On classical upper bounds for slice genera
We introduce a new link invariant called the algebraic genus, which gives an upper bound for the topological slice genus of links. In fact, the algebraic genus is an upper bound for another version of the slice genus proposed here: the minimal genus of a surface in the four-ball whose complement has infinite cyclic fundamental group. We examine the relationship of the algebraic genus and the Seifert form, the algebraic unknotting number, and other knot invariants. One of the strengths of the algebraic genus is that it yields simply computable good upper bounds for the topological slice genus. The paper also contains a discussion of the optimality of slice genus bounds, employing Casson-Gordon invariants.
Key words and phrases:Slice genus, Seifert form, Casson-Gordon invariants, algebraic unknotting number
2010 Mathematics Subject Classification:57M25, 57M27
In this paper we introduce the notion of the algebraic genus of links in , denoted by . The main interest in is that it provides an upper bound for the topological slice genus of a link —the smallest genus of an oriented connected properly embedded locally flat surface in the 4–ball with oriented boundary . The upper bounds on in this paper all invoke Freedman’s celebrated Disk Theorem. In fact, we only apply the following theorem of Freedman.
[Fre82] For a knot , the following are equivalent:
has Alexander polynomial 1.
arises as the boundary of a properly embedded locally flat disk in with . ∎
We define the algebraic genus of link with components by
We postpone a more conceptual definition to Section 2. Our new link invariant bundles earlier work by Baader, Liechti, McCoy and the authors on upper bounds for [Fel16, BL15, FM16, BFLL15]. Indeed, the definition of together with [BFLL15, Prop. 3] imply that, for all links ,
Our treatment clarifies the strengths and limitations of this upper bound, while also examining as a link invariant of independent interest. For example, we show that is a classical link invariant in the following sense: for all links , only depends the S-equivalence class of Seifert forms for .
We also provide the following strengthening of creftypecap 1. Let the –slice genus of a link denote the smallest genus of an oriented connected properly embedded locally flat surface in the 4–ball with oriented boundary and . Of course, for all links .
For all links ,
In light of creftypecap 1, it is natural to wonder whether the inequality in creftypecap 2 is an equality, which would constitute a completely classical characterization of . We note that such a characterization is impossible for since, for example, there are pairs of knots with the same Seifert form such that is topologically slice while is not; i.e. and .
We also give a characterization of as the smallest genus of a cobordism between and a knot with Alexander polynomial 1 that can be taken to be in –space.
For all links with components, equals the smallest genus among Seifert surfaces for links with components such that the first components form and the last component forms a knot with Alexander polynomial 1.
This characterization of is the reason for naming the invariant “algebraic genus”, in parallel to the algebraic unknotting number : for a knot , both and can be defined either purely in terms of the Seifert form, or as a –dimensional distance (using the genus of Seifert surfaces and unknotting, respectively) to knots that have Alexander polynomial 1. The term “algebraic slice genus”, on the other hand, would be more fitting for Taylor’s invariant (see below).
We summarize the relation between and other knot invariants in Figure 1. The algebraic genus is naturally defined for multi-component links; this is not true for the other invariants we consider. Therefore, for the rest of the introduction we only consider knots rather than multi-component links.
1.1. Algebraic genus and algebraic unknotting number
For the algebraic unknotting number of a knot, introduced by Murakami [Mur90], we find the following: \thmt@toks\thmt@toks For all knots ,
Let denote the Alexander polynomial of . We understand its degree to be the breadth of ; e.g. for the trefoil. Then, using that , the above theorem yields
For all knots, .
1.2. The Taylor invariant
The algebraic genus can be understood as a measure of how strongly a knot fails to have Alexander polynomial . Indeed, if and only if . Taylor introduced a knot invariant that generalizes algebraic sliceness [Tay79]: a knot is algebraically slice, i.e. has a metabolic Seifert form, if and only if . Explicitly, if is a Seifert form of a knot , then is defined as minus the maximal rank of a totally isotropic subgroup of . Taylor’s invariant provides a lower bound for the slice genus. Taylor established this in the smooth respectively PL category [Tay79]; however, is known to hold for all knots by experts. For example, this can be obtained from a minor generalization of the classical proof that slice knots are algebraically slice (compare e.g. [Lic97, Proposition 8.17]) using only one locally flat input: all locally flat oriented surfaces in are the boundary of a compact oriented –dimensional submanifold. The latter can be deduced from topological transversality as established in [FQ90, Chapter 9]; compare [Ran98, page XXI]. In [LM16], this argument is given with some more details.
