Immersed concordances of links

Immersed concordances of links and Heegaard Floer homology

Maciej Borodzik Institute of Mathematics, Polish Academy of Science, ul. Śniadeckich 8, Warsaw, Poland Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warsaw, Poland  and  Eugene Gorsky Department of Mathematics, UC Davis, One Shields Avenue Davis CA 95616 USA National Research University Higher School of Economics, Vavilova 7, Moscow, Russia

An immersed concordance between two links is a concordance with possible self-intersections. Given an immersed concordance we construct a smooth four-dimensional cobordism between surgeries on links. By applying -invariant inequalities for this cobordism we obtain inequalities between the -functions of links, which can be extracted from the link Floer homology package. As an application we show a Heegaard Floer theoretical criterion for bounding the splitting number of links. The criterion is especially effective for L-space links, and we present an infinite family of L-space links with vanishing linking numbers and arbitrary large splitting numbers. We also show a semicontinuity of the -function under -constant deformations of singularities with many branches.

Key words and phrases:
L-space link, splitting number, immersed concordance, –invariant, surgery, semigroup of a singular point, Hilbert function
2010 Mathematics Subject Classification:
primary: 57M25, secondary: 14B07, 14H20
The second author was partially supported by the grants DMS-1559338 and RFBR-13-01-00755

1. Introduction

1.1. Overview

An immersed cobordism between two links and in is a smoothly immersed surface in , whose boundary is and . An immersed concordance is an immersed cobordism, whose all the components have genus 0. The notion of an immersed cobordism gives a unified approach for studying smooth four genus, clasp number, splitting number and unlinking number of links. Recently many papers using this technique appeared [3, 5, 19, 25, 27]. Generalizing the construction of [5] we can use an immersed concordance as a starting point in constructing a four-dimensional cobordism between large surgeries on and with precisely described surgery coefficients. Under some extra assumptions we can guarantee that the four-dimensional cobordism is negative definite. We apply the the -invariant inequality of Ozsváth and Szabó, see (4.3), to relate the -invariants of the corresponding surgeries on and . These inequalities are best expressed in terms of the -functions.

The -function is a function that is used to calculate the -invariant of large surgeries on links (see Theorem 4.10, which can be thought of as an informal definition of ). For knots it was first defined by Rasmussen in his thesis [31] (as an analogue of the Frøyshov invariant in Seiberg-Witten theory), who used it to obtain nontrivial bounds for the slice genus of knots. For L-space knots, the -function can be easily reconstructed from the Alexander polynomial. For L-space links with several components (see Section 2.2), the -function was introduced by the second author and Némethi [14] (denoted by small there), who showed that for algebraic links it coincides with the Hilbert function defined by the valuations on the local ring of the corresponding singularity.

Unfortunately, apart from different notations of in the literature there are at least three different “natural” conventions on the definition of , all differing by some shift of the argument. This can be seen in [4], where three different functions , and denote very similar objects. In the link case the situation will be similar. The function called will take as an argument the levels of the Alexander filtration in the chain complex , that is, its arguments will be from some lattice. Shifting the argument of by half the linking numbers will yield a function from to . The normalization of the -function makes it suit very well for studying link concordances. Finally, we will have a function , defined for algebraic singularities, which resembles the most the semigroup counting function from [4] and agrees with the Hilbert function from [14].

We define the -function for general links and find inequalities between the -functions of two links related by an immersed concordance (under some assumptions on the concordance). The following theorem is one of the main results of the paper. The statement is easier in terms of the -function than in terms of the -function.

Theorem (Theorem 6.20).

Let and be two -component links differing by a single positive crossing change, that is, arises by changing a negative crossing of into a positive one. Let and be the corresponding -functions and let , .

  • If the crossing change is between two strands of the same component , then

  • If the crossing change is between the -th and -th strand of , then


As an application we provide new criteria for splitting numbers of links.

Theorem (Theorem 7.7).
  • Suppose that a two component link can be unlinked using positive and negative crossing changes. Let denote the slice genus of . Define vectors

    Define the region by inequalities:

    Then for .

  • If, in addition, is an L-space link, then

    In particular, all coefficients of the Alexander polynomial vanish in .

In the examples we focus on a family of two-bridge links which were shown in [23] to be L-space links.

Theorem (Theorem 7.12).

The splitting number of the two-component two-bridge link

equals , although the linking number between the components of vanishes.

We compare this theorem with the work of Batson and Seed [1] which provides a different bound for the splitting number in terms of Khovanov homology. It turns out their lower bound is quite weak in this case and is at most three for all .

