Signature and concordance of positive knots
We derive a linear estimate of the signature of positive knots, in terms of their genus. As an application, we show that every knot concordance class contains at most finitely many positive knots.
Algebraic links arise in the context of plane curve singularities. In terms of knot theory, they are certain iterated cables of torus links. Their classification is well-understood, even on the level of smooth concordance: concordant algebraic links are equal . This feature is believed to extend to various larger classes of knots, in particular to closures of positive braids. As pointed out by Baker, this would follow from the Slice-Ribbon Conjecture . Stoimenow proposed a weaker formulation that is supposed to hold for the much larger class of positive knots and proved it for closures of positive braids , as well as special alternating knots . As we will see, an extension of his technique works for positive knots.
Every topological, locally-flat knot concordance class contains at most finitely many positive knots.
No larger class of knots is likely to share this feature. Rudolph’s construction of transverse -links yields infinite families of smoothly slice quasipositive knots . Furthermore, Baker describes infinite families of smoothly concordant (in fact, ribbon concordant) strongly quasipositive knots , based on Hedden’s work on Whitehead doubles . Our proof relies on a careful analysis of the signature function of positive knots. The main ingredient is a linear estimate of the signature invariant of positive links.
The following inequalities hold for all positive links :
Here denotes the minimal first Betti number of all Seifert surfaces of the link . The earliest result in this direction is due to Rudolph : closures of positive braids have positive signature111There exist two opposite conventions concerning the sign of the signature. Here we adopt Rudolph’s convention: positive links have positive signature.. This was independently extended to positive knots by Cochran and Gompf  and to positive links by Traczyk and Przytycki [21, 15]. Later Stoimenow improved Rudolph’s result by showing that the signature is bounded by an increasing function of the first Betti number . The first linear bound for positive braid links was derived by Feller :
Recently, the lower bound has been improved to by the same author . He conjectures the optimal bound to be and proves this for closures of positive -braids. An extension of this conjecture to the larger class of positive links is conceivable, as confirmed by examples with small crossing number.
Theorem 2 yields a linear bound for the topological 4-genus of positive knots, in terms of their genus:
This follows from Kauffman and Taylor’s signature bound , , combined with the equality , valid for all knots . Again, this does not extend to (strongly) quasipositive knots, thanks to a result by Rudolph : every Seifert form can be realized by a quasipositive surface. In particular, there exist topologically slice, strongly quasipositive knots of arbitrarily high genus.
The following section contains basic facts about the signature function, non-orientable spanning surfaces and the signature formula of Gordon and Litherland . The proofs of Theorems 1 and 2 are contained in Sections 4 and 3, respectively.
Acknowledgements. We would like to thank Peter Feller for all the inspiring discussions we have had.
2. The signature function
Let be a link and . Levine and Tristram defined the -signature of to be the signature of the hermitian matrix
where is a Seifert matrix for the link [12, 22]. This does not depend on the choice of Seifert matrix and for , it equals Trotter’s definition of the classical signature invariant . Furthermore, since
the signature function is locally constant except at zeroes of the Alexander polynomial , i.e. at finitely many points. A crossing of an oriented link diagram is positive if, when following the bottom strand, the top strand comes from the left, and negative otherwise (see Fig. 1). An oriented link is positive if it admits a diagram with only positive crossings. We will use that changing a positive crossing to a negative one does not increase the -signature for any .
2.1. Gordon and Litherland’s formula for the signature
Given a surface (orientable or not) embedded in the 3-sphere and a curve on , the normal displacement of is the multi-curve in obtained by slightly pushing normally off the surface in both directions. It is connected if and only if is connected and has a non-orientable annular neighbourhood. The Goeritz form of is the symmetric bilinear form on defined by . If is orientable, the Goeritz form coincides with the symmetrised Seifert form.
The complement of every link diagram in the plane admits two checkerboard colourings. To each of them is associated a spanning surface for the link obtained by lifting the black faces to 3-space (see Fig. 2). Since they need not be orientable, the surfaces may have much smaller genus than the link. Building on the work of Goeritz , Gordon and Litherland gave a formula for computing the signature of the link using this data only . For an oriented link diagram with a given checkerboard colouring of , we denote by the associated checkerboard surface. A crossing of is of type I if the orientation of the two strands induce the same local orientation on and of type II otherwise (see Fig. 1). The formula of Gordon and Litherland, stated below, contains a quantity . In the case of positive diagrams , it is simply the number of type II crossings.
Theorem 3 (compare  Thm ).
For a positive link diagram and the surface associated to a checkerboard colouring, the signature of the associated link is equal to .
