Tau invariants for balanced spatial graphs

Tau invariants for balanced spatial graphs

Abstract.

In 2003, Ozsváth and Szabó defined the concordance invariant for knots in oriented 3-manifolds as part of the Heegaard Floer homology package. In 2011, Sarkar gave a combinatorial definition of for knots in and a combinatorial proof that gives a lower bound for the slice genus of a knot. Recently, Harvey and O’Donnol defined a relatively bigraded combinatorial Heegaard Floer homology theory for transverse spatial graphs in , extending HFK for knots. We define a -filtered chain complex for balanced spatial graphs whose associated graded chain complex has homology determined by Harvey and O’Donnol’s graph Floer homology. We use this to show that there is a well-defined invariant for balanced spatial graphs generalizing the knot concordance invariant. In particular, this defines a invariant for links in . Using techniques similar to those of Sarkar, we show that our invariant is an obstruction to a link being slice.

This work was partially supported National Science Foundation grant DMS-1309070.

1. Introduction

1.1. Background

A graph is a one-dimensional CW-complex whose edges (one-cells) may be oriented. A spatial graph is a smooth or piecewise linear embedding , where is an (oriented) graph. One way to think of spatial graphs is as a generalization of the classical study of knots and links, which are embeddings of one or more ordered components into . Just as for knots and links, we consider spatial graphs up to ambient isotopy.

Figure 1.1. A spatial graph

Knot Floer homology is a package of invariants which was independently defined in 2002 by Ozsváth and Szabó [OS04b] and by Rasmussen [Ras03]. One invariant from the knot Floer homology package is the invariant, which was defined by Ozsváth and Szabó in 2004 [OS04a].

One reason the invariant is important is its relationship to knot concordance. The invariant is a concordance invariant and its absolute value is a lower bound for slice genus [OS04b]. In 2011, Sarkar gave a combinatorial proof of the relationship between and slice genus [Sar11]. Recently, Harvey and O’Donnol have defined graph Floer homology for a certain class of spatial graphs in using a grid diagram construction analogous to that used for knots and links [HO17]. However, while knot Floer homology is filtered by the integers, Harvey and O’Donnol’s graph Floer homology is not; rather it is relatively graded graded by the first homology group of the spatial graph complement.

1.2. Summary of main results

In this paper, we define a filtered version of graph Floer homology for balanced transverse spatial graphs whose associated graded object is Harvey and O’Donnol’s and prove that it is a spatial graph invariant. We prove that the filtered graph Floer chain complex is, up to filtered quasi-isomorphism, an invariant of balanced spatial graphs. Thus we have the following theorem.

Theorem 3.15.

For grid diagrams representing , there exist filtered quasi-isomorphisms and which preserve the symmetrized filtration .

This allows us to define a invariant for balanced spatial graphs and prove that it is an invariant.

Definition 3.13.

For a graph grid diagram representing a balanced spatial graph , define the invariant of to be

where is the map induced by inclusion.

Corollary 3.17.

If and are graph grid diagrams representing a balanced spatial graph , then .

Considering links as spatial graphs with one vertex and one edge in each link component, we obtain the following result relating the invariant to link cobordisms.

Theorem 4.5.

If and are - and -component links, respectively, and is a connected genus cobordism from to , then

As a corollary, we see that the invariant can be an obstruction to a link being slice.

Corollary 4.6.

If an -component link has or , then is not slice.

Recently, Cavallo independently defined a invariant for links and proved a result similar to Theorem 4.5 [Cav18].

2. Graph Floer Homology

In this section we give an overview of Harvey and O’Donnol’s graph Floer homology, which is defined for transverse spatial graphs. For precise definitions of spatial graphs and transverse spatial graphs, see [HO17].

Definition 2.1.

A spatial graph is an embedding of a -dimensional CW-complex into . An oriented spatial graph is a spatial graph with an orientation given for each edge. For each vertex of an oriented spatial graph, the incoming edges of are the edges incident to whose orientation points toward , and the outgoing edges of are the edges incident to whose orientation points away from . A disk graph is one which has a standard disk at each vertex, attached to the graph by identifying the center point of with the vertex. A transverse spatial graph is an embedding of an oriented disk graph , such that at each vertex the standard disk is embedded in a plane that separates the incoming and outgoing edges, as shown in Fig. 2.1.