Since , it would be of interest to relate and . It appears that aside from the two invariants are rather independent. However, we can prove the following: if a genus 2 fibered knot is algebraically slice (i.e. ), then (and thus also ’s topological slice genus) is at most 1. In fact, we show the following.
If a knot is algebraically slice and has monic Alexander polynomial of degree , then .
In contrast, such results are not available for knots with Alexander polynomials of higher degree. For example, there exist algebraically slice knots with monic Alexander polynomial of degree and ; compare Example 29.
1.3. Optimality of slice genus bounds
Taylor’s lower bound to the slice genus is known to be the best classical lower bound; that is to say, every Seifert form is realized by a knot whose slice genus equals . One may ask the analogous question about classical upper bound for the topological slice genus. As a first step towards determining the best classical upper bound for the topological slice genus of knots—for which the algebraic genus is a candidate—we prove the following. \thmt@toks\thmt@toks Every Seifert form of a knot is realized by a knot with
Here, denotes the minimum number of generators of the first integral homology group of the –fold branched cover of a knot realizing (note that only depends on ). The relevant knots in the proof are constructed via infection, following Livingston [Liv02]. Gilmer’s work on Casson-Gordon invariants [Gil82] provides the necessary non-classical lower bound for the topological slice genus.
1.4. Computability and effectiveness
It is a virtue of that it—or at least upper bounds for it—can be explicitly calculated using Seifert matrix manipulation. It turns out that is a very effective tool for determining the topological slice genus of small knots; see for example Example 11. Using computer calculations for , one can determine for all prime knots with 11 or less crossings111The second author has implemented the calculation of upper bounds for on a computer, determining some previously unknown values of the topological slice genus of small knots. These results will be available together with McCoy’s computer calculation on the smooth slice genus of small knots and other slice genera calculations as [LM16].:
Proposition 8 ([Lm16]).
If is a prime knot of crossing number 11 or less and denotes the smooth slice genus of , then
1.5. Structure of the paper
Section 2 contains the definition of and the proof for its alternative characterization in Proposition 3, as well as some basic examples and an overview over known results involving . creftypecap 2 and Section 1.1 on the –slice genus and the algebraic unknotting number are proven in Sections 4 and 3, respectively. Section 5 is concerned with the algebraic genus of knots with monic Alexander polynomial, and contains the proof of creftypecap 6. In Section 6, optimality of slice genus bounds is discussed and Section 1.3 is proved.
We thank Danny Ruberman for pointing us to [Tay79]. We thank Sebastian Baader and Livio Liechti for valuable inputs; in particular, concerning creftypecap 6. We thank Mark Powell for comments on a first version of this paper. Both authors gratefully acknowledge support by the SNSF and thank the MPIM Bonn for its hospitality.
2. The algebraic genus
2.1. Definition and Examples
In this section, we define the algebraic genus of an oriented link using the Seifert form defined on , where is a genus Seifert surface with boundary the component link . By a Seifert surface for a link , we mean an oriented connected embedded surface in with boundary .
Let us start with some notations on bilinear forms (which we will readily use for Seifert forms). For integers and , let be a bilinear form on an abelian group such that its antisymmetrization, denoted by , satisfies the following: the radical of —the subgroup of elements that pair to 0 with all other elements—is isomorphic to as a group and restricted to has determinant 1. These are precisely the bilinear forms that arise as Seifert forms of genus Seifert surfaces of links with components. If is a matrix representing such a form , we call
the Alexander polynomial of , and denote this by . This is independent of the choice of basis and hence indeed a well-defined; in fact, it is invariant under S-equivalence. We call a subgroup Alexander-trivial if is a unit in for a matrix representing . One obtains
by substituting . It follows that is a summand of , and the rank of is even and at most . Furthermore, is Alexander-trivial if and only if is the Seifert form of a knot with .
In practice, establishing that a subgroup is Alexander-trivial may be done by finding a basis of with respect to which is given by a matrix of the form
where , , , and denote square matrices of half the dimension of that are zero, the identity, lower triangular with zeros on the diagonal, upper triangular with zeros on the diagonal, and arbitrary, respectively. For this we note that, if a matrix is of the form creftypecap 2, then . The following lemma implies that Alexander-triviality of a subgroup can always be established by finding such a basis.
If a Seifert form on has Alexander polynomial 1, then there exists a basis of with respect to which is given by a matrix of the form where , , and denote matrices that are zero, identity, and upper triangular with zeros on the diagonal, respectively.
