Graph Homologies and Functoriality
We follow the same technics we used before in [AZ] of extending knot Floer homology to embedded graphs in a 3-manifold, by using the Kauffman topological invariant of embedded graphs by associating family of links and knots to a such graph by using some local replacements at each vertex in the graph. This new concept of Graph Floer homology constructed to be the sum of the knot Floer homologies of all the links and knots associated to this graph and the Euler characteristic is the sum of all the Alexander polynomials of links in the family. We constructed three pre-additive categories one for the graph under the cobordism and the other one is constructed in [AMM] the last one is a category of Floer homologies for graph defined by Kauffman. Then we trying to study the functoriality of graphs category and their graph homologies in two ways, under cobordism and under branched cover, then we try to find the compatibility between them by the idea of Hilden and Little [HL] by giving a notion of equivalence relation of branched coverings obtained by using cobordisms, and hence define a a functor from the graph Floer-Kauffman Homology category to the graph Khovanov-Kauffman Homology category.
Key words and phrases:Floer knot homology, embedded graphs, kauffman replacements, graph cobordism group, graph homology
Of course, no introduction can be completely self-contained, so let me say a few words
about the background knowledge the reader is assumed to have.
Many publications appeared studying Homology in Low dimensional Topology during last two decades.
These studies depended on the concept of categorification, which originally arose in representation theory
and was coined by L. Crane and I. Frenkel [CF]. In the context
of topology, categorification is a process of upgrade algebraic invariants of topological
objects to algebraic categories with richer structure.
The idea of categorification the Jones polynomial is known by
Khovanov Homology for links which is a new link invariant introduced by
Khovanov [kh], [Ba], [pt1]. For each link in Khovanov defined a graded chain complex,
with grading preserving differentials, whose graded Euler characteristic is
equal to the Jones polynomial of the link .
In 2003 Ozsváth and Szabó and independently by Rasmussen defined a new useful knot invariant
called Knot Floer Homology by categorify the Alexander polynomial [os], [Rs].
Using grid diagrams for the Heegaard splittings, knot Floer
homology was given a combinatorial construction by Manolescu, Ozsváth and Sarkar (2009) [MOS].
The idea of Khovanov Homology for graphs arises from the same idea of Khovanov homology for links by
the categorifications the chromatic polynomial of graphs.
This was done by L. Helme-Guizon and Y. Rong [laur],
for each graph G, they defined a graded chain complex whose graded
Euler characteristic is equal to the chromatic polynomial of G.
In [AZ] the author defined Khovanov-Kauffman Homology for embedded Graphs by using the
Kauffman topological invariant of embedded graphs by associating family of links and
knots to a such graph where the Euler characteristic is the sum of all the Jones polynomials of links in the family.
Jonathan Hanselman in his work [Jh] described an algorithm to compute of any graph
manifold by using the result of two explicit computations of bordered Heegaard Floer
invariants. The first is the type D trimodule associated to the trivial -bundle
over the pair of pants . The second is a bimodule that is necessary for self-gluing,
when two torus boundary components of a bordered manifold are glued to each
Yuanyuan Bao in [Yb] defined the Heegaard diagram for a balanced bipartite graph in a rational homology 3-sphere, by introducing a base point for each edge. Then he defined the minus-version and hat-version of the Heegaard Floer complexes for a given Heegaard diagram. The hat-version coincides with the sutured Floer complex for the complement of the graph, the sutures of which are defined by using the meridians of the edges. Bao proved that the homology modules of both versions are topological invariants of the given graph and discussed some basic properties of the homology. He studied the Euler characteristic of the hat-version complex. In particular, when the ambient manifold is the 3-sphere, we give a combinatorial description of the Euler characteristic by using the states of a given graph projection.
