The signature of a splice
We study the behavior of the signature of colored links [Flo05, CF08] under the splice operation. We extend the construction to colored links in integral homology spheres and show that the signature is almost additive, with a correction term independent of the links. We interpret this correction term as the signature of a generalized Hopf link and give a simple closed formula to compute it.
The splice of two links is an operation defined by Eisenbud and Neumann in [EN85], which generalizes several other operations on links such as connected sum, cabling, and disjoint union. The precise definition is given in Section 2.1 (see Definition 2.1), but the rough idea is as follows: the splice of two links and along the distinguished components and is the link in the -manifold obtained by an appropriate gluing of the exteriors of and . There has been much interest in understanding the behavior of various link invariants under the splice operation. For example, the genus and the fiberability of a link are additive, in a suitable sense, under splicing [EN85]. The behavior of the Conway polynomial has been studied in [Cim05], and more recently the relation between the -spaces in Heegaard–Floer homology and splicing has been addressed in [HL12]. The goal of this paper is to obtain a similar (non-)additivity statement for the multivariate signature of oriented colored links. As a consequence, we show that the conventional univariate Levine–Tristram signature of a splice depends on the multivariate signatures of the summands.
In Section 3.2 we define the signature of a colored link in an integral homology sphere. This is a natural generalisation of the multivariate extension of the Levine-Tristram signature of a link in the -sphere, considered in [Flo05, CF08]. The principal result of the paper is Theorem 2.2, expressing the signature of the splice of two links in terms of the signatures of the summands. We show that the signature is almost additive: there is a defect, but it depends only on some combinatorial data of the links (linking numbers), and not on the links themselves. Geometrically, this defect term appears as the multivariate signature of a certain generalized Hopf link, which is computed in Theorem 2.10. At the end of Section 2, we discuss a few applications of Theorem 2.2 and relate it to some previously known results: namely, we compute the signature of a satellite knot (see Section 2.4 and Theorem 2.12) and that of an iterated torus link (see Section 2.5 and Theorem 2.13). More precisely, we reduce the computation to the signature of cables over the unknot. We also show that the multivariate signature of a link can be computed by means of the conventional Levine–Tristram signature of an auxiliary link (see Section 2.6 and Theorem 2.15).
The paper is organized as follows. Section 2 is devoted to the detailed statement of main results, and the computation of the defect. In Section 3, we introduce the necessary background material on twisted intersection forms and construct the signature of colored links in integral homology spheres. The proofs of the main theorems are carried out in Section 4 and Section 5, where the signature of the generalized Hopf links is computed.
We would like to thank S. Orevkov, who brought the problem to our attention. We are also grateful to the anonymous referees of this paper who corrected a mistake in the original version of Corollary 2.6 and a sign in Theorems 2.2 and 2.10; Example 2.5 was also suggested by a referee. This work was partially completed during the first and third authors’ visits to the University of Pau, supported by the CNRS, and the first author’s visit to the Abdus Salam International Centre for Theoretical Physics.
2. Principal results
2.1. The set-up
A -colored link is an oriented link in an integral homology sphere equipped with a surjective function , referred to as the coloring. The union of the components of given the same color is denoted by .
The signature of a -colored link is a certain -valued function defined on the character torus
see Definition 3.5 below for details. We let . Note that is an abelian group. If , the link is monochrome and coincides with the restriction (to rational points) of the Levine–Tristram signature [Tri69] (whose definition in terms of Seifert form extends naturally to links in homology spheres). Given a character and a vector , we use the common notation .
Often, the components of are split naturally into two groups, , on which the coloring takes, respectively, and values, . In this case, we regard as a function of two “vector” arguments . We use this notation freely, hoping that each time its precise meaning is clear from the context.
Clearly, in the definition of colored link, the precise set of colors is not very important; sometimes, we also admit the color . As a special case, we define a -colored link
as a -colored link in which is the only component given the distinguished color . Here, we assume connected; this component, considered distinguished, plays a special role in a number of operations.
