑_{k} kp_{λ}^{k}} .

Lemma 4.1.1. The Alexander polynomial Δ(t) is zero if S_{0} ≠ 0, and it equals P (t) (up to an invertible element of ℝ[t, t^{−1}]) otherwise. In this second case, the degree of Δ(t) is equal to the cardinality of the extended spectrum ESp.

Proof. Use Δ(t) = det(S_{ndeg} − tS_{ndeg}^{T}) = det S_{ndeg}^{T} · det(h − tI). □

As the Alexander polynomial of a knot has no root at t = 1, we get:

Corollary 4.1.2. If L is a knot then p_{1}^{k}(±1) = 0.

The symmetry property of H–numbers (Lemma 2.2.13) explain (once again) the wellknown property of the Alexander polynomial, namely, if we write Δ = a_{0} +a_{1}t+⋯+a_{m}t^{m}, then a_{n} = (−1)^{m}a_{m−n}.

4.2. Higher Alexander polynomials. Let us recall briefly the construction of the higher order Alexander polynomials (see [BZ, Definition 8.10]) via higher order elementary ideals of the matrix S − tS^{T}. We remark that our construction differs slightly from the standard one, because we consider ideals in ℝ[t, t^{−1}] instead of ℤ[t, t^{−1}] (hence we loose some information about ℤ–torsion elements).

Let ℓ be a positive integer. Consider an ℓ × ℓ matrix H depending on a variable t. For 0 ≤ n < ℓ, let E_{n} be the ideal in ℝ[t] generated by the determinants of all (ℓ − n) × (ℓ − n) minors of H. As ℝ[t, t^{−1}] is a principal ideal domain the ideal E_{n} is generated by a single element Δ_{n}^{H}(t) ∈ ℝ[t, t^{−1}]. Δ_{n}^{H}(t) is defined only up to an invertible element in ℝ[t, t^{−1}], multiplying it t in an appropriate power we can guarantee that Δ_{n}^{H} is in fact a polynomial and, unless it is zero, that Δ_{n}^{H}(0) ≠ 0.

Definition 4.2.1. The polynomial Δ_{n}^{H}(t) for H = S − tS^{T} is called the n-th Alexander polynomial of the link L and denoted by Δ_{n}(t).

The indexing was chosen so that Δ_{0} is the standard Alexander polynomial. If m_{0} is the rank of S_{0}, then Δ_{n} = 0 for 0 ≤ n < m_{0} and Δ_{m_{0}} = P, cf. (4.1.1).

Our goal is to express Δ_{n} in terms of the primitive numbers. Notice that multiplying H by a non-degenerate matrix independent of t or taking its transpose does not change the polynomials Δ_{n}. Therefore, Δ_{n+m_{0}}(S −tS^{T}) = Δ_{n}(h−tI). Let the second term be denoted by Δ_{n}^{h}. By choosing a suitable basis of U we may also assume that h is in the Jordan form. Moreover, if H = λJ_{k} − It, then Δ_{0}^{H} = (t − λ)^{k} and Δ_{1}^{H} = 1. Next, we need to see what happens when we take a direct sum of several matrices.

Lemma 4.2.2. Let H^{1} and H^{2} be two square matrices, H = H^{1} ⊕ H^{2}, and Δ_{i}^{H_{1}}, Δ_{j}^{H_{2}}, Δ_{k}^{H} the corresponding higher Alexander polynomials. For fixed λ ∈ ℂ^{∗}, let a_{i} (respectively b_{j}, c_{k}) be the multiplicity of (t − λ) in Δ_{i}^{H_{1}} (respectively Δ_{j}^{H_{2}}, Δ_{k}^{H_{3}}). Then

c_{k} = min{a_{i} + b_{j} : i + j = k}.

Proof. Use the fact that for arbitrary minors A_{1}, A_{2} of H^{1} and H^{2}, A_{1} ⊕ A_{2} is a minor of H. Moreover, any minor of H, with non-zero determinant, arises in this way. □

Fix μ ∈ ℂ^{∗} and set for each k ≥ 1

(4.2.3)  s_{k}(μ) := \{ \begin{matrix}p_{μ}^{k}(+1) + p_{μ}^{k}(−1) & if μ ∈ S^{1}, \\ p_{λ}^{k} & if μ ∈ \{λ, 1/λ̄\}, (|λ| < 1). \end{matrix}

I.e., s_{k}(μ) is the number of Jordan blocks of size k with eigenvalue μ. Now let Θ := {θ_{1}, …, θ_{r}}, θ_{1} ≤ ⋯ ≤ θ_{r}, be a set of integers, such that each k ∈ ℤ is contained in Θ precisely s_{k}(μ) times (hence r = r(μ) = #Θ = ∑ s_{k}(μ)). Define the function

I(n) = \begin{cases}∑_{i=1}^{r(μ)−n} θ_{i} & for n < r(μ) \\ 0 & otherwise. \end{cases}

The above facts combined provide:

Proposition 4.2.4. The multiplicity of the root μ in the n–th Alexander polynomial Δ_{n}^{h} is I(n).