Another application is a topological proof of semicontinuity of the Hilbert function of singularities under -constant deformations. The result was proved in [5] for unibranched singular points (there is also an algebraic proof of a more general version in [14] for one component links). Our result is for multibranched singularities under the assumption that the number of branches does not change.

1.2. Structure of the paper

The paper uses a lot of background facts about Heegaard Floer homology and L-space links, most of them were discussed in [14, 23] using slightly different set of notations. For the reader’s convenience, we repeat these facts and introduce the functions and in full generality in Sections 2 and 3. In Section 4, we relate the Ozsváth-Szabó -invariants of large surgeries on a link to the -function. Section 5 is the technical core of the paper: for an immersed cobordism between two links we construct a cobordism between the surgeries of the 3-sphere on these links, and prove that it is negative definite under certain assumptions. In the negative definite case, we apply the classical inequality for -invariants of , and obtain in Section 6 an inequality for and -functions for the links stated in Theorem 6.1. We use this result to prove Theorem 6.20.

In Section 7 we apply these results to obtain more concrete inequalities for two–component links, and prove Theorems 7.7 and 7.12. Finally, in the last section we apply the inequalities to algebraic links and compare them with the algebro-geometric results on deformations of singularities.

1.3. Notations and conventions

All links are assumed to be oriented. For a link , we denote by its components. This allows us to make a distinction between and . The former denotes two distinct links, the latter stays for two components of the same link .

We will mark vectors in –dimensional lattices in bold, in particular, we will write . Given , we write if for all , and if and . We will write (resp. ) if (resp. ) for all . We denote the -th coordinate vector by .

For a subset and , we denote by the vector . For a link we denote by the corresponding sublink.

We will always work with coefficients.


The authors would like to thank to David Cimasoni, Anthony Conway, Stefan Friedl, Jennifer Hom, Yajing Liu, Charles Livingston, Wojciech Politarczyk and Mark Powell for fruitful discussions. The project was started during a singularity theory conference in Edinburgh in July 2015. The authors would like to thank the ICMS for hospitality.

2. Links and L-spaces

2.1. Links and their Alexander polynomials

Let be a link. Denote by its components. Throughout the paper, the multivariable Alexander polynomial (see [17] for definition) will be symmetric:

The sign of a multivariable Alexander polynomial can be fixed using the interpretation of the Alexander polynomial via the sign refined Reidemeister torsion; see [17, Section 4.9] for discussion and [36] for an introduction to Reidemeister torsion.

Example 2.1.

The Alexander polynomial for the Whitehead link equals

For the Borromean link the Alexander polynomial equals

In some examples we will consider algebraic links, defined as intersections of complex plane curve singularities with a small 3-sphere. The Alexander polynomials of algebraic links were computed by Eisenbud and Neumann [12]. In Section 8 below we also discuss more recent results of Campillo, Delgado and Gusein-Zade [6], relating the Alexander polynomial to the algebraic invariants of a singularity, such as the multi-dimensional semigroup.

Example 2.2.

The link of the singularity consists of 2 unknots with linking number . The corresponding Alexander polynomial equals

For future reference we recall the Torres formula, proved first in [34]. It relates the Alexander polynomial of a link with the Alexander polynomial of its sublink.

Theorem 2.3 (Torres Formula).

Let be an component link and let . The Alexander polynomials of and of are related by the following formula.

where is the linking number between and .

2.2. L-spaces and L-space links

We will use the minus version of the Heegaard Floer link homology, defined in [30]. To fix the conventions, we assume that is supported in degrees . To every 3-manifold this theory associates a chain complex which naturally splits as a direct sum over Spin structures on : . The homology , as a graded -module, is a topological invariant of .

Definition 2.4.
  • A 3-manifold is called an L-space if and its Heegaard Floer homology has minimal possible rank: for all .

  • A link is called an L-space link if , the integral surgery of on the components of with coefficients , is an L-space for .

For a link and a vector we define the framing matrix :


It is well known that if then . We recall the following result of Liu.

Theorem 2.6 (see [23, Lemma 2.5]).

Suppose is a link. Let be a framing such that

  • The framing matrix is positive definite.

  • For every the surgery on is an L-space.

Then for any integer vector the surgery on is an L-space. In particular, is an L-space link.

We will generalize this result for rational surgeries.

Proposition 2.7.

Suppose and are as in the statement of Theorem 2.6. Then for any rational framing vector , the surgery on is an L-space.


For a surgery vector denote by the number of non-integer entries in the vector .

Let us make the following statement.

() For any with , if , then is an L-space.

The statement is covered for all by Theorem 2.6. Moreover, the statement is standard. Our aim is to show that implies .