Shelly Harvey and Danielle O’Donnol [Ho] extend the theory of combinatorial link Floer homology to a class of oriented spatial graphs called transverse spatial graphs. They defined the notion of a grid diagram representing a transverse spatial graph, which we call a graph grid diagram. They proved that two graph grid diagrams representing the same transverse spatial graph are related by a sequence of graph grid moves, generalizing the work of Cromwell for links. For a graph grid diagram representing a transverse spatial graph , they defined a relatively bigraded chain complex (which is a module over a multivariable polynomial ring) and showed that its homology is preserved under the graph grid moves; hence it is an invariant of the transverse spatial graph. In fact, they defined both a minus and hat version. Taking the graded Euler characteristic of the homology of the hat version gives an Alexander type polynomial for the transverse spatial graph. Specifically, for each transverse spatial graph , we define a balanced sutured manifold . They show that the graded Euler characteristic is the same as the torsion of defined by S. Friedl, A. Juhász, and J. Rasmussen.
We discuss the question of extending Floer homology from links to embedded graphs. This is based on a result of Kauffman that constructs a topological invariant of embedded graphs in the 3-manifold by associating to such a graph a family of links and knots obtained using some local replacements at each vertex in the graph. He showed that it is a topological invariant by showing that the resulting knot and link types in the family thus constructed are invariant under a set of Reidemeister moves for embedded graphs that determine the ambient isotopy class of the embedded graphs. We build on this idea and simply define the Floer homology of an embedded graph to be the sum of the Floer homologies of all the links and knots in the Kauffman invariant associated to this graph. Since this family of links and knots is a topologically invariant, so is the Floer-Kauffman homology of embedded graphs defined in this manner.
The author would like to express his deeply grateful to Prof.Matilde Marcolli for her advices and numerous fruitful discussions. Some parts of this paper done in Max-Planck Institut für Mathematik (MPIM), Bonn, Germany, the author is kindly would like to thank MPIM for their hosting and subsidy during his study there.
3. Floer-Kauffman Homology for graphs
In this part we study a new concept which called Floer-Kauffman Homology for graphs by using Kauffman procedure [kauff] of associating a family of links to an embedded graph by making some local replacement to each vertex in the graph. We will start recalling the concept of Floer homology for links (Knots) but not in details and then we combine the definitions of Floer Homology for link with kauffman idea to produce Floer-Kauffman Homology for graphs. We close this paper by giving an example of computation of Floer-Kauffman homology for an embedded graph using this definition.
3.1. Link Floer Homology
Ozsváth - Szabó and Rasmussen around 2003 Introduced Floer Homology which is an invariant of knots and links in three manifolds. This invariant contain many information about several properties of the knot (genus, slice genus fiberedness, effects of surgery). Ozsváth - Szabó
used Atiyah Floer Conjecture to develop Heegaard Floer Theory as a symplectic geometric replacement for gauge theory by using Gromov’s theory of pseudo-holomorphic curves to construct an invariant of closed 3-manifolds called Heegaard Floer homology. Knot Floer homology is a relative version of Heegaard Floer homology, associated to a pair consisting of a 3-manifolds and a nullhomologous knot in it. Seiberg-Witten equation is the origin of the knot floer homology which is play an important role in 3 and 4 dimensional topology. An invariant called Seiberg-Witten Floer homology of a 3-dimensional manifold constructed by studying the equation of Seiberg-Witten on the 4-dimensional manifold . Heegaard Floer homology and Seiberg-Witten Floer homology are isomorphic. Knot Floer homology can be thought of as encoding something about the Seiberg-Witten equations on times the knot complement and it is very similar in structure to knot homologies coming from representation
theory, such as those introduced by Khovanov and Khovanov-Rozansky in addition to that it can also be calculated for many small knots using combinatorial methods.
Let be an oriented knot. There are several different variants of the knot Floer homology of . The simplest is the hat version, which takes the form of a bi-graded, finitely generated Abelian group
Here, is called the Maslov (or homological) grading, and is called the Alexander grading. The graded Euler characteristic of is the Alexander-Conway polynomial
[RA] The filtered Poincaré polynomial of a knot homology is given by
It is a Laurent polynomial in and .
If we substitute , the filtered Poincaré polynomial reduces to the filtered
Euler characteristic. When , we will often use the shorthand to refer to a generator of this group.