In Lemma 4.1.1 and Proposition 4.2.4 the exponents of the monomials (t − λ) depended on the sums p_{λ}^{k}(+1)+p_{λ}^{k}(−1). This, together with Lemma 3.1.4, explains in this terminology too why the higher Alexander polynomials of a link and its mirror are the same.

4.3. Rational Nakanishi index. We begin with recalling the definition of the Nakanishi index. Let Λ = ℤ[t, t^{−1}] be a ring of Laurent polynomials with integer coefficient and Λ_{ℚ} = ℚ[t, t^{−1}]. For a knot K, set X = S^{3} \ K and let us consider the Alexander module of K, i.e. the homology group of X

H = H_{1}(X; Λ)

with coefficients in Λ. This group can be regarded as the homology group of the universal abelian cover of X. This group has a natural structure of a Λ module, where t and t^{−1} are deck transformations (cf. [Kaw, Section 5.4])

Definition 4.3.1. A square presentation matrix for H is a square matrix A with entries in Λ such that H = Λ^{n}/AΛ^{n}, where n is the size of A. The Nakanishi index n(K) is the minimal size of a square presentation matrix of the module H.

Since we are allowed to perform row operations on a square presentation matrix and, independently, column operations, we can always assume that A is diagonal.

It is well known [Kaw, Proposition 5.4.1], that if S is a Seifert matrix of K, then tS − S^{T} is a square presentation matrix for H. However, in general, its size is not minimal. For example, for all torus knots n(K) = 1.

We show a relationship between our primitive numbers and the Nakanishi index defined over rational numbers instead of integers.

Definition 4.3.2. The rational Nakanishi index n_{ℚ}(K) is a minimal size of a square matrix A_{ℚ} with entries in Λ_{ℚ} such that

H ⊗ ℚ = Λ_{ℚ}^{n}/A_{ℚ}Λ_{ℚ}^{n}.

Obviously we have n(K) ≥ n_{ℚ}(K). In [Nak] is proved that n(K) is a lower bound for the unknotting number, hence n_{ℚ}(K) is a lower bound too. Moreover we have the next result.

Proposition 4.3.3. If Δ_{0}(K), …, Δ_{n}(K) are higher order Alexander polynomials with Δ_{0} the ordinary Alexander polynomial, then

n_{ℚ} = min{k: Δ_{k}(K) ≡ 1}.

In particular n_{ℚ} is the maximal number of Jordan blocks of the monodromy matrix with the same eigenvalue

(4.3.4)  n_{ℚ} = max_{λ} r(λ) = max (max_{|λ|=1} ∑_{k,u} p_{λ}^{k}(u), max_{0<|λ|<1} ∑_{k} p_{λ}^{k}) .

Proof. First of all observe that given a square presentation matrix A_{ℚ} of size n, Δ_{l}(K) is the generator of ideal spanned by all (n − l) × (n − l) minors of l (see Section 4.2). Hence, if the l−th Alexander polynomial Δ_{l} is non-trivial, it follows that the size of A_{ℚ} is at least l.

Conversely, if Δ_{k−1} ≢ 1 and Δ_{k} ≡ 1 we may define A_{ℚ} to be a diagonal k × k matrix with Δ_{l−1}/Δ_{l} on the (l, l)-th place. Then Λ_{ℚ}^{k}/A_{ℚ}Λ_{ℚ}^{k} is easily seen to be isomorphic as a Λ_{ℚ}-module to Λ_{ℚ}^{n}/(tS − S^{T}).

Equation (4.3.4) follows now from Proposition 4.2.4. □

4.4. Signatures. Besides Alexander polynomials, the Tristram–Levine signatures also can be computed from the H–numbers. We begin by recalling their definition.

Definition 4.4.1. The Tristram–Levine signature (or the signature function) is the function associating to each ζ ∈ S^{1} \ {1} the signature σ(ζ) of the Hermitian form given by

(4.4.2)  M_{S}(ζ) := (1 − ζ)S + (1 − ζ̄)S^{T} .

The nullity n(ζ) is the nullity of the above form.

Remark 4.4.3. Some authors, like [Mur1, Definition 3.11], define n(ζ) as the nullity increased by 1. It is merely a matter of convention, we stick to the notation we find more natural.

Clearly, in the definition of σ one can replace S by S_{ndeg}. Hence, in the sequel, for the simplicity of the notations, S will denote S_{ndeg}. Then, M_{S}(ζ) equals

(4.4.4)  S((ζζ̄ − ζ)I + (1 − ζ̄)S^{−1}S^{T})) = (1 − ζ̄)S · (h^{−1} − ζI).

It is not hard to express these signatures by H–numbers: we compute the signature function associated with each irreducible simple HVS and then we use the additivity of signatures. Notice that for non–real matrices S, M_{S}(ζ) in (4.4.2) should be replace by (1 − ζ)S + (1 − ζ̄)S̄^{T}.

Lemma 4.4.5. Let V be the variation operator of V_{λ}^{2k} (see Example 2.2.1). Let S = (V̄^{T})^{−1}. Then the signature of M_{S}(ζ) is zero and the form is non-degenerate for any ζ.