For general , there is no basis such that is the zero matrix, since the rank of a matrix of is an invariant of the form. Note that a significantly stronger statement holds: there are knots which have Alexander-polynomial 1, yet do not admit a Seifert matrix as above with the zero matrix [GT04].
Proof of Lemma 9.
By a rather direct calculation provided in [Fel16, Lemma 6 and Remark 7] there is a basis such that the corresponding matrix is of the form where is some matrix. The statement follows by applying the following base change
where is the unique matrix that satisfies the equation . To be explicit, is inductively given as follows. Set . For the induction step, we fix and assume is defined whenever , and thus we can set
Suppose is the maximal rank of an Alexander-trivial subgroup for a bilinear form . Then we define the algebraic genus of to be . If denotes the S-equivalence class of , we define as the minimum for .
For all oriented links , we define the algebraic genus of to be
Clearly, if is a link and some fixed Seifert form of , then
Note that reversing the orientation of all components of , or taking the mirror image of does not change , since the Alexander-trivial subgroups with respect to and are the same.
We consider the 12–crossing alternating knot and one of its Seifert matrices (i.e. a matrix of a Seifert form of a Seifert surface of that knot, written in terms of a basis for the Seifert surface’s integral first homology group)
as provided by KnotInfo [CL]. There exists an Alexander-trivial subgroup of rank in with respect to the bilinear form represented by . Indeed,
which shows that the columns of are the basis of an Alexander-trivial subgroup of rank in (note that is of the form creftypecap 2). Furthermore, no Seifert form of has an Alexander-trivial subgroup of full rank, since the Alexander polynomial of is different from 1 (it is ). Thus, by Definition 10. In fact, , where denotes the signature of the knot , and so
The genus of is 3 (since the degree of the Alexander polynomial of is 6), the smooth slice genus is 2 (by an argument based on Donaldson’s Diagonalization Theorem; compare [LM16]), and the algebraic unknotting number is [BF15, BF]. Therefore, there is no immediate way via the smooth slice genus or the algebraic unknotting number to find that ; while the above calculation of is quite explicit.
Definition 10 is made such that the inequality creftypecap 1, that is for all links, follows immediately from Freedman’s creftypecap 1 and the following proposition, which is proved in detail in [BFLL15, Proof of Prop. 3] (compare also [Fel16, Proposition 2]). For the sake of completeness, we nevertheless include a concise version of the proof below. Proposition 12 will be used in the proof of creftypecap 2, which subsumes creftypecap 1.
Let be a link, a genus Seifert surface for , and the Seifert form on . If is an Alexander-trivial subgroup of of rank , then there exists a separating simple closed curve on with the following properties:
the curve (viewed as a knot in ) has Alexander polynomial 1,
such that is of genus with boundary .
Sketch of the proof.
As discussed at the beginning of the section, is a summand, i.e. . One may choose a separating simple closed curve on such that , , and . It turns out that there is a group automorphism with that preserves the intersection form and that maps homology classes given by components of to homology classes given by components of . The group automorphism is realized as the action of a diffeomorphism of since the mapping class group of surjects onto the symplectic group. Take and . Then is a genus Seifert surface of and the corresponding Seifert form is given by , hence . ∎
The next proposition shows that only depends on the S-equivalence class of Seifert forms of .
For all links ,
Clearly holds, since the minimum ranges over a larger class of bilinear forms. For the other direction, we recall that any bilinear form S-equivalent to a Seifert form, can be stabilized to become a Seifert form. Indeed, let be a Seifert form for and any bilinear form S-equivalent to . There exists a bilinear form that arises as the stabilization of both and . Since stabilizations of a Seifert form can be realized geometrically by a stabilization of the corresponding Seifert surface, the bilinear form arises as a Seifert form. Now, the statement follows from the following lemma about stabilizations. ∎
Let be a bilinear form arising as Seifert form of a link, and let be obtained from by a stabilization. Then .
For the bilinear form on , let be an Alexander-trivial subgroup of maximal rank, and denote this rank by . We view the stabilization of as a bilinear form on such that with respect to the standard basis is given by
where represents and is some element of . The statement follows from the fact that is an Alexander-trivial subgroup of with respect to . ∎
We have already seen that Taylor’s invariant is a lower bound for the algebraic genus. Indeed, this is evident from the definitions, because an Alexander-trivial subgroup of rank contains a totally isotropic subgroup of rank . Let us consider another lower bound for , which will be generalized considerably by Section 1.3. As mentioned in the introduction, let be the minimum number of generators of , the first integral homology group of the double branched covering of a knot with the Seifert form . Note that if is a matrix for , then is isomorphic to the cokernel of . Let us now prove that
Indeed, by the classification of finite abelian groups, there is a prime such that . If is an Alexander-trivial subgroup of rank , and is a matrix of , then . So has full rank over , and thus has rank at least over . Thus has dimension at most over , where is the dimension of .