In order to compute the signature of the Goeritz form, we assume that the diagram is reduced. This means that smoothing any crossing of does not split off a link component. Furthermore, we use a particular basis of : for every white face of the checkerboard colouring except one, there is a generator that runs around it (see Fig. 2). For such a generator , we denote by the number of type I crossings minus the number of type II crossings along . For two generators and , we denote by their signed number of intersection, that is, the number of type I common crossing points minus the number of type II common crossing points. Reducedness guarantees that no generator runs twice through the same crossing. In this basis, the coefficients of the Goeritz form are given by for and otherwise .
2.2. The first Betti number of a positive link
Given a positive, non-split diagram of a link , Seifert’s algorithm yields a canonical orientable surface . Cromwell proved that this surface is genus-minimising . Let be the number of Seifert circles and let be the number of crossings of . By Cromwell’s theorem, we have
3. Proof of Theorem 2
Let be a reduced positive diagram of a link . In particular, there are no faces with only one edge. We will modify the diagram at certain faces with two edges. There are two types of these depending on whether the boundary builds a Seifert circle or not (see Fig. 3). If yes, there are still two distinct possibilities. Untwisting the full twist given by both crossings does or does not reduce the first Betti number (see Fig. 4). Actually, by the formula , untwisting does not reduce the genus if and only if the Seifert circle in question is connected to two distinct other Seifert circles. We choose to untwist those full twists where untwisting does not reduce the first Betti number. By this procedure, we obtain a reduced diagram of a new positive link such that . Furthermore we have , since the untwisting can be realised by positive-to-negative crossing changes. We now show for the new link . This implies the same inequality for the link .
Choose a checkerboard colouring of the diagram and let be the surface defined by the black regions. We further distinguish the basis curves of described above. We say that the curve is of type if it runs through crossings of type I and crossings of type II, so . By considering the boundary orientation along a white face, we get that is always even (see Fig. 5). By our simplifications of the link diagram, we removed all curves of type . Furthermore, all curves of type correspond to Seifert circles with two edges we did not untwist before. If two such Seifert circles meet at a crossing, then they meet at two crossings (otherwise untwisting a corresponding full twist would not reduce the first Betti number) and the corresponding part of corresponds to a 2-component Hopf link split from the rest of the link. Hence we can assume without loss of generality that no two such Seifert circles meet at a crossing, so that the number of crossings of type II is at least twice the number of curves of type .
Writing for the number of curves of type , we get that and Furthermore, writing for the number of with , we have
Since every crossing of type II is met by at most two curves , we get
and, by combining with the equation above, we obtain
Now we estimate the signature of using the formula of Gordon and Litherland (Theorem 3) discussed in Section 2.1. We actually need to estimate . In order to do this, we consider the curves with . Let be the graph with a vertex for each curve with and an edge between two distinct vertices and if and only if . Being defined via a planar construction, the graph is planar. From the four-colour theorem222One could also use five-colourability instead of four-colourability. This would cause a drop in the constant of Theorem 2 to . it follows that one can choose a bipartite subgraph containing at least one half of the vertices . The Goeritz form restricted to the subspace generated by the curves corresponding to the vertices of is given by a block matrix
where and are non-negative diagonal matrices. Since the signature of such a matrix is non-negative, this gives a subspace of dimension at least
restricted to which the Goeritz form has non-negative signature, where is the number of white regions in the checkerboard colouring of , so . Plugging this into the formula of Gordon and Litherland, we obtain
Using the other checkerboard surface, we have that white faces become black faces and vice versa, and crossings of type I become crossings of type II and vice versa. By exactly the same procedure, we get
Finally, we sum both inequalities for , observe , use and obtain
4. Proof of theorem 1
Suppose that there exists a topological, locally-flat concordance class containing infinitely many positive knots . The average signature function depends only on the topological, locally-flat concordance class of a knot, so all the have identical signature function outside the zeroes of their Alexander polynomials . In particular, all the have the same signature . Thus, by Theorem 2, the genera of the knots are bounded from above by . It then follows from a theorem of Stoimenow that there exist finitely many positive knot diagrams , such that every knot is obtained from one of the by inserting a certain number of positive full twists at the crossings of (see Fig. 6) [18, Theorem 3.1].
Since there are infinitely many but only finitely many , we can assume without loss of generality that all the knots are obtained from one single reduced diagram by inserting a certain number of positive full twists at the crossings of . Again, since there are infinitely many but only finitely many crossings, we can assume that the number of inserted full twists at one of the crossings, say , becomes arbitrarily large as tends to infinity. Now let be the knot obtained by starting from the diagram and inserting positive full twists at and let be the link obtained from by smoothing the crossing as in Fig. 7.