Proof. The non–degeneracy follows from (4.4.4). For vanishing of the signature notice that

M_{S}(ζ) has the block form (\begin{matrix}0 & A \\ Ā^{T} & 0 \end{matrix}) with A non–degenerate. Hence the signature is zero

by elementary linear algebra. □

The case of HVS W_{λ}^{k}(u) for |λ| = 1 is slightly more complicated.

Lemma 4.4.6. Set S = (V̄^{T})^{−1}, where V is the variation operator of W_{λ}^{k}(u). Then the form M_{S}(ζ) is non-degenerate for all ζ ≠ λ̄. If ζ = λ̄ then it has a one–dimensional kernel. In particular, the nullity of the variation structure V_{L} is

n(ζ) = ∑_{k,u} pk ; ζ̄(u).

Proof. Use (4.4.4). □

The next lemma is rather computational. To formulate it we need the next

Convention 4.4.7. Let α, β ∈ S^{1}. We say that

α < β

if α = e^{2πix}, β = e^{2πiy} with x, y ∈ [0, 1) and x < y.

Proposition 4.4.8. Let L be a link, and consider the primitive numbers of the variation structure V_{L} associated with its Seifert for as above. Let ζ ∈ S^{1} \ {1}. Then the Tristram– Levine signature of L is equal to

σ(ζ) = σ(ζ̄) = − ∑_{\begin{matrix}λ<ζ \\ k odd \\ u=±1 \end{matrix}} up_{λ}^{k}(u) + ∑_{\begin{matrix}λ>ζ \\ k odd \\ u=±1 \end{matrix}} up_{λ}^{k}(u) + ∑_{\begin{matrix}k even \\ u=±1 \end{matrix}} up_{ζ}^{k}(u).

The proof is slightly long and computational, but straightforward; it is left to the reader.

This can partly be expressed from the spectrum, a fact which emphasizes more the different roles of the sums in the expression (4.4.8).

Corollary 4.4.9. (a) Let Sp be the (mod 2)–spectrum of a variation structure V_{L} (see Definition 2.3.1). Let ζ = e^{2πix}, where x ∈ (0, 1). Then

σ(ζ) = −#Sp ∩ (x, x + 1) + #Sp \ (x, x + 1) + ∑_{\begin{matrix}k even \\ u=±1 \end{matrix}} up_{ζ}^{k}(u).

In particular, if ζ is not an eigenvalue of the monodromy h then

σ(ζ) = −#Sp ∩ (x, x + 1) + #Sp \ (x, x + 1).

(b) Let ESp be the extended spectrum of the variation structure V_{L} and let ζ = e^{2πix}, x ∈ (0, 1). Let H_{x} = (x, x + 1) × iℝ and assume that ESp ∩ ∂H_{x} = ∅. Then

σ(ζ) = −#ESp ∩ H_{x} + #ESp \ H_{x}.

Remark 4.4.10. (a) Corollary 4.4.9(a) can be compared with [Li, Proposition 1], where the signatures of the iterated torus knots is be computed. In fact, the spectrum of the torus knot (k, l), or equivalently, the spectrum of the singularity {x^{k} + y^{l} = 0}, is

Sp = \{\frac{i}{k} + \frac{j}{l} , 1 ≤ i ≤ k − 1, 1 ≤ j ≤ l − 1\}.

(b) The equivariant signatures of any HVS were computed in [Nem2] and are expressible in terms of H–numbers (primitive equivariant Hodge numbers). However, in general, they is not expressible in terms of the spectrum alone.

E.g., for plane curve singularities, the equivariant signature σ_{−1} cannot be determined, in general, from the spectrum. This is the case with the Tristram–Levine signature σ(−1) too. In fact, σ(−1) is the signature of S + S^{T}. If S is the Seifert form of the plane curve singularity f(x, y), then σ(S + S^{T}) is the signature of the suspension surface singularity f(x, y) + z^{2}. For a pair of singularities with the same spectrum but with different σ(−1) see e.g. [Nem2, (6.10)].

This also shows that H–numbers of a link are not determined by Tristram–Levine signatures and orders of higher Alexander polynomials alone. This can be exemplified by a model situation as follows. Take some λ ∈ S^{1} \ {1} and consider structures such that

p_{λ}^{3}(+1) = p_{λ}^{1}(−1) = 1 respectively p_{λ}^{3}(−1) = p_{λ}^{1}(+1) = 1,

and all other p^{k}’s for this λ are zero. The two structures are different, but they provide the same contribution to signatures (actually 0), and orders of zeros of subsequent Alexander polynomials at t = λ are in both cases 4, 1, 0, . . ..

(c) On the other hand, the higher Alexander polynomials with a set of ‘higher equivariant signatures’ determines the set of H–numbers. They are defines as follows, cf. [Nem2, (4.4)].