Choose with . Take with . Suppose is such that and let . Let . As , the assumption (which is contained in ) implies that is an L-space. The component can be regarded as a knot in . Let be the set of surgery coefficients such that if and only if is an L-space. By the inductive assumption all integers belong to , indeed is the surgery on with coefficient , where is the vector with at the -th position. Furthermore as well, because itself is an L-space.

In [32] possible shapes of were classified. The result allows us to conclude that if belong to , then all rational numbers greater than are in . This shows . ∎

As an application of Proposition 2.7 we will prove the following result, which generalizes [16, Theorem 1.10].

Proposition 2.8.

Suppose is an L-space link. Let be coprime positive integers and let be the link , where is the cable on . If is sufficiently large, than is also an L-space link. More precisely, if is an integer vector satisfying the conditions of Theorem 2.6 then is an L-space link if .


The proof is a direct generalization of [16, Proof of Theorem 1.10]. Choose and coprime and suppose that satisfies the conditions of Theorem 2.6 and . First we will show that the surgery on is an L-space, where . By [16, Section 2.4] we know that , where we set and is the lens space. As is an L-space and since a connected sum of L-spaces is an L-space, it is enough to show that is an L-space. But , so by Proposition 2.7 we conclude that is an L-space. Hence the surgery on is an L-space. The same proof applies to any sublink of which contains , and for a sublink not containing the –surgery is an L-space by assumption.

Let be the framing matrix for with framing , let be the framing matrix for with framing . By assumption, is positive definite. The matrix differs from only at the first column and at the first row. As for , we conclude that can be obtained from by multiplying the first row and the first column by (the element in the top-left corner is multiplied by ) and then adding to the element in the top-left corner. The first operation is a matrix congruence so it preserves positive definiteness of the matrix. Adding an element can be regarded as taking a sum with a matrix with all entries zero but in the top-left corner. This matrix is positive semi-definite, because we assumed that . Now a sum of a positive definite matrix and a positive semi-definite one is a positive definite matrix. Therefore is positive definite.

By Theorem 2.6 applied to with framing we conclude that is an L-space link. ∎

To make Proposition 2.8 more concrete, we have to present an explicit vector satisfying the conditions of Theorem 2.6. This is done in the following theorem.

Theorem 2.9.

Let denote the maximal degree of in the multivariable Alexander polynomial of an L-space link , . Assume that and for all . Then satisfies the conditions of Theorem 2.6.


Since the degrees of the multivariable Alexander polynomials of the sublinks of are less than , it is sufficient to prove that is an L-space and the framing matrix is positive definite. The former is proved below as Lemma 3.21. To prove the latter, remark that by Theorem 2.3 one has:



Now is a sum of positive definite matrices

with the only nonzero block at -th and -th rows and columns, and a diagonal nonnegative definite matrix with entries

so it is positive definite. ∎

Remark 2.11.

This bound is far from being optimal for links with many components. For example, it is proved in [13] that the point satisfies the conditions of Theorem 2.6 for the torus link, while in the above bound one has for . On the other hand, for the torus link we get , so Theorem 2.9 gives , and the two bounds agree.

3. Heegaard Floer link homology and the -function for links

In this section define the -function for links and collect some useful facts about it.

3.1. Alexander filtration

A knot in a 3-manifold induces a filtration on the Heegaard Floer complex . Similarly, a link with components in induces different filtrations on , which can be interpreted as a filtration indexed by an -dimensional lattice. For a link in , it is natural to make this lattice different from .

Definition 3.1.

Given an -component oriented link , we define an affine lattice over :

We also define the linking vector:

We have .

For define a subcomplex corresponding to the filtration level . The filtration is ascending, so for . The Heegaard Floer link homology can be defined as the homology of the associated graded complex:


The Euler characteristic of this homology was computed in [30]:


where, as above, denotes the symmetrized Alexander polynomial of .

One can forget a component in and consider the -component link . There is a natural forgetful map defined by the equation:


In general, one defines a map for every sublink :


Furthermore, for the subcomplexes stabilize, and by [30, Proposition 7.1] one has a natural homotopy equivalence . More generally, for a sublink one gets:


There is an action of commuting operators on the complex . The action of drops the homological grading by and drops the -th filtration level by . In particular, . This action makes the complexes modules over the polynomial ring . It is known [30] that is a finitely generated module over , and all the are homotopic to each other on . In particular, all the act in the same way in the homology , which can therefore be naturally considered as –module, where a single variable acts as .

3.2. The -function

It is known (see [24], this is also a consequence of the Large Surgery Theorem 4.7 below) that the homology of is isomorphic as an -module to the Heegaard Floer homology of a large surgery on equipped with a certain Spin structure. Therefore it always splits as a direct sum of a single copy of and some -torsion. We begin with the following fact.