A more indirect proof of creftypecap 4 uses that , where denotes the Nakanishi-index.
Consider a Seifert form of dimension with maximal , i.e. . Clearly, that condition is equivalent to the existence of an odd prime modulo which vanishes ( cannot vanish modulo , since its determinant is odd). Such a form may be realized as Seifert form of a knot with topological slice genus (Section 1.3). Moreover, all knots admitting such a Seifert form have some peculiar properties: for example, the unknotting number of is bounded below by ; and all Alexander ideals are non-trivial over , since they are sent to by the substitution .
2.2. Overview of previous results
By finding a separating curve with Alexander polynomial 1 on a Seifert surface and using Freedman’s creftypecap 1, one can show that is smaller than the genus for some knots and links. This was used by Rudolph to show that for most torus knots; in fact even [Rud84]. Baader used this idea to show that if a minimal genus Seifert surface for a knot contains an embedded annulus with framing , then if and only if [Baa16]. We will generalize the latter result at the end of the next subsection (Proposition 16).
It turns out that the existence of separating Alexander polynomial 1 knots on Seifert surfaces is completely determined by the Seifert form; compare Proposition 12. The following results by Baader, Liechti, McCoy and the authors were proven using some version of this. We present them rewritten in the language of introduced above, while suppressing the inequalities .
For all knots , one has
For prime homogeneous knots that are not positive or negative:
There is an infinite family of 2–bridge knots with
The algebraic genus of torus links satisfies
and for , and :
For positive braid knots one has
except if .
2.3. The stable algebraic genus
If are Seifert forms with respective Alexander-trivial subgroups , then has the Alexander-trivial subgroup . This implies that is subadditive with respect to the connected sum of links:
for all links and . Let us construct an example demonstrating that is in general not additive. Take to be the Seifert form given by
for which we have , since . On the other hand, the form admits an Alexander-trivial subgroup of rank generated by and . Indeed
and thus . Following Livingston’s definition of the stable slice genus [Liv10], one may define the stable algebraic genus of a knot as
It is then an immediate consequence of creftypecap 2 that
where the stable –slice and the stable topological slice genus are defined in the same fashion. This inequality gives some motivation for studying the stable algebraic genus. We will refrain from doing so here, with the exception of the following characterization of knots whose stable algebraic genus is strictly less than their genus. This characterization results from strengthening Baader’s result mentioned at the beginning of the last subsection by recasting his argument algebraically and making connections to .
For all knots ,
holds if and only if and .
creftypecap 15 discussed that is a lower bound for . So both and are additive lower bounds for , hence also lower bounds for . This implies the ‘only if’ part of the statement.
Let us now prove the ‘if’ part. Fix a Seifert matrix of of size , where is the genus of . Denote by the gcd of the entries of , i.e. . Note that is odd, and so is as well. Hence
Let us denote by the quadratic form defined by , i.e. . Let be the union of the images of for all .
In a first step, we prove that . To show ‘’, we notice that by definition elements are sums of elements of the form
for , which are –linear combinations of and . To show ‘’, it is sufficient to show that is a subgroup of and contains all and . If , and , then
This proves that is closed under addition. Now, let a non-zero integer be given. We are going to show that , which will complete the proof that is a subgroup. Since , the form is indefinite, and so there exists with the opposite sign of . We first consider the case and . Since is closed under addition and thus also under multiplication with positive integers, we find
If instead, we have and , then we similarly find
Finally, let us check that all and are in . Firstly, , where denotes –th standard basis vector. Secondly,
As a second step, note that is a presentation matrix of the first integral homology group of the double branched covering of . So, if were non-trivial, then that homology group would be the sum of groups of the form for some , which would contradict . It follows that , and so .
To finish, pick two vectors with . Pick another two vectors with such that and . Then
generate an Alexander-trivial subgroup since
This implies that , and therefore . ∎
2.4. Three-dimensional characterizations of
We conclude this section with two alternative characterizations of .
Let be a Seifert surface of . Let be a simple closed curve such that with and , and equals 1. Then is the minimal genus of surfaces such as above.