Let (U; b, h, V) be a variation structure, let U_{λ} ⊂ U be the generalized λ–eigenspace of h, and for each integer k ≥ 1 consider U_{λ}^{(k)} := ker((h−λI)^{k}|U_{λ}). On U_{λ}^{(k)}/U_{λ}^{(k−1)} one defines a (±1)–hermitian form by B_{λ}^{(k)}(x, y) = B(x, λ^{1−k}(h − λI)^{k−1}y), where B(x, y) = b(x)(ȳ). Let n_{λ}^{(k)} be the dimension of U_{λ}^{(k)}, while σ_{λ}^{(k)} the signature of B_{λ}^{(k)}. Then the collection of the integers {n_{λ}^{(k)}}_{k,λ} is equivalent with the collection of higher order Alexander polynomials, while the collection of integers {n_{λ}^{(k)}, σ_{λ}^{(k)}}_{k,λ} characterizes completely the variation structure (i.e the real Seifert form).

5. Some examples

5.1. Let us consider a (right-handed) trefoil with non-degenerate Seifert matrix

S = (\begin{matrix}−1 & 0 \\ −1 & −1 \end{matrix}) .

The variation matrix V = (S^{T})^{−1} and monodromy matrix h = V · (V^{T})^{−1} = V · S are

V = (\begin{matrix}−1 & 1 \\ 0 & −1 \end{matrix}) h = (\begin{matrix}0 & −1 \\ 1 & 1 \end{matrix}) .

The eigenvalues of h are λ_{1} = \frac{1}{2} − \frac{1}{2}i\sqrt{3} and λ_{2} = \frac{1}{2} + \frac{1}{2}i\sqrt{3}. We need to diagonalise h. Let us put

A = −\frac{1}{i\sqrt[4]{3}} (\begin{matrix}−\frac{1}{2} − \frac{1}{2}i\sqrt{3} & −1 \\ \frac{1}{2} − \frac{1}{2}i\sqrt{3} & 1 \end{matrix}) .

Then AhA^{−1} is diagonal with diagonal entries λ_{1} and λ_{2}, and

AV Ā^{T} = \frac{1}{\sqrt{3}} (\begin{matrix}−\frac{3}{2} + i\frac{1}{2} \sqrt{3} & 0 \\ 0 & −\frac{3}{2} − i\frac{1}{2} \sqrt{3} \end{matrix}) = (\begin{matrix}−i(λ_{1} − 1) & 0 \\ 0 & i(λ_{2} − 1) \end{matrix}) .

Thus the HVS of a treefoil is

V = V_{λ_{1}}^{1}(−1) ⊕ V_{λ_{2}}^{1}(+1).

The spectrum is \{\frac{5}{6}, \frac{7}{6}\}.

5.2. Let us consider a knot 8_{20}. We have.

S = (\begin{matrix}−1 & −1 & −1 & −1 \\ 0 & 0 & −1 & −1 \\ 0 & −1 & 0 & −1 \\ 0 & 0 & −1 & 0 \end{matrix}) , V = (S^{T})^{−1} = (\begin{matrix}−1 & 0 & 0 & 0 \\ 0 & 1 & 0 & −1 \\ 1 & −1 & 0 & 0 \\ 1 & −1 & −1 & 1 \end{matrix}) .

And

h = V · S = (\begin{matrix}1 & 1 & 1 & 1 \\ 0 & 0 & 0 & −1 \\ −1 & −1 & 0 & 0 \\ −1 & 0 & −1 & 1 \end{matrix}) .

The monodromy h has eigenvalues λ_{1} = \frac{1}{2} − \frac{1}{2}i\sqrt{3} and λ_{2} = \frac{1}{2} + \frac{1}{2}i\sqrt{3}. It has two Jordan blocks of size 2. Let A be such matrix that AhA^{−1} is in the Jordan form. E.g.:

A = (\begin{matrix}−1 & 2 & −2 & \frac{3}{2} − \frac{3}{2} \sqrt{3}i \\ \sqrt{3}i & \sqrt{3}i & −\frac{3}{2} + \frac{1}{2} \sqrt{3}i & 0 \\ 1 & −2 & 2 & −\frac{3}{2} − \frac{3}{2} \sqrt{3}i \\ \sqrt{3}i & \sqrt{3}i & \frac{3}{2} + \frac{1}{2} \sqrt{3}i & 0 \end{matrix}) .

Then we have

W = AV \overline{A}^{T} = (\begin{matrix}\frac{3}{2} − \frac{1}{6}i\sqrt{3} & \frac{1}{2} − \frac{1}{2}i\sqrt{3} & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & \frac{3}{2} + \frac{1}{6}i\sqrt{3} & \frac{1}{2} + \frac{1}{2}i\sqrt{3} \\ 0 & 0 & 1 & 0 \end{matrix}) .

Then W_{12} = (−1) · (λ_{1} − 1), W_{34} = (−1) · (λ_{2} − 1). Now the size of each Jordan block is k = 2 and i^{k} = −1. Hence both signs in the direct sum decompositions are ’+’ and

V = V_{λ_{1}}^{2}(+1) ⊕ V_{λ_{2}}^{2}(+1).

This is with the agreement with the fact that the Tristram–Levine signature of W is zero, only it is +1 at ζ = λ_{1,2}. The knot 8_{20} is also reversible.