In the following definition, for a -colored link , or , we denote by a small tubular neighborhood of disjoint from and let be, respectively, its meridian and longitude. (The latter is well defined as is a homology sphere.)
Given two -colored links , or , their splice is the -colored link in the integral homology sphere
where the gluing homeomorphism takes and to and , respectively.
2.2. The signature formula
Given a list (vector, etc.) , the notation designates that the -th element (component, etc.) has been removed. The complex conjugation is denoted by . The same notation applies to the elements of the character torus , where we have .
The linking number of two disjoint oriented circles , in an integral homology sphere is denoted by , with omitted whenever understood. For a -colored link , we also define the linking vector , where .
The index of a real number is defined via . The -function sends to . This function extends to via ; in other words, we specialize each argument to the interval and add the arguments as real numbers (rather than elements of ) afterwards. For any integral vector , , we define the defect function
For short, if for all , we simply denote the defect , and omit the subscript. The reader is referred to Figure 1 for a few examples of the defect function on .
The following statement is the principal result of the paper.
For or , consider a -colored link , and let be the splice of the two links. For characters , introduce the notation
Then, assuming that , one has
Eisenbud and Neumann [EN85, Theorem 5.2] showed that the Alexander polynomial is multiplicative under the splice. For a -colored link , we denote the Alexander polynomial of . Similar to Theorem 2.2, let . One has
unless (ie. is a knot) and , in which case
Note that this formula were refined by Cimasoni [Cim05] for the Conway potential function. Moreover, in relation with the signature of a colored link, one may consider the nullity, related to the rank of the twisted first homology of the link complement. This nullity is also additive under the splice operation, in the suitable sense. Detailed statements can be found in [DFL].
Consider two copies and of the (1,1)-colored generalized Hopf link , see Section 2.3, where and are the single components. Then, is a (1,1)-colored link, and for , we show by using C-complexes that
This illustrates trivially that a defect appears.
For the reader convenience we add the following example. Notice the use of the formula in Theorem 2.2 when (cf. Remark 3.6). Let be the (2,4)-torus link and be the (4,2)-cable over the unknot with the core retained (cf. Section 2.5). Then, the splice of these two links along the components and is the (3,6)-torus link, which we shall denote .
In the notation of Theorem 2.2, we have and . For the -complexes bounded by these three links one can take those depicted in Figure 2. To simplify the resulting Hermitian matrices , we re-denote by their arguments (in the order listed) and, for an index set , introduce the shortcut . Then
so that, up to units and factors of the form , , the Alexander polynomials are
The computation of the signature of these matrices is straightforward: on the respective open tori, they are the piecewise constant functions given by the following tables:
Note, however, that is the unknot and is homeomorphic to ; hence,
Now, it is immediate that the identity
given by Theorem 2.2 holds whenever or . (It suffices to compare the values at all triples of 8-th roots of unity.) If , we obtain an extra discrepancy of ; this phenomenon will be explained in [DFL].
As an immediate consequence of Theorem 2.2, we see that the Levine–Tristram signature of a splice cannot be expressed in terms of the Levine–Tristram signature of its summands: in general, the multivariate extension is required.
Let be the splice of -colored links and , and denote and . Consider as a -colored link. Then, for a character such that , one has
where is the Levine–Tristram signature of .
Let be the splice of a -colored link and a -colored link , and let . Then, for any character , one has
The assumption in Theorem 2.2 is essential. If , the expression for the signature acquires an extra correction term, which can be proved to take values in . In many cases, this term can be computed algorithmically, and simple examples show that typically it does not vanish. Indeed, consider two copies of the Whitehead link and . If , then , but and there is a non-zero extra term. (Addendum 2.7 states that the extra term does vanish whenever one of the links , is empty.) The general computation of this extra term, related to linkage invariants (see, e.g., [Mur70]), is addressed in a forthcoming paper [DFL].