Let be a knot in . There are various Heegaard Floer homology groups of such as , , , and , but our study will be just with . The knot Floer homology is a bigraded chain complex equipped with a homological grading and a filtration grading , which is also known as the Alexander grading. Conventionally, the Alexander grading is chosen so as to define a downward filtration on . The filtered Euler characteristic of is the Alexander polynomial , and the homology of the complex is a single copy of in homological grading . We can extend Knot Floer Homology to Links Floer Homology and the Euler characteristic for it multiply by the factor where this the number of the components of the link. Let be an oriented -component link. Ozsváth - Szabó [os1] show how can naturally be thought of as a knot in connected sum of copies of . This construction gives rise to a knot Floer homology group , which is again a filtered complex. Its filtered Euler characteristic is given by
and its total homology has rank . The Poincaré polynomial of the total homology is given by
(when is odd, the homological grading on is naturally an element of ( rather than of .) In this study, we will not speak more about the many definitions of Knot (Link) Floer homology, but one can find more details in [os], [os1], and [Rs].
[RA] Let be a link in . Here we give some properties of knot Floer homology.
where denotes with the orientations of all components reversed.
where the is the mirror image of , and denotes the operation of taking the dual complex.
For two oriented links and , the Knot Floer homology of the disjoint union satisfies
where is the rank two complex Poincaré and trivial deferential.
For two oriented links and , the Knot Floer homology of the oriented connected sum satisfies
A natural question is what knot types can be distinguished by knot Floer homology. If and are distinguished by the Alexander polynomial, then they are also distinguished by knot Floer homology. However, knot Floer homology is a strictly stronger invariant. For example:
If denotes the mirror of , then . On the other hand, for the trefoil, and for many other knots;
If differ from each other by Conway mutation, then . A well-known example of mutant knots, the Conway knot and the Kinoshita-Terasaka knot, have different knot Floer homologies.
knot Floer homology is generally an effective invariant for distinguishing between two small knots. Nevertheless, it has its limitations: we can find examples of different knots with the same knot Floer homology (and, in fact, with the same full knot Floer complex up to filtered homotopy equivalence). The alternating knots and are the simplest such example. A related question is what knots E are distinguished from all other knots by knot Floer homology. At present, the only known examples are the four simplest knots: the unknot, the two trefoils, and the figure-eight;
3.3. Homology theories for embedded graphs
In [kauff] Kauffman introduced an idea relating graphs with links. He by making some local replacements at each vertex in the graph (Figure1)
can get a family of links associate to graph which is an invariant under expanded Reidemeister moves defined by Kauffman (Figure 2).
In [AZ] we used kauffman technique to introduce Khovanov-Kauffman homology for embedded graphs. In this part we intend to define similar homology for graphs by using the concept of Floer homologies for links associated to the graph , to get Floer-Kauffman homology for graphs.
3.4. Definition of Floer homology for embedded graphs
We define the concept of Floer homology for embedded graphs by using Floer homology for the links (knots) and Kauffman theory of associate a family of links to an embedded graph , as described above.
3.5. Kauffman’s invariant of Graphs
We give now a survey of the Kauffman theory and show how to associate to an embedded graph in -Manifold a family of knots and links. We then use these results to give our definition of Floer homology for embedded graphs. In [kauff] Kauffman introduced a method for producing topological invariants of graphs embedded in -Manifold. The idea is to associate a collection of knots and links to a graph so that this family is an invariant under the expanded Reidemeister moves defined by Kauffman and reported here in figure (2).
He defined in his work an ambient isotopy for non-rigid (topological) vertices. (Physically, the rigid vertex concept corresponds to a network of rigid disks each with (four) flexible tubes or strings emanating from it.) Kauffman proved that piecewise linear ambient isotopies of embedded graphs in -Manifold correspond to a sequence of generalized Reidemeister moves for planar diagrams of the embedded graphs.
Let be a graph embedded in -Manifold. The procedure described by Kauffman of how to associate to a family of knots and links prescribes that we should make a local replacement as in figure (1) to each vertex in . Such a replacement at a vertex connects two edges and isolates all other edges at that vertex, leaving them as free ends. Let denote the link formed by the closed curves formed by this process at a vertex . One retains the link , while eliminating all the remaining unknotted arcs. Define then to be the family of the links for all possible replacement choices,
[kauff] Let be any graph embedded in -Manifold, and presented diagrammatically. Then the family of knots and links , taken up to ambient isotopy, is a topological invariant of .