5.3. Consider, for any n ≥ 1, the following link

n full twists

This link represents two unlinks with linking number n. The shaded part between the two strands forms a Seifert surface of genus 1. The Seifert matrix in (n). Hence the variation structure is simply Ṽ_{1}^{1}(−1) and does not depend on n. Therefore we do not see the linking numbers, or the integral Seifert form from the H–numbers.

5.4. According to [CL], there are five knots with up to 12 crossings with the Alexander polynomial

1 − 4t + 10t^{2} − 16t^{3} + 19t^{4} − 16t^{5} + 10t^{6} − 4t^{7} + t^{8} = (t − μ)^{4}(t − μ̄)^{4},

where μ = e^{πi/3}. These are L_{1} = 10_{99}, L_{2} = 12_{n106}, L_{3} = 12_{n508}, L_{4} = 12_{n604} and L_{5} = 12_{n666}. Their monodromy matrices (S^{T})^{−1}S are respectively h_{1}, h_{2}, h_{3}, h_{4} and h_{5}.

Here we consider h_{1} and h_{2}:

h_{1} = (\begin{matrix}0 & 0 & −1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & −1 & 0 \\ 2 & 1 & 2 & −1 & 0 & −1 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 1 & −1 \\ −1 & −1 & −1 & 1 & 1 & 1 & −1 & 1 \\ 1 & 0 & 1 & −1 & 0 & 0 & 0 & 0 \\ −1 & 1 & 0 & 0 & 0 & 0 & 1 & −1 \\ 0 & 0 & 0 & 0 & −1 & 0 & 0 & 0 \end{matrix})

h_{2} = (\begin{matrix}0 & −1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & −1 & 0 & −1 & −1 & −1 & −1 \\ 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & −1 & 1 & −1 & −1 & −1 & −1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & −1 \\ −1 & −1 & 0 & −1 & 1 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 & −1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 & −1 & 0 & 0 \end{matrix})

The matrix h_{1} has four Jordan blocks of size two, we easily get in this case

p_{μ}^{2}(+1) = p_{μ}^{2}(−1) = p2 ; μ̄(+1) = p2 ; μ̄(−1) = 1.

The matrix h_{2} has a single Jordan block for each eigenvalue. We have

p_{μ}^{4}(−1) = p4 ; μ̄(−1) = 1.

The matrices h_{3} and h_{5} have two Jordan blocks of size one and two of size 3. We conclude that

p_{μ}^{1}(−1) = p_{μ}^{3}(−1) = p1 ; μ̄(+1) = p3 ; μ̄(+1) = 1.

In case of h_{4} we have similarly two Jordan blocks of size one and two of size 3. We can compute that

p_{μ}^{1}(+1) = p_{μ}^{3}(+1) = p1 ; μ̄(−1) = p3 ; μ̄(−1) = 1.

Hence, only the knots 12_{n508} and 12_{n666} are undistinguishable by H–numbers.

Observe that if we take a connected sum of three left–handed treefoils and one right– handed one, then the Alexander polynomial and the signatures for ζ ≠ μ, μ̄, shall be the same as in the case of the knots 12_{n508} and 12_{n666}, but the Jordan block structure is different.

6. Skein relations for H–numbers

6.1. Signatures. Although we do not have a precise Skein relation for H–numbers, there are several constrains from them, coming mostly from relations for classical invariants.

As usually in skein relation, we consider the three links L_{0}, L_{+} and L_{−}. Their Seifert matrices S_{0}, S_{+} and S_{−} can be chosen (see [Kau, Proof of Theorem 7.10]) so that S_{+} and S_{−} are (n + 1) × (n + 1) matrices, such that

(6.1.1)  S_{+} − S_{−} = (\begin{matrix}& & & 0 \\ & \underline{0} & & ⋮ \\ & & & 0 \\ 0 & … & 0 & 1 \end{matrix}) ,

where \underline{0} denotes an n × n zero matrix.

Moreover S_{0} arises from S_{+} by deleting the (n + 1)-st row and (n + 1)-st column.

The following fact is classical (see [Kaw, Lemma 12.3.4] or [Mur1]).

Proposition 6.1.2. For any ζ ∈ S^{1} \ {1}, we have the following bounds

|σ_{L_{±}}(ζ) − σ_{L_{0}}(ζ)| + |n_{L_{±}}(ζ) − n_{L_{0}}(ζ)| ≤ 1.

Proof. We follow the proof of [Kaw]. For fixed ζ, let M_{+}, M_{−} and M_{0} denote the forms (4.4.2) for S_{+}, S_{−} and S_{0}, respectively. Let p_{+}, p_{−}, p_{0}, q_{+}, q_{−}, q_{0} and n_{+}, n_{−}, n_{0} be the maximal dimension of subspaces on which M_{+}, M_{−}, M_{0} are, respectively, positive definite, negative definite and zero. As S_{0} is submatrix of S_{+}, M_{0} is a restriction of M_{+} onto an n−dimensional subspace. It follows that

p_{0} ≤ p_{+} ≤ p_{+} + 1
q_{0} ≤ q_{+} ≤ q_{+} + 1
n_{0} ≤ n_{+} ≤ n_{+} + 1.