Figure 2.1. The standard disk separating incoming and outgoing edges at a vertex of a transverse spatial graph

In contrast to spatial graph ambient isotopy, in which any combination of edges incident to a vertex can move freely, ambient isotopy of transverse spatial graphs only allows free movement of incoming edges with other incoming edges or outgoing edges with other outgoing edges at each vertex. This is because the edges may not pass through the standard disk at the vertex.

Graph Floer homology is defined using grid diagrams, like the combinatorial definition of knot Floer homology. The definition of spatial graph grid diagrams is very similar to the definition of grid diagrams for knots and links.

An index graph grid diagram for a transverse spatial graph is an by grid in which each grid square may contain an -marking, an -marking, or be empty, such that there is exactly one in each row and in each column. We make a distinction between standard -markings, which are those which are in the interior of a graph edge when we recover the spatial graph from the graph grid diagram, and special -markings, which are vertices of the graph when it is recovered from the graph grid diagram. We mark special ’s with an asterisk in the graph grid diagram. Standard -markings have exactly one in their row and column, while vertex ’s may have any number of -markings in their row and column. If a transverse spatial graph has more than one connected component, we require that there be at least one special -marking in each component. A toroidal graph grid diagram is one in which we think of the grid as being a torus, with the leftmost and rightmost gridlines identified and the top and bottom gridlines identified.

Figure 2.2. Graph grid diagram (note the starred vertex )

To recover the spatial graph from a grid diagram, connect the ’s to the ’s vertically and the ’s to the ’s horizontally. At each crossing, the vertical strand is the overpass and the horizontal strand is the underpass. At vertex ’s (those with more than one in their row or column) use a straight line to connect the closest in the row or column to the vertex and a curved line to connect the more distant ’s to the vertex , observing the same conventions with regard to the crossings created, so that the line connecting two markings within a column is always the overstrand. See Fig. 2.2. Just as is the case for knots and links, every transverse spatial graph can be represented by a graph grid diagram.

For knots and links, Cromwell’s theorem [Cro95] gives a sequence of grid moves connecting any two grid diagrams representing equivalent links. Harvey and O’Donnol have proved a similar theorem for transverse spatial graphs.

Theorem 2.2 ([Ho17]).

Any two graph grid diagrams for a given transverse spatial graph are related by a finite sequence of cyclic permutation, commutation’, and (de-)stabilization’ moves.

A cyclic permutation moves the top (resp. bottom) row of a grid diagram to the bottom (resp. top) or moves the left (right) column to the far right (left) of the diagram. See the example in Fig. 2.3. Thinking of the grid as a torus, this equates to changing which gridline we “cut” the torus along to get the square diagram.

Figure 2.3. A cyclic permutation move

Two adjacent columns (or rows) may be exchanged using a commutation’ move if there are vertical (horizontal) line segments and on the torus such that contain all the ’s and ’s in the two adjacent columns (rows), the projection of to a single vertical circle (horizontal circle ) is (), and the projection of their endpoints, , to a single () is precisely two points. See the example in Fig. 2.4.

Figure 2.4. A commutation’ move: notice the dotted helper arcs in the left-hand grid

A row (column) stabilization’ at an -marking is performed by adding one new row and one new column to the grid next to that . The is then moved to the new row (column), remaining in the same column (row), with the and any other -markings in which were in the same row (column) as the being stabilized remaining in the old row (column). A new -marking is placed in the intersection of the new column (row) and the row (column) previously occupied by the -marking, and a new is placed in the intersection of the new row and column. See the example in Fig. 2.5. A destabilization’ is the opposite of a stabilization’.

Figure 2.5. A row stabilization’

Harvey and O’Donnol’s graph Floer homology is defined for transverse spatial graphs without sinks or sources. A sink is a vertex with no outgoing edges and a source is a vertex with no incoming edges. In other words, graph Floer homology is defined for spatial graphs whose underlying graph has at least one incoming edge and at least one outgoing edge at every vertex. This corresponds to a requirement that a graph grid diagram representing the spatial graph has at least one -marking in every row and column.