Lemma 3.7.

For the natural inclusion

is injective on the free parts of the homology, hence it is a multiplication by a nonnegative power of .


It is sufficient to prove that

is injective on the free parts. The latter holds because contains the image of acting on . Indeed, if , where is -torsion, then . Consider the inclusions . Since the composite inclusion of into is injective on free parts, we conclude that is injective and


Definition 3.9.

We define a function by saying that is the maximal homological degree of the free part of .

We will gather now some important properties of the -function.

Proposition 3.10.

The function has nonnegative integer values. Furthermore, for all one has or .


By Lemma 3.7 the inclusion of in induces an injective map on the free parts of the homology, so it sends a generator of the free part to times a generator of the free part for some . Since the inclusion preserves the homological grading (and the generator of has grading 0), the generator of the free part of has grading , and . The last statement immediately follows from (3.8). ∎

Proposition 3.11.

If is a split link then , where is the -function for the -th component of the link.


For a split link by [30, Section 11] one has

and the isomorphism preserves the homological gradings. Note that all the linking numbers of a split link vanish, so , and the projections to sublattices do not require any shifts as in (3.6). ∎

Proposition 3.12.

For a sublink , one has


Follows from (3.6). ∎

3.3. The -function for L-space links

By Theorem 4.7 (see also [23]), a link is an L-space link if and only if for all . It turns out that for L-space links the -function is determined by the Alexander polynomial.

Throughout Section 3.3 we will assume that is an L-space link. Since for all , by (3.2) and by the inclusion-exclusion formula one can write:


where denotes the characteristic vector of the subset ; see [13, formula (3.3)]. For equation (3.14) has the form , so can be easily reconstructed from the Alexander polynomial: . For , one can also show that equation (3.14) together with the boundary conditions (3.13) has a unique solution, which is given by the following theorem:

Theorem 3.15 ([14]).

The -function of an L-space link is determined by the Alexander polynomials of its sublinks as following:


where .

There is a formula for the -function in terms of the multivariable Alexander polynomial. Consider the generating function:


Note that is a Laurent series in : by (3.13) vanishes if for some , but it does not vanish for . Then [14, Theorem 3.4.3] implies


where the sublink has components and is defined by (3.3). This immediately follows from (3.16) and the identity

which itself follows from (3.3) and (3.5).

As above, let denote the maximal -degree of the Alexander polynomial of , .

Lemma 3.19.

Assume that , then


By Theorem 2.3 the degree of the Alexander polynomial of a sublink in variable is less than or equal to . Therefore if then

and if then . Therefore the summands contributing to (3.16) nontrivially correspond to subsets such that for . Applying (3.16) to , one gets exactly the same summands. ∎

Corollary 3.20.

For one has .

Lemma 3.21.

Let be as above, then for the surgery yields an L-space.


Consider the parallelepiped in with opposite corners at and . To compute the Heegaard Floer homology of , we use the surgery complex of Manolescu-Ozsváth [24]. Every Spin structure on corresponds to an equivalence class of modulo the lattice generated by the columns of . For this equivalence class has at most one point in , and the whole surgery complex can be contracted to a single copy of supported at that point. For the precise description of the “truncation” procedure, we refer to [24, Section 8.3, Case I], where the constant in [24, Lemma 8.8] can be chosen equal to by Lemma 3.19. ∎

The following symmetry property of , which generalizes the symmetry in the case of knots [26, 18], is proved in [23, Lemma 5.5].

Proposition 3.22.

For an L-space link one has


The symmetry (3.23) and the projection formula (3.13) imply a useful “dual projection formula”.

Corollary 3.24.

Let be an L-space link, consider a set and the sublink . Then, as long as for all , the following holds:


For set . By (3.23), . Since the projection formula implies

and (3.23) for implies

3.4. The -function

The -function of a link is essentially the same object as the -function, only it differs from by a shift in variables. This shift makes a function on instead of . It is therefore more convenient to study changes of the -function under some changes (like crossing changes) of the link : these changes might affect the lattice . Yet another variant is the -function, which turns out to be useful for bounding the splitting number of L-space links; see Section 7 for details.

Definition 3.26.

The -function of a link with components is a function given by

With this definition the projection formula (3.13) takes a particularly simple form.

Lemma 3.27.

Let and . Consider a sublink of and suppose that for . Then we have


Indeed, by (3.5) and (3.13):

In particular, the -function of a component can be reconstructed from the values of -function for evaluated on vectors whose all components but the -th one are sufficiently large.

Definition 3.28.

For define