-foam homology calculations
We exhibit a certain infinite family of three-stranded quasi-alternating pretzel knots which are counterexamples to Lobb’s conjecture that the –knot concordance invariant (suitably normalised) should be equal to the Rasmussen invariant . For this family, . However, we also find other knots for which . The main tool is an implementation of Morrison and Nieh’s algorithm to calculate Khovanov’s –foam link homology. Our C++-program is fast enough to calculate the integral homology of e.g. the –torus knot in six minutes. Furthermore, we propose a potential improvement of the algorithm by gluing sub-tangles in a more flexible way.
During the last decade, –polynomials were categorified one after the other: beginning, of course, with Khovanov’s categorification of the Jones polynomial [khovanov] using cobordisms, followed by his categorification of the –polynomial [khovanovsl3] using foams; and finally, Khovanov and Rozansky’s categorification of the –polynomials for arbitrary [roz] and of the Homflypt–polynomial itself [roz2] (see also Khovanov [khsoergel] and Webster [homflycan]) using matrix factorisations.
All these homologies are completely combinatorial in nature – unlike e.g. the original knot Floer homology – meaning that their definition is in itself a description how to compute them. By hand, this direct way of computation is hardly feasible for any but the smallest knots, and using skein long exact sequences and criteria for thinness is much more efficient (see e.g. a computation of Mackaay and Vaz [terasaka]). On a computer, however, even the straight-forward method, as implemented with some tweaks in the program KhoHo [khoho] by Shumakovitch, could already compute the –homology of knots with up to ca. crossings. Matrix factorisations, on the other hand, are more reluctant to efficient treatment by a computer. Nevertheless, Carqueville and Murfet [carqueville] wrote a program that is able to compute –homology of links with up to six crossings and . Webster’s program [krm2] computes Homflypt-homology of short braids, but is largely limited to 3–stranded ones.
Yet it is desirable to be able to compute the Khovanov–Rozansky homologies of much larger knots, since some phenomena require a certain complexity of the knot to occur; see e.g. Hedden and Ording’s 15–crossing knot which demonstrates that the Rasmussen and Floer concordance invariants differ [sneqtau]. Or to mention an extreme example, see Freedman et al. [manandmachine] for a link with 222 crossings whose Rasmussen invariant was worthwhile to compute at the time, because it could have provided a counterexample to the four-dimensional smooth Poincaré conjecture (this approach was later rebuted by Akbulut [akbulut]).
Bar-Natan’s extension of –homology from link diagrams to tangles [tanglescobordisms] (see also Khovanov [khtangles]) led subsequently to a divide-and-conquer algorithm to compute –homology [fastcompu]. The speed of this algorithm depends primarily on the girth of the link diagram: this is the maximal number of intersection points of a horizontal line with the diagram (see e.g. [freedman], and cf. section LABEL:sec:algo for details). An implementation by Green and Morrison called JavaKh [javakh] is able to compute the homology of knots of girth up to 14, e.g. the -torus knot. Mackaay and Vaz [vazuniverse] and Morrison and Nieh [su] then extended –homology to tangles, and the latter describe in detail the ensuing algorithm. In this text, we present an implementation of this algorithm as a C++-program called FoamHo [foamho].
The algorithm can be improved by gluing sub-tangles in a more flexible way, along a sub-tangle tree instead of one after the other. This leads to the notion of the recursive girth of a link, which replaces the girth as main factor limiting calculation speed. Even though this improvement is not implemented in FoamHo yet, the program is still fast enough to calculate the integral homology, reduced or unreduced, of links with girth up to 10, such as the –torus knot. The –concordance invariant , as defined by Lobb [lobbconcordance] (see also Wu [wu3] and Lobb [lobbonroz]), may (in most cases) be extracted from the –homology by means of the spectral sequence converging to the filtered version of homology. This method was used for –homology by Freedman et al. [manandmachine]. It does not depend on the conjectured convergence of the spectral sequence on the second page.