As p_{0} + q_{0} + n_{0} + 1 = p_{+} + q_{+} + n_{+}, the statement follows. □

The above proposition and (4.4.8) give restriction for possible H–numbers of L_{±} and L_{0}, when the primitive numbers of one of them are known.

6.2. Semicontinuity of the extended spectrum. The inequality of (6.1.2) can be used to prove a variant of the semicontinuity of spectra.

Here some comments are in order. The semicontinuity property of (genuine) spectrum of hypersurface singularities says the following: if {f_{t} : t ∈ (ℂ, 0)} is a family of isolated singularities, then for any interval I = [α, α+1) the spectral numbers {Spec(f_{t})}_{t} associated with f_{t} satisfies: # Spec(f_{0}) ∩ I ≥ # Spec(f_{t≠0}) ∩ I, see [Steen2, Var]. The semicontinuity principle is codified in the very geometric substance of the deformation. In particular, several invariants behave semicontinuously, e.g. for the Milnor number μ(f_{0}) ≥ μ(f_{t≠0}). If one tries to generalize this phenomenon to the level of some surgeries, one needs to assume that the geometric situation mimics in the right way the presence of the deformation. In the next proposition we will assume that deg Δ (i.e. the Milnor number in algebraic case) is controlled.^{1}

Proposition 6.2.1. Let L_{1} and L_{2} are two links. Let ESp_{1} and ESp_{2} be the corresponding extended spectra. Let Δ_{L_{i}} (i = 1, 2) be the characteristic polynomial of the monodromy operator associated with the non–degenerate part of the Seifert form; in other words, the first non–zero higher order Alexander polynomial. (If the Seifert forms are non–degenerate then Δ_{L_{i}} is just the Alexander polynomial. See subsections (4.1) and (4.2).)

Assume also that x ∈ (0, 1) is such that ∂H_{x} ∩ (ESp_{1} ∪ ESp_{2}) = ∅ (where H_{x} = (x, x + 1) × iℝ as above). Moreover assume that one of the following holds.

(a) L_{1} arises from L_{2} by changing a negative (or left-handed) crossing to positive (or right-handed) crossing (see [Liv1, Exercise 3.2.5] for the necessary definitions) and deg Δ_{L_{1}} ≥ deg Δ_{L_{2}};

(b) L_{1} arises from L_{2} by changing one crossing and deg Δ_{L_{1}} > deg Δ_{L_{2}};

(c) L_{1} arises from L_{2} by a hyperbolic transformation (i.e. L_{1} and L_{2} can play a role of L_{0} and L_{∞} at some diagram, see [Kaw, Definition 12.3.3]) and deg Δ_{L_{1}} > deg Δ_{L_{2}}.

Then

#ESp_{1} ∩ H_{x} ≥ #ESp_{2} ∩ H_{x}.

1[MB]: I changed ’decreasing’ to ’controlled’, since we deal with two links L_{1} and L_{2}

Proof. Let a_{1} = #ESp_{1} ∩ H_{x}, a_{2} = #ESp_{2} ∩ H_{x}, b_{1} = #ESp_{1} \ H_{x}, b_{2} = #ESp_{2} \ H_{x}. Then by Corollary 4.4.9(b) we have

a_{1} + b_{1} = deg Δ_{L_{1}}  a_{2} + b_{2} = deg Δ_{L_{2}}
−a_{1} + b_{1} = σ_{L_{1}}(e^{2πix})  −a_{2} + b_{2} = σ_{L_{2}}(e^{2πix}).

Thus

a_{1} − a_{2} = \frac{1}{2} (deg Δ_{L_{1}} − deg Δ_{L_{2}} − σ_{L_{1}}(e^{2πix}) + σ_{L_{2}}(e^{2πix})) .

Now, in the case (a), as deg Δ_{L_{1}} > deg Δ_{L_{2}} and L_{1} and L_{2} have the same number of components, the degrees differ at least by 2. The signatures cannot differ by more than 2 by Proposition 6.1.2. In the case (b), signature of L_{1} is not larger than that of L_{2}, in case (c), both degree of Alexander and signature cannot differ by more than one. □

The above results is enough to prove a variant of Theorem 6.7 from [Bo] with ESp ∩ H_{x} instead of Tristram–Levine signatures. Since for algebraic links the ESp is the same as the ordinary spectrum, we can relate the spectra of singularities of a plane curve with the spectrum of the singularity at infinity. This subject involves mathematical machineries not discussed in this article, and it will be investigated in a forthcoming paper.

Remark 6.2.2. ^{2} In singularity theory the signature is not semicontinuous, see e.g. [KN]. Similar behavior can be observed in the knot theory: the condition deg Δ_{L_{1}} > deg Δ_{L_{2}} (in the notation from Proposition 6.2.1) alone is not sufficient to determine the sign of σ_{L_{1}}(ζ) − σ_{L_{2}}(ζ), so we do not have a strong ’semicontinuity property’ for signatures of knots.

6.3. Higher Alexander polynomials and rational Nakanishi index. Next, we wish to connect the higher order Alexander polynomials associated with S_{±} and S_{0}, cf. (6.1). In order to formulate the result, we need to introduce some additional notation.