We expect that the conclusion of Theorem 2.2 would still hold without the assumption that the characters should be rational. In fact, all ingredients of the proof would work once recast to the language of local systems, and the main difficulty is the very definition of the signature in homology spheres, where the link does not need to bound a surface and the approach of [CF08] does not apply. (If all links are in , an alternative proof can be given in terms of -complexes.) This issue will also be addressed in [DFL].
2.3. The generalized Hopf link
A generalized Hopf link is the link obtained from the ordinary positive Hopf link by replacing its components and with, respectively, and parallel copies. This link is naturally -colored; its signature, which plays a special role in the paper is given by Theorem 2.10 below. Observe the similarity to the correction term in Theorem 2.2; a posteriori, Theorem 2.10 can be interpreted as a special case of Theorem 2.2, using the identity (which is easily proved independently) and the fact that is the splice of and . However, the Hopf links and their signatures are used essentially in the proof of Theorem 2.2.
For any character , one has .
Certainly, Theorem 2.10 computes as well the signature of a generalized Hopf link equipped with an arbitrary coloring and orientation of components. First, one can recolor the link by assigning a separate color to each component (cf. Proposition 3.7 below). Then, one can reverse the orientation of each negative component ; obviously, this operation corresponds to the substitution . For example, the orientation of the original link can be described in terms of a pair of vectors, viz. the linking vector of the -part of with the -component of the original Hopf link and the linking vector of the -part with the -component. Then, assuming that any two linked components of are given distinct colors, we have
For future references, we state a few simple properties of the defect function and, hence, of the signature . All proofs are immediate.
The defect function has the following properties:
; if or ;
for all ;
is preserved by the coordinatewise action of the symmetric group ;
commutes with the coordinate embeddings , ;
commutes with the embeddings , for any .
2.4. Satellite knots
As was first observed in [EN85], the splice operation generalizes many classical link constructions: connected sum, disjoint union and satellites among others.
Our first application is Litherland’s formula for the Levine–Tristram signature of a satellite knot, which is a particular case of Addendum 2.7.
Recall that an embedding of a solid torus in into another solid torus in another copy of is called faithful if the image of a canonical longitude of the first solid torus is a canonical longitude of the second one. Let be an unknotted solid torus in , and let be a knot in the interior of , with algebraic winding number , i.e., is times the class of the core in . Given any knot , the satellite knot is defined as the image under a faithful embedding sending the core of to .
The isotopy class depends of course on the embedding (and even its concordance class, see [Lit84]). Nevertheless, its Levine–Tristram signature is determined by the signatures of the constituent knots and the winding number:
Theorem 2.12 (cf. [Lit79, Theorem 2]).
In the notation above, the Levine–Tristram signatures of , and are related via
Let be the core of the solid torus . The satellite can be written as the splice of and . By Addendum 2.7, we have
where . By assumption, , and the statement follows. ∎
2.5. Iterated torus links
Our next application is another special case of Theorem 2.2, which provides an inductive formula for the signatures of iterated torus links. In particular, this class of links contains the algebraic ones, i.e., the links of isolated singularities of complex curves in . Note that partial results on the equivariant signatures of the monodromy were obtained by Neumann [Neu87].
Iterated torus links are obtained from an unknot by a sequence of cabling operations (and maybe, reversing the orientation of some of the components). In order to define the cabling operations (we follow the exposition in [EN85]), consider two coprime integers and (in particular, if one of them is , the other is ), a positive integer , a -colored link , and a small tubular neighbourhood of disjoint from . Let be the meridian and longitude of , and be the oriented simple closed curve in homologous to . More generally, let be the disjoint union of parallel copies of in . We say that the link (resp. ) is obtained from by a -cabling with the core removed (resp. retained).