The most striking calculatory result obtained with FoamHo concerns this –invariant, which is the –analogue of the Rasmussen invariant . Those two invariants share many properties, e.g. they agree on homogeneous and quasi-positive knots, and until now it was not known whether they were actually equal or not (see Lobb [lobbconcordance, Conjectures 1.5, 1.6]).
Let be the –concordance invariant, normalised to take the same value as the Rasmussen invariant on the trefoil. If , , and , then the –pretzel knot has –invariant
where and for .
This statement is called a “conjecture” since it is established by FoamHo-calculations only for small values of , and . In addition, however, we have a prove for the case which does not rely on computer calculations [lewarkprepare]. See section LABEL:mainproof for further details.
If one appends to the hypotheses of the conjecture, then the –pretzel knot is quasi-alternating (see Champanerkar and Kofman [quasialtpretzels] and Greene [greene]), and hence its –invariant equals its classical knot signature (see Manolescu and Ozsváth [quasialt]): . This value of the signature can be easily computed using Göritz matrices and the formula of Gordon and Litherland [gordon]. So for this infinite family of pretzel knots, the – and –invariant differ, and the latter gives a weaker bound for the slice genus than the former. However, there are knots for which this is different, e.g. and .
The rest of the paper is organised as follows: section 2 provides a brief, but essentially self-contained definition of the –foam homology as defined by Khovanov [khovanovsl3], Mackaay and Vaz [vazuniverse], and Morrison and Nieh [su]. We also define reduced homology in the framework of foams. In section LABEL:sec:3.1, we give an account of the divide-and-conquer algorithm, including the computation of integral homology and the acceleration of the algorithm using a sub-tangle tree. In section LABEL:sec:3.2, we discuss how to extract from the -homology. Section LABEL:sec:3.3 addresses some particular implementation issues and their resolution in FoamHo, and section LABEL:sec:3.4 presents the usage and characteristics of the program itself. Finally, section LABEL:sec:3.5 states some results of FoamHo calculations, and compares them against previously known results.
Acknowledgements I would like to thank Pedro Vaz and Alexander Shumakovitch for encouraging me to pursue this subject and write a program, Louis-Hadrien Robert for all the inspiring discussions, and Christian Blanchet for his continuous support and advice. Thanks to Nils Carqueville, Pedro Vaz, Scott Morrison and Christian Blanchet for comments on the first version of the paper.
2. The -foam homology of tangle diagrams
This section gives a definition of the –polynomial and its categorification, the –foam homology. Except for the use of a generalised definition of planar algebras (sec. 2.2) to formalise localness, and the definition of reduced homology using foams (sec. LABEL:sec:red), this section contains nothing essentially new. We just review the parts of [khovanovsl3, su] which are relevant to the purpose of this section – which is to provide a self-contained definition of –homology with the objective of calculation in mind. Instead of choking tori we use dots, like Khovanov [khovanovsl3] and Mackaay and Vaz [vazuniverse]. The origin of webs and the –polynomials lie in representation theory [RT, kuperberg], an aspect we we will not dwell on.
2.1. The –polynomial, naively
The –polynomial can be defined by a single skein relation involving only link diagrams. We will instead use the two skein relations (Sk) and (Sk), see below, because this allows a categorification using foams. These skein relations involve webs: a closed web is a plane oriented trivalent graph whose every vertex is either a source or a sink, and that may have vertex-less circles as additional edges.
A tangle diagram is the generic intersection of a link diagram with a disc; generic means that the disc’s border intersects the diagram’s strands transversely, and does not pass through a crossing. Let us define a map from the set of smooth isotopy classes of tangle diagrams to the free –module on the set of smooth isotopy classes of closed webs. The map is uniquely determined by the following two local relations, which are interpreted naively for now (i.e. apply these relations to all crossings of the link at once, then expand):
Next, we define an evaluation of closed webs, called the Kuperberg bracket [kuperberg, jaeger], which associates to a closed web a Laurent polynomial in . This evaluation is given by the four relations
The –polynomial, which associates a Laurent polynomial in to a link diagram, can now be obtained by composing with , and identifying the empty web with . Categorifying this construction is going to yield the –homology. However, it is advantageous to formalise the localness of the relations (Sk, C, D, S) before proceeding.