Since the number of inserted positive full twists at the crossing becomes arbitrarily large as tends to infinity, for every , the knot can be obtained from one of the by applying some positive-to-negative crossing changes. In particular, we have
for all non-algebraic and . Here and in the following, we consider non-algebraic numbers , guaranteeing that as defined in Section 2 is a concordance invariant of knots. Since is the concordance class of a knot, for close enough to . Choose such that and . The second condition can be achieved since the coefficient of the linear term of the Conway polynomial of a two-component link equals the linking number of its components and thus the Alexander polynomial is not identically zero. Here we use that is non-split (otherwise would not be reduced) and positive. We now show that can be chosen so that . This will be the contradiction we are looking for.
Since a Seifert surface for can be obtained from a Seifert surface of by glueing a suitably twisted ribbon, a Seifert matrix for can be chosen to be
where is a Seifert matrix for , is a column vector, is a row vector and is a natural number. In fact, nothing depends on except for and . We are interested in , which is the signature of the matrix
The determinant of this matrix equals , where does not depend on and since . From this it follows that for large enough , the determinant of has the same sign as the determinant of . In particular, the matrix has one more positive eigenvalue than . A theorem by Przytycki implies since is a positive link . Therefore the signature of is strictly positive for large enough. In particular, this implies , a contradiction.
-  K. Appel and W. Haken, Every planar map is four colorable, Contemporary Mathematics 98, American Mathematical Society, Providence RI, 1989.
-  K. L. Baker, A note on the concordance of fibered knots, J. Topol. 9 (2016), no. 1, 1–4.
-  T. D. Cochran and R. E. Gompf, Applications of Donaldson’s theorems to classical knot concordance, homology 3-spheres and Property P, Topology 27 (1988), 495–512.
-  P. Cromwell, Homogeneous links, J. London Math. Soc. 39 (1989), 535–552.
-  P. Feller, The signature of positive braids is linearly bounded by their genus, Internat. J. Math. 26 (2015), no. 10, 1550081.
-  P. Feller, A sharp signature bound for positive four-braids, Q. J. Math. 69 (2018), no. 1, 271–283.
-  L. Goeritz, Knoten und quadratische Formen, Math. Z. 36 (1933), 647–654.
-  C. McA. Gordon and R. A. Litherland, On the Signature of a Link, Invent. math. 47 (1978), 53–69.
-  M. Hedden, Knot Floer homology of Whitehead doubles, Geom. Topol. 11 (2007), 2277–2338.
-  L. H. Kauffman and L. R. Taylor, Signature of links, Trans. Amer. Math. Soc. 216 (1976), 351–365.
-  P. B. Kronheimer and T. S. Mrowka, The genus of embedded surfaces in the projective plane, Math. Res. Lett. 1 (1994), no. 6, 797–808.
-  J. Levine, Knot cobordism groups in codimension two, Comment. Math. Helv. 44 (1969), 229–244.
-  R. A. Litherland, Signatures of iterated torus knots, Topology of low-dimensional manifolds (Proc. Second Sussex Conf., Chelwood Gate, 1977), pp. 71–84, Lecture Notes in Math. 722, Springer, Berlin, 1979.
-  M. Powell, The four-genus of a link, Levine-Tristram signatures and satellites, J. Knot Theory Ramifications 26 (2017), no. 2, 1740008, 28 pp.
-  J. H. Przytycki, Positive knots have negative signature, Bull. Polish Acad. Sci.: Math. 37 (1989), 559–562. http://arxiv.org/abs/0905.0922.
-  L. Rudolph, Nontrivial positive braids have positive signature, Topology 21 (1982), 325–327.
-  L. Rudolph, Constructions of quasipositive knots and links, I, Noeuds, Tresses, et Singularités (ed. C. Weber) L’Enseign. Math. 31, (Kundig, Geneva, 1983), 233–246.
-  A. Stoimenow, Knots of genus one or on the number of alternating knots of given genus, Proc. Amer. Math. Soc. 129 (2001), no. 7, 2141–2156.
-  A. Stoimenow, Bennequin’s inequality and the positivity of the signature, Trans. Amer. Math. Soc. 360 (2008), 5173–5199.
-  A. Stoimenow, Application of braiding sequences III: Concordance of positive knots, Internat. J. Math. 26 (2015), no. 7, 1550050.
-  P. Traczyk, Nontrivial negative links have positive signature, Manuscripta Math. 61 (1988), 279–284.
-  A. G. Tristram, Some cobordism invariants for links, Proc. Cambridge Philos. Soc. 66 (1969), 251–264.
-  H. F. Trotter, Homology of group systems with applications to knot theory, Ann. of Math. (2) 76 (1962), 464–498.