Convention 6.3.1.

(a) Let us fix λ ∈ ℂ for this section. For any matrix H with coefficients in ℂ[t] we define

d(H) = ord_{t=λ} det H.

(b) For any m × m matrix K and 1 ≤ i, j ≤ m, we define K^{i,j} as the (m − 1) × (m − 1) minor resulting from K by removal of i−th row and j−th column.

The next easy lemma will be important in the sequel.

Lemma 6.3.2. For any matrix H of size m × m and for any 1 ≤ j ≤ m one has

d(H) ≥ min{d(H^{i,j}): 1 ≤ i ≤ m}.

Proof. det H = ∑_{i}(−1)^{i+j}H_{i,j} det H^{i,j} by the Laplace expansion of det H along the j−th column. (Recall that H_{i,j} denotes the element of H at i−th row and j−th column.) It follows that, if all det H^{i,j} are divisible by (t − λ)^{d}, so will be det H. □

The definition of d(H) is motivated by the higher order Alexander polynomials. Namely, if S is an m×m Seifert matrix and H = S −tS^{T}, then the multiplicity of a root of the k−th Alexander polynomial (see Definition 4.2.1) of H at t = λ can be expressed as the minimum of d(K), where K runs through all (m − k) × (m − k) minors of H:

(6.3.3)  ord_{t=λ} Δ_{k}^{H} = min{d(K): K is an (m − k) × (m − k) minor of H}.

2[MB] I added this remark

For Seifert matrices S_{+}, S_{−} and S_{0} of links L_{+}, L_{−} and L_{0} (notation from Section 6.1) let us define H_{∗} = S_{∗} − tS_{∗}^{T} (’∗’ is one of ’+’, ’−’ or ’0’) and

d_{k}^{∗} = ord_{t=λ} Δ_{k}^{H_{∗}}.

An immediate consequence of Lemma 6.3.2 is that for any k and ∗ ∈ {+, −, 0}:

(6.3.4)  d_{k}^{∗} ≥ d_{k+1}^{∗}

The skein relation gives the following restrictions for values of d_{k}^{∗}.

Proposition 6.3.5. The integers d_{k}^{+}, d_{k}^{−} and d_{k}^{0} satisfy the following relations:

(6.3.6a)  d_{k}^{0} ≥ d_{k+1}^{±}
(6.3.6b)  d_{k}^{±} ≥ d_{k+1}^{0}
(6.3.6c)  d_{k}^{±} ≥ d_{k+1}^{∓}  if λ ≠ 1
(6.3.6d)  d_{k}^{±} ≥ min(d_{k+1}^{∓} + 1, d_{k}^{∓})  if λ = 1.

Proof. Let m be the size of H_{0}. H_{0} can be regarded as an m × m minor of both H_{+} and H_{−} (cf. (6.1.1)). Then any (m − k) × (m − k) minor of H_{0} is also an ((m + 1) − (k + 1)) × ((m + 1) − (k + 1)) minor of H_{+} and of H_{−}. It follows that d_{k+1}^{±} ≤ d_{k}^{0}, because in d_{k+1}^{±} we take a minimum over larger set. Equation (6.3.6a) follows.

As for (6.3.6b), we can divide all possible (m + 1 − k) × (m + 1 − k) minors of H_{+} in three categories.

• A_{α}^{k}, α ∈ A, will denote minors lying entirely in H_{0};

• B_{β}^{k}, β ∈ B, will denote minors containing a part of the last column or row of H_{+} but not containing the corner;

• C_{γ}^{k}, γ ∈ C, will denote minors containing the element of H_{+} lying in (m + 1)st row

and (m + 1)st column. Graphically we can present these minors like that

[  ]

H_{+}

[  ]

H_{0}

A B

C

By (6.3.3)one has (where d(α) = d(A_{α}^{k}), and similarly for d(β) and d(γ)):

(6.3.7)  d_{k}^{+} = min(min_{α∈A} d(α), min_{β∈B} d(β), min_{γ∈C} d(γ)).

We need to show that all three minima are greater than d_{k+1}^{0}.

First of all min_{α∈A} d(α) is precisely d_{k−1}^{0}.

Let B_{β}^{j,m+1−k} be the minor of H_{+} resulting by removing the last column and j−th row from B_{β} (assuming that B_{β} contains a part of (m + 1)st column of H_{+} as on the picture, not a part of (m + 1)st row). Remark that B_{β}^{j,m+1−k} is an (m − k) × (m − k) minor of H_{0}. By Lemma 6.3.2 we have

d(B_{β}) ≥ min_{j=1,...,m+1−k} d(B_{β}^{j,m+1−k}).

Thus

min_{β∈B} d(B_{β}) ≥ d_{k}^{0}.