2.2. Planar algebras
While 2–categories do give a framework for webs and foams, they make sense only if one aims at interpreting webs as maps between oriented 0–manifolds; this aspect is not essential to the calculation of –homology, and so we use planar algebras instead. Planar algebras were introduced by Jones [jones] to identify subfactors. They were subsequently used to describe locally defined knot invariants such as the Jones polynomial. Bar-Natan [tanglescobordisms] introduced a categorified version of planar algebras called canopolis to describe Khovanov homology, a method adaptable to –homology [su]. We will use a slightly generalised version of planar algebras, working over arbitrary monoidal categories instead of over the category of vector spaces over a fixed field. In this way, a canopolis is a planar algebra as well.
Let be a closed disc, and be pairwisely disjoint smaller closed discs. Punching out these discs yields a disc with holes, . Let be a compact oriented one-dimensional smooth submanifold of with ; in other words, is a collection of circles and of intervals whose endpoints lie on the boundary of the big discs or one of the smaller discs. An input diagram consists of , , and on each boundary component of a base point which is not in . We consider input diagrams up to smooth isotopy, in the course of which boundary points of may not cross the base points.
For every , the intersection is a finite set; at each of its points, the corresponding interval of is either oriented towards the boundary ( for , for ), or away from it ( for , for ). Moreover, these points have a canonical order, given by starting from the base point and walking once around the circle in the counterclockwise direction. Thus the isotopy type of may be written as a sign-word , i.e. a word over the alphabet . The boundary of is the tuple .
Now suppose and are two input diagrams, such that for a fixed . Then may be shrunk and glued into , base point on base point and boundary points on boundary points, resulting in a new input diagram with holes.
Let be a subset of the set of all sign-words. Let be a monoidal category, in the easiest case just the category of sets, and in the classical case the category of vector spaces over a fixed field. Then a planar algebra over and consists of the following data:
For each , an object .
For each input diagram with boundary such that , a –morphism
This data is required to satisfy the following axioms:
Suppose is an input diagram with and that consists only of appropriately oriented radial strands. Then is the identity morphism.
Let and be two input diagrams with boundary and , respectively. Suppose that for a fixed , . Let be the input diagram obtained from gluing into the –th hole of . Then the morphism is equal to the composition of the morphisms and , i.e.
If is a monoidal functor, one may define the planar algebra over and by and
to be the composition , where is the natural transformation
which comes with the functor because it is monoidal. Examples of this construction include, for a planar algebra over , replacing for all the set by the free –modules for some ring by means of applying the left-adjoint of the forgetful functor from the category of –modules to ; or the quotient by an equivalence relation on , by which we mean a collection of equivalence relations on all the which respect the planar algebra structure.
Suppose and are planar algebras over , and , ,
respectively, such that . A planar algebra morphism from to
consists of a functor and an –indexed collection of
which respect the planar algebra structure, i.e. commute with the maps and
. The functor will typically be a forgetful functor.
Tangle diagrams with a base point on the boundary, considered – as input diagrams – up to smooth isotopy, form a planar algebra over and the set of sign-words with . Let us elaborate this example: the planar algebra associates to a sign-word with an equal number of both signs the (countably infinite) set of all tangles diagrams with boundary , modulo smooth isotopy; and to an input diagram with holes (see e.g. fig. 2) a function which maps a tuple of tangle diagrams with appropriate boundaries to a new, bigger tangle diagram, by gluing each of the tangle diagrams into the corresponding hole of . One easily verifies that the planar algebra axioms are satisfied.