For example, in the figure (3) the graph is not ambient isotopic to the graph , since contains a non-trivial link.
Let be an embedded graph with the family of links associated to by the Kauffman procedure. Let be the usual link Floer homology of the link in this family. Then the Floer homology for the embedded graph is given by
Its graded Euler characteristic is the sum of the graded Euler characteristics of the Floer homology of each link, i.e. the sum of the Alexander polynomials,
We show some simple explicit examples.
4. Category of embedded graphs cobordism
It is interesting to define a cobordism category for graphs by using the idea of kauffman mentioned before. Our construction depends on associating a family of cobordisms for links to a cobordism for graphs, and hence can define a homomorphism for Khovanov-Kauffman and Floer-Kauffman homologies for graphs. In the next section by using the idea of kauffman of associating family of links to a graph, we try to define graph cobordism category whose objects are oriented embedded graphs in , and whose morphisms are isotopy classes of 2-complex oriented, graph cobordisms in and hence discuss the functoriality with the category of group of Kauffman-Floer homologies, whose morphisms are isomorphisms.
4.1. Graph Cobordance
Here we will define two types of the cobordance between two graphs, one by PL cobordism defined in (4.2) and the second is by a family of smooth cobordisms defined in (4.3). We will prove that these two definitions are compatible and equivalent by the associativity of the smooth cobordisms to the PL cobordism . The second definition will help us to define an additive category of embedded Kauffman-graphs under cobordism. We begin by recalling the definitions of cobordisms in surfaces (smooth and PL surface) and set of smooth surfaces associated to the PL cobordism that will be useful in the rest of our work.
Two links and are called cobordic if there is a surface have the boundary with , . Here by ”surfaces” we mean -dimensional compact differentiable manifold embedded in . We define the identity cobordism to be for a link . denotes the cobordism class of the link .
[AZY] Two graphs and are called cobordic if there is a PL surface have the boundary with , . Here by ”surfaces” we mean -dimensional simplicial complexes that are PL-embedded in . We define the identity cobordism to be for a graph . denotes the cobordism class of the graph .
Let two embedded graphs have associated families of links and respectively according to Kauffman construction, and let be a family of smooth cobordisms have links boundaries of and . These graphs are said to be cobordant if there is a family of smooth cobordisms in such that and are the boundary of . These family of smooth cobordisms defined in (4.1) can be associated to the PL cobordism between and defined in (4.2).
The family of smooth cobordisms is associated to the PL cobordism .
We show that the union is a cobordism between the Kauffman invariants of the graph and of the graph . Let be a 2-complex embedded in , with boundaries and . Define the map to be the projection on the second factor. Then, for each , consider . This is an embedded graph in , You can apply to the construction of Kauffman to get (a collection of links associated to for each ). By using Kauffman procedure we can associate for all and a family of links and this means we can obtain many cobordisms between and in different ways but they are all equivalent and hence we can make a class of cobordisms for links between and . In case the 2-complex for some be a point or line or even a graph but has no family of links associate this means that the set is empty and for some where and graphs have families of links then we can use unit and counit to close the smooth surface. ∎
4.2. Composition of Cobordisms
We can define the composition between two PL cobordisms and (where is the cobordism with boundary , is the cobordism with boundary in ) to be which is a PL cobordism with boundary . For the second type of the cobordance by family of smooth cobordisms we can define the composition as follows : Let and be embedded graphs in with and links families associated to each graph respectively. We have a family of smooth cobordisms have boundary and another family with boundary . Let and be a smooth cobordism with boundary links from the sets and and let for be a smooth cobordism with boundary links from the sets and . is a smooth cobordism with boundary links from the sets and and this defines the composition of the second type of the graphs cobordance. Let be a graph cobordism, we can think of the class of link cobordisms as a morphism between and
is not unique, we can get another with the same class of link cobordism with the addition of coefficients. Let be a category, whose objects are embedded graphs that have a family of links according to the kauffman definition and morphisms are -dimensional simplicial complex surface with graphs boundary. As we maintained before, to each graph cobordism we can associate a family of link cobordisms with boundary family of links associated to each graph. There are many link cobordism can be associated but they are all equivalent.