Let be the ordinary Hopf link. The link can be regarded as either -colored or -colored. We denote the corresponding multivariate and bivariate signature functions by and , respectively. Note that, by Proposition 3.7 below,
In the case of core-removing, the link obtained by the cabling is nothing but the splice of and . (Similarly, in the core-retaining case, is the splice of and .) Hence, the following statement is an immediate consequence of Theorem 2.2.
Let be obtained from a -colored link by a -cabling with the core removed. For a character , let
Then, assuming that , one has
With the evident modifications, this last corollary can be adapted to give a formula for a -cabling with the core retained.
The Levine–Tristram signature of the torus link (which coincides with in our notation) was computed by Hirzebruch. For the reader’s convenience, we cite this result in the next lemma. Unfortunately, we do not know any more general statement.
Lemma 2.14 (see [Bri66]).
Let and let . Consider
Then one has for .
2.6. Multivariate vs. univariate signature
The last application is the computation of the multivariate signature of a link in terms of the Levine–Tristram signature of an auxiliary link. (One obvious application is the case where the latter auxiliary link is algebraic, so that its Seifert form can be computed in terms of the variation map in the homology of its Milnor fiber , see [AGZV88].) This result is similar to [Flo05, Theorem 6.22] by the second author and is related to the computation of signature invariants of -manifolds by Gilmer, see [Gil81, Theorem 3.6].
Let be a -colored link. For simplicity, we assume that the coloring is maximal, i.e., each component of is given a separate color. Let be the linking matrix of , i.e., for and .
Consider a character and assume that , where , for some integers and . (In particular, all .) For , denote
, the weighted linking number of and ;
, where is the -th row of .
Fix an integral vector and consider the monochrome link obtained from by the -cabling along the component for each . In other words, each component of is regarded -fold, and it is replaced with “simple” components, possibly linked (if ).
In the notation above, one has the identity
If , the second term in Theorem 2.15 vanishes and one has
For small values of , this identity simplifies even further:
if , then ;
if and , then .
Proof of Corollary 2.16.
Proof of Theorem 2.15.
Denote and, for , let be the link obtained from by the -cabling along the component . Each link is naturally -colored; we assign to this link the character . In this notation, is the monochrome version of and, by Proposition 3.7,
Introduce the following characters:
, obtained from by replacing each with copies of , if , or copies of , if ;
, obtained from by replacing each with copies of .
By definition, is the splice of and . Then Theorem 2.2 applies and, for each ,
We have ; since , this implies
One can show that . Indeed, is obtained from by operations of replacement of a single copy of with copies of for all ; as in (2.5), one such operation increases the value of by . The character has all entries equal to or , with the exponent sum equal to . Using Lemma 2.11(5) and (3) to cancel the pairs , , we get ; hence,
and the statement of the theorem follows from (2.3). ∎
3. Signature of a link in a homology sphere
In the early sixties Trotter introduced a numerical knot invariant called the signature [Tro62], which was subsequently extended to links by Murasugi [Mur70]. This invariant was generalized to a function (defined via Seifert forms) on by Levine and Tristram [Tri69, Lev69]. It was then reinterpreted in terms of coverings and intersection forms of -manifolds by Viro [Vir73, Vir09]. Our definition of the signature of a colored link follows Viro’s approach and the -signature theorem, see also [GLM81, Flo05].
3.1. Twisted signature and additivity
We start with recalling the definition and some properties of the twisted signature of a -manifold.
Let be a compact smooth oriented -manifold with boundary and a finite abelian group. Fix a covering , possibly ramified, with the group of deck transformations. If the covering is ramified, we assume that the ramification locus is a union of smooth compact surfaces such that
each surface is transversal to , and
distinct surfaces intersect transversally, at double points, and away from .
Items (1) and (2) above mean that each component of is a properly embedded surface. For short, a compact surface satisfying all conditions (1)–(3) will be called properly immersed. Under these assumptions, is an oriented rational homology manifold and we have a well-defined Hermitian intersection form