2.3. The –polynomial in the context of planar algebras
Suppose the unit circle intersects a closed web generically; as in the definition of tangle diagrams, this means that the circle intersects the edges of the closed web transversely, and does not pass through a vertex. Then the intersection of the closed web with the unit disc is called a web. As for input diagrams, we fix a base point on the boundary of a web, and encode the isotopy type of the boundary by a sign-word, in which stands for a strand oriented away from the boundary, for a strand oriented towards it. Note that a sign-word is the boundary of some web if and only if . Denote by the set of all such sign-words. Webs, up to smooth isotopy, form a planar algebra over and .
Let be the free –module on . Then forms a planar algebra over and the category of –modules. In , we may interpret the relations (C), (D) and (S) as relations on , or and or , respectively. Denote by the quotient by the generated equivalence relation. In this context, the relation (U) is implied by the compatibility of the equivalence relation with the planar algebra structure.
Let be two webs and a diffeomorphism – just of the webs themselves, not taking into account the ambient discs. We call a web diffeomorphism if it preserves the order of the boundary points, and the cyclic ordering of edges around vertices. Note that in the quotient , the equivalence class of a web is already determined by its web diffeomorphism type. In , this distinction is slightly coarser than the isotopy type, since e.g. web diffeomorphisms do not take the orientation and relative position of closed components into account.
The two skein relations (Sk) determine a unique morphism of planar algebras, being the planar algebra of tangles. A link diagram may be seen as element of . The equivalence class has a unique member that is a –multiple of the empty web. The coefficient equals the –polynomial of the link diagram. Reidemeister invariance may be shown by proving that the tangle diagrams with two, four and six boundary points corresponding to the Reidemeister moves I, II and III have in each case the same image under .
2.4. The –homology in the context of a special kind of planar algebras: canopolis
To categorify the –polynomial, one needs to understand foams, the cobordisms of webs. Suppose that for all , are compact oriented smooth (generally not connected) 2–manifolds with boundary components each. Let and be orientation preserving diffeomorphisms. Consider the quotient of by the equivalence relation generated by for all . The images of the in the quotient are called singular circles, and the images of connected components of the are called facets. There are three facets adjacent to each singular circle. Associate a non-negative integer to each facet. Such an integer will graphically be represented by drawing dots on the facet, which may roam the facet freely, but may not cross a singular circle. Such a quotient, together with the dots and with a choice of cyclic ordering of the three facets around each singular circle is called a prefoam.
Now consider a smooth embedding of a prefoam into , i.e. an embedding that is smooth on the facets and on the singular circles. Such an embedding induces cyclic orderings of the facets around each singular circle by the left-hand rule (see figure 3).
If these cyclic orderings agree with those given by the prefoam, the image of the embedding is called a closed foam. Under the following conditions, the intersection of a closed foam with the cylinder is called a foam:
The boundary of the cylinder intersects the facets and singular circles of the closed foam transversely.
The side of the cylinder intersects the closed foam in finitely many vertical lines, and is disjoint from all singular circles.
The intersections with the top and bottom of the cylinder are webs.
The base point of the top and the base point of the bottom web have the same – and –coordinates.
We consider foams up to isotopies which, on the side of the cylinder, do not depend on the –coordinate. A connected component of the intersection of a singular circle with the cylinder is called a singular edge. The tangle diagrams on the bottom of the cylinder is called domain of the foam, and the codomain of the foam is defined as the tangle diagram on the top of the cylinder, with the orientation of each edge reversed. As usual, let denote the Euler characteristic. Then the degree of a foam is defined by
Foams can be glued in two ways: if the domain of one foam agrees with the codomain of another, by stacking them on top of each other. Or, by gluing them into the cylindrical holes of a thickened input diagram. The degree is additive with respect to both of these operations.
Webs with a fixed boundary and the foams between them thus constitute a graded category,
i.e. a category whose morphisms have an integral rank
which is additive under composition.
Let us define a planar algebra
over and the category
of small graded categories:
Next, may be constructed by applying a functor from to , the category of small additive categories: replace webs by –linear combinations of webs, and foams by matrices of –linear combination of foams, where morphisms from to are the foams with degree . So the categories are not graded, but have instead a shift operator for their objects. In this planar algebra, consider the following morphisms:
- The superscript stands for categorification.