A 3D-cobordism between links and with and components, respectively, is a Seifert surface for a link with components such that the link given by the first components is and the link given by other components is with reversed orientation. For all links , equals the minimal genus of a 3D-cobordisms between and a knot with Alexander polynomial 1.
These should be viewed in light of similar characterizations for the algebraic unknotting number of a knot , which can be defined purely algebraically using the Seifert form, but is most quickly defined as the smallest number of crossing changes needed to turn into an Alexander polynomial 1 knot.
Let two links and , a Seifert surface for , and a genus 3D-cobordism between and be given. Then can be stabilized such that there exists a 3D-cobordism with genus between and with .
The surface defines a framing of ; i.e. a disjoint union of embedded annuli, given as a small closed neighborhood of in . We first modify such that that the induced framing on agrees with the framing induced by : for every component of take a properly embedded interval in with one boundary point on and the other on , which is possible since is connected (by the definition of a Seifert surface). By inserting full twists along into if necessary, we get a new genus 3D-cobordism that induces the correct framing on , which we denote by .
Next we observe that the cobordism can be viewed as arising by adding 1–handles to . More precisely: let be ; in other words, is the parallel copy of that forms the other part of the boundary of . For some non-negative integer , there exist pairwise disjoint disks in such that
where the are pairwise disjoint disks in such that consists of two closed intervals contained in . Let denote the core of the handle ; i.e. a properly embedded interval in such that its two boundary points lie in the interior of , one in each component.
Now, we study the intersection between and . Since and induce the same framing on , we may isotope them such that . We also arrange that the cores intersect transversely.
We now modify such that the algebraic intersection number between and becomes . Indeed, by modifying as depicted in Figure 2, we change the algebraic intersection number between and by . Thus, by modifying several times as described in Figure 2, we obtain a genus 3D-cobordism such that the corresponding cores have algebraic intersection number with . We note that is still a genus 3D-cobordism between and since the operation described in Figure 2 does not change the isotopy type of or (however, in general, it does change the isotopy type of ).
In a last step we show that can be stabilized such that it no longer intersects . This is done by inductively doing stabilizations on to reduce the geometric intersection between the cores and to . Indeed, if is non-empty, then we find two consecutive (on ) intersection points of opposite orientation. The subinterval of connecting and defines a stabilization of that reduces by . The stabilization of which does not intersect any can now be isotoped (rel ) away from except for the intersection at . ∎
3. The –slice genus
We recall from the introduction:
Let the –slice genus of a link denote the smallest genus of an oriented connected properly embedded locally flat surface in the 4–ball with boundary and .
We now proof creftypecap 2; i.e. we establish that is an upper bound for for all links.
Proof of creftypecap 2.
Given an –component link , let be a Seifert surface such that , where denotes the genus of and is the rank of an Alexander-trivial subgroup of . By Proposition 12, there exists a separating curve with Alexander polynomial 1 on such that can be written as the following union of surfaces:
where is a Seifert surface for of genus and the are closed disks that are pairwise disjoint and each disk intersects in two closed intervals that lie in . In other words, is given by attaching many 1–handles to ; compare Figure 3. Compare also with the proof of Lemma 19, where we started with a similar setup.
By creftypecap 1, bounds a properly embedded locally flat disk in such that its complement has fundamental group . We may arrange that close to the disk is smooth (say we arrange for the disk to agree with the linear cone over for all points of distance from the origin of more than ; i.e. for , , where denotes the curve in the –sphere of radius given by stretching by the factor ).
Let be the following locally flat surface of genus in the –ball of radius .
where denotes the link in obtained by stretching by . In particular, is a witness for ; i.e. we have established creftypecap 1. To get the stronger statement , it suffices to establish the following claim.
The fundamental group of is isomorphic to .
For this, we consider the topological 4–manifold with boundary , where denotes an open tubular neighborhood of , rather than . We first remark that (as a topological manifold with boundary) can be obtained from by attaching many 2–handles: one 4–dimensional 2–handle corresponding to each ; compare [GS99, Proposition 6.2.1]. In particular, the inclusion induces a surjection on . In Figure 4,
we illustrate the situation one dimension lower: for a knot in the –ball rather than a surface in the –ball.
We now describe the attaching spheres for the handles in more detail. Let be a core of the handle ; i.e. a properly embedded interval in such that its two boundary points lie in the interior of , one in each component; compare Figure 3. Choose closed disks in such that each intersects