A pre-additive category is a category such that, for any the set of morphisms is an abelian group and the composition of maps is a bilinear operation, that is, for the composition
is a bilinear homomorphism.
is a pre-additive category.
In fact, we can write PL morphisms between graphs in term of smooth morphisms between links as follows:
where the sum ranges over the set of all PL cobordisms and all but finitely many of the coefficients are zero. Then, for
The composition rule given by composition of cobordisms. We can compose cobordisms by gluing graphs. This gives a bilinear homomorphism
This shows that is a pre-additive category. In our case, the set of morphisms is an abelian group
Suppose given a pre-additive category . Then the additive category is defined as follows (cf. [Ba]).
The objects in are formal direct sums of objects , where we allow for the direct sum to be possibly empty.
If is a morphism in with objects and then is a matrix of morphisms in . The abelian group structure on is given by matrix addition and the abelian group structure of .
The composition of morphisms in is defined by the rule of matrix multiplication and the composition of morphisms in .
Then is called the additive closure of . For more details see for instance [Ba].
In the following, for simplicity of notation, we continue to use the notation for the additive closure of the category of Definition 4.13. Define an equivalence relation between two graphs and as follows: two graphs are said to be equivalent if they have the same set of links associated to both and for each local replacement to each vertex in and , and hence their Floer-Kauffman homologies are isomorphic . is the equivalence class of graphs under the Kauffman idea. This equivalence built a new class for Khovanov-Kauffman and Floer-Kauffman homologies. These definitions of cobordance of and can define an equivalence classes for graphs. Two graphs and are said to be equivalent if there is PL cobordism (or family of smooth cobordisms between ) with boundary . and are equivalent (and have the same set of links ) then graph Khovanov-Kauffman and Floer-Kauffman homologies are equivalent (i.e. or ), and this class induce a Khovanov morphism also this class of cobordisms can be associate to the 2-complex with boundary and .
4.3. Category of embedded graphs under branched covers
In [AMM] we constructed a Statistical Mechanical System of 3-manifolds as branched coverings of the 3-sphere, branched along embedded graphs (or in particular knots) in the 3-sphere and where morphisms are formal linear combinations of 3-manifolds. Our definition of correspondences reliesed on the Alexander branched covering theorem, which shows that all compact oriented 3-manifolds can be realized as branched coverings of the 3-sphere, with branched locus an embedded (not necessarily connected) graph. The way in which a given 3-manifold is realized as a branched cover is highly not unique. It is precisely this lack of uniqueness that makes it possible to regard 3-manifolds as correspondences.
[pr] A branched covering of 3-manifolds is defined as a continuous map such that there exists a one-dimensional subcomplex in whose inverse image is a one-dimensional subcomplex on the complement to which, , the restriction of is a covering. In this situation is called the covering manifold, is the base, and is the branching set.
Using cyclic branched coverings as the starting point, it is possible to construct other examples of branched coverings by performing surgery along framed links. In the base of the branched covering , let us choose a framed link and do surgery along , producing a manifold . Consider the inverse image of under the map and perform surgery along it, obtaining another manifold . The branched covering induces the branched covering ; we shall also call such a branched covering cyclic.
(Poincaré homology sphere): Let denote the Poncaré homology sphere. This smooth compact oriented 3-manifold is a 5-fold cover of branched along the trefoil knot (that is, the torus knot), or a 3-fold cover of branched along the torus knot, or also a 2-fold cover of branched along the torus knot. For details see [pr], [ks].
We constructed an additive category whose objects are embedded graphs in the
3-sphere and where morphisms are formal linear combinations of 3-manifolds.
In this section we will recall the definitions of morphisms between graphs and trying
to establish later a functor with the graph homologies by using the idea of Hilden and Little ([HL]).
Define between graphs as formal finite linear combinations
with and compact oriented smooth 3-manifolds with submersions
that are branched covers, respectively branched along and .
(Cyclic branched coverings): We represent as . Let be a straight line chosen in . Consider the quotient map that identifies the points of obtained from each other by a rotation by an angle of about the axis . Upon identifying , this extends to a map which is an -fold covering branched along the unknot and with multiplicity one over the branch locus.