In order to deal with d(C_{γ}) we need to do expand det C_{γ} first along the last row and then along the last column. Applying Lemma 6.3.2 twice we get

d(C_{γ}) ≥ min_{1≤j,j′≤m+1−k} d(C_{γ}^{(j,m+1−k),(j′,m+1−k)}),

where C_{γ}^{(j,m+1−k),(j′,m+1−k)} arises from C_{γ} by deleting j−th and (m+1−k)−th column and j−th and (m+1−k)−th row. It is thus an (m−(k+1))×(m−(k+1)) minor of H_{+} and also of H_{0}, because it does not contain neither the last column, nor the last row of H_{+}. Hence, d(C_{γ}) ≥ d_{k+1}^{0}. Finally, using (6.3.3) and (6.3.7) we obtain d_{k}^{+} ≥ min(d_{k−1}^{0}, d_{k}^{0}, d_{k+1}^{0}) = d_{k+1}^{0}. Hence the inequality (6.3.6b) is proved.

In order to prove the last two inequalities, let us consider two (m + 1 − k) × (m + 1 − k) minors K_{+} and K_{−} of H_{+} and H_{−} obtained by removing the same columns and the same rows from matrices H_{+} and H_{−}.

As H_{+} − H_{−} is a matrix with (t − 1) in the place (m + 1, m + 1), and zeros everywhere else, K_{+} = K_{−} unless they contain the element at the bottom right corner of H_{+} and H_{−}. If they do not contain,

det K_{+} = det K_{−}, so d(K_{+}) = d(K_{−})

If they do,

det K_{+} = det K_{−} + (t − 1) det K_{0},

where K_{0} = K_{±}^{m+1−k,m+1−k} arises from K_{±} by removing the last column and the last row. In this case we deduce that

d(K_{+}) ≥ min(d(K_{−}), d(K_{0}) + s),

where s = 1 if λ = 1 and 0 otherwise.

This shows in particular that

d_{k}^{+} ≥ min(d_{k}^{−}, s + d_{k}^{0}).

Now it is enough to observe that by (6.3.6b) d_{k}^{0} ≥ d_{k−1}^{−}. □

In order to apply this skein relation, let us fix λ with 0 < |λ| ≤ 1 and consider the set Θ (defined after the proof of Lemma 4.2.2) associated with λ. For any N ≥ 1 set

(6.3.8)  P_{N} = #{θ ∈ Θ : θ ≥ N}.

P_{N} can be interpreted as the number of the Jordan blocks of size at least N with eigenvalue λ of the monodromy matrix h; i.e., with the notation of (4.2.3), one has:

P_{N} = ∑_{k≥N} s_{k}(λ).

Corollary 6.3.9. Fix some λ. Let P_{N}^{+} and P_{N}^{−} be the P_{N} numbers associated to the links L_{+} and L_{−} respectively as in (6.3.8). Then for any N ≥ 2 one has

|P_{N}^{+} − P_{N}^{−}| ≤ 2N

while for N = 1, |P_{1}^{+} − P_{1}^{−}| ≤ 1.

Proof. For N > 1, assume that P_{N}^{+} − P_{N}^{−} = a > 0. By Proposition 4.2.4 for any i < P_{N}^{+} we get

(6.3.10)  d_{i}^{+} − d_{i+1}^{+} ≥ N,

and for i ≥ P_{N}^{−}

(6.3.11)  d_{i}^{−} − d_{i+1}^{−} ≤ N − 1.

Therefore, we obtains the next sequence of inequalities:

d_{P_{N}^{+}−1}^{+} ≥^{(∗)} d_{P_{N}^{+}}^{−} ≥^{(∗∗)} d_{P_{N}^{−}}^{−} − a(N − 1) ≥^{(∗)}

≥ d_{P_{N}^{−}+1}^{+} − a(N − 1) ≥^{(∗∗∗)} d_{P_{N}^{+}−1}^{+} + (a − 2)N − a(N − 1).

Here the inequalities denoted by (∗) follow from (6.3.6c), (∗∗) from (6.3.11) and (∗ ∗ ∗) from (6.3.10). Hence (a − 2)N − a(N − 1) ≤ 0, or 2N ≥ a.

So now assume that N = 1. Then P_{1}^{±} = min{i > 0: d_{i}^{±} = 0}. So let us take k such that d_{k}^{−} > 0 = d_{k+1}^{−} (i.e. P_{1}^{−} = k + 1). By (6.3.6c) we have d_{k−1}^{+} ≥ d_{k}^{−}, hence d_{k−1}^{−} > 0, so P_{1}^{+} ≥ k. The argument follows from symmetry. □

For knots, the maximum of the values P_{1} for all λ ≠ 1 is, by Proposition 4.3.3, equal to the Nakanishi index n_{ℚ}. Therefore, Corollary 6.3.9 implies that

Corollary 6.3.12. Let K_{+} and K_{−} be two knots differing by one change of crossing. Then

|n_{ℚ}(K_{+}) − n_{ℚ}(K_{−})| ≤ 1.

In particular, we reprove a theorem of Nakanishi in a weaker version:

Corollary 6.3.13. The rational Nakanishi index of a knot K is bounded from above by the unknotting number of K.

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Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warsaw, Poland

   E-mail address: mcboro@mimuw.edu.pl

A. Rényi Institute of Mathematics, 1053 Budapest, Reáltanoda u. 13-15, Hungary.

   E-mail address: nemethi@renyi.hu