We use the notation
for a 3-manifold that is realized in two ways as a covering of , branched along the graph or . These cyclic branched coverings are useful to construct other more complicated branched coverings by performing surgeries along framed links ([pr]).
[AMM] Suppose given
One defines the composition as
where the fibered product is defined as
The composition defined in this way satisfies the following property.
[AMM] Assume that the maps of (4.3) have the following multiplicities. The map is of order for and of order for ; the map is of order for and for ; the map is of order for and of order for ; the map is of order for and for . For simplicity assume that
Then the fibered product is a smooth 3-manifold with submersions
This definition makes sense, since the way in which a given 3-manifold is realized as a branched cover of branched along a knot is not unique.
[AMM] We let denote the category whose objects are graphs and whose morphisms are -linear combinations of 3-manifold with submersions and to , including the trivial (unbranched) covering in all the .
[AMM] The category is a small pre-additive category.
The proof is in [AMM].
4.4. Category of graph Khovanov and Floer homologies
In this section we define 4 homology groups categories (two for links homologies and two graphs homologies). Let be a category of Khovanov homology for links whose objects are khovanov homology groups and morphisms are group homomorphisms for some links. In a parallel direction we can define a category of Floer homologies for links, whose objects are links floer homology group defined in section (3.3) and morphisms are group homomorphisms for some links. Clearly and are Additive categories. In [AZ] we introduced the notation of Khovanov-Kauffman homology for graphs. To each embedded graph who can be under a certain of local replacements at each vertex associated by a family of links or knots, this definition help us to define a new category of Khovanov-Kauffman homology for groups whose objects are Khovanov-Kauffman homology groups and morphisms are group homomorphisms for some graphs. Here refereing to the kauffman. In definition (3.5) we defined Floer-Kauffman homology for graphs and hence we can define and new category of Khovanov-Kauffman homology for groups whose objects are Khovanov-Kauffman homology groups and morphisms are group homomorphisms for some graphs.
The categories and are small pre-additive (Additive) categories.
In the next section we need to study the Functoriality between the cobordisms categories and the homology categories.
5.1. Functoriality under PL Cobordisms
In this part we want to study the Functoriality between link, graph categories in this side and their homology groups categories in the other side.
5.1.1. Functoriality under Khovanov homology
An important problem in this theory is to extend the Khovanov homology to a monoidal functor from the category of cobordisms of oriented links. The first attempt by Khovanov gave a negative answer because of many problems of signs. Functoriality up to sign was conjectured by Khovanov and proved later by Jacobson [Ja], Bar Natan [BN2] and Khovanov [Kh3]. This functoriality up to sign was used by Rasmussen [jr] to prove a conjecture of Milnor about the slice genus. Let denoted the link cobordism category whose objects are oriented links in , and whose morphisms are isotopy classes of oriented, link cobordisms in . In this section we trying to introduce the functoriality between graphs cobordism category and Khovanov-Kauffman homology under the concept of PL cobordism. We give now the definition of cobordism class of the graph . We can define the morphism of Khovanov-Kauffman homology for graphs to induce the functoriality by using the same procedure prove the functoriality of Floer-Kauffman homologies obtain a morphism between the homologies using the cobordism L and existing results on functoriality of Khovanov or Floer theory of links with respect to link cobordisms. Let and be two embedded graphs in and is a PL cobordism in with . In [Jaco] Jacobsson showed that a movie for a cobordism in with starting and ending diagrams and for links and respectively can induces a map
Where is the khovanov homology. Two movies from to supposed to be equivalent if their lifts, for fixed links and , represent the same morphism in , and this means Khovanov homology is really a functor
where is the category of Vector spaces.
If we have a family of smooth cobordism then we can get a family of morphisms where is a homomorphism between khovanov homologies for links and . Since and are two embedded graphs in , then according to kauffman construction we can associate a family of links to both and and denoted . We defined Khovanov-Kauffman homology for graphs as: for and for where . In theorem (4.4) we showed that, for a PL cobordism we can associate a family of smooth cobordism to and hence a family of khovanov homology homomorphisms induced by these smooth cobordisms, this family of homomorphisms induced by smooth cobordisms will induce a homomorphism from a PL cobordism with boundary .