For a spatial graph represented by an graph grid diagram , the graph Floer chain complex is freely generated as a module over , where and the ’s are formal variables corresponding to the -markings in the graph grid diagram. The generating set of is

where is the symmetric group on letters.

The map counts empty rectangles in the toroidal graph grid diagram . An embedded rectangle in connects a generator to another generator if for all but two , if are the two indices for which and are not equal, and if the corners of are, clockwise from the bottom left, and . We say that is empty if the interior of does not contain any points of or . The set of empty rectangles from to is denoted . The map is defined as follows on the generating set and then extended to all of as an -module homomorphism:

where is zero if is not in and one if is in . Note that counts rectangles that contain any of the -markings in but does not count any rectangles that contain -markings. This is because does not have a natural filtration, so Harvey and O’Donnol’s graph Floer homology is graded rather than filtered.

Proposition 2.3 ([Ho17] Proposition 4.10).

For as defined above, .

Before we can define the Alexander grading we need to define weights of the edges of . We define a weight function , where is the set of edges of , by mapping each edge to the homology class of the meridian of , oriented according to the right-hand rule, as shown in Fig. 2.6.

Figure 2.6. The weight of an edge

For -markings and -markings associated to the interior of an edge , the weights are or . For -markings associated to a graph vertex , the weight is , where and are, respectively, the sets of incoming and outgoing edges of .

We can now define the Alexander grading on the generating set :

This grading is not well-defined on toroidal graph grid diagrams, but Harvey and O’Donnol show that the relative grading is well-defined on toroidal graph grid diagrams ([HO17] Corollary 4.14).

The graph Floer chain complex is bigraded, with an absolute -valued grading (the Maslov grading) and a relative -valued grading (the Alexander grading). The graph Floer homology is for any graph grid diagram representing , and it is also absolutely -graded and relatively -graded.

3. Filtered Graph Floer Homology and the Invariant

3.1. Spatial Graphs and the Chain Complex

In this section, we will define our filtered graph Floer homology chain complex. It is defined for balanced spatial graphs.

Definition 3.1.

A transverse spatial graph is balanced if there is an equal number of incoming and outgoing edges at each vertex.

Figure 3.1. A balanced spatial graph

For an index grid diagram representing a spatial graph , we choose an ordering for the -markings of and denote them . The chain complex is freely generated over , where and each is a formal variable corresponding to the -marking . It is generated by the set of unordered -tuples of intersection points in with one point on each horizontal and vertical gridline. The generating set is in bijection with , the set of permutations of elements, so . See Fig. 3.2 for an example of a generator.

Figure 3.2. A generator of
Definition 3.2.

A rectangle in the grid diagram connects a generator to another generator if its lower left and upper right corners are points in , its upper left and lower right corners are points in , and all other points in and coincide. Such a rectangle is empty if its interior does not contain any points of and . An empty rectangle may contain - and -markings. The set of empty rectangles from to is denoted .

Figure 3.3. An empty rectangle connecting the black generator to the white generator

The boundary map is defined as follows on the generators and extended linearly to :

where if is contained in and otherwise.

If is a graph grid diagram representing a balanced spatial graph, the chain complex is bigraded over . The gradings are defined using the following bilinear map .

For a point and a finite set of points in the plane, define to be half of the number of points in which lie either above and to the right of or below and to the left of . That is, . By extending bilinearly to formal sums and differences of sets of points in the plane, we can make the following definition, which is the same as the Maslov grading defined in [MOST07] and [HO17].

Definition 3.3.

The Maslov grading, also known as the homological grading, is defined as follows on the generators of the chain complex:

where and are the sets whose points are the - and -markings, respectively. The Maslov grading is extended to the rest of the chain complex by

For example, the Maslov grading of the element is .

Definition 3.4.

The -valued Alexander grading is defined as follows for grids which represent balanced spatial graphs (for grids representing spatial graphs that are not balanced, an -valued Alexander grading can be defined, as in [HO17]):

where is the weight of : the number of -markings in the same column (or equivalently, since we are restricting to balanced graphs, row) as . The Alexander grading is extended to the rest of the chain complex by

We can also view the Alexander grading as a relative grading, namely , where are elements of the chain complex, computed using rectangles. Any two generators in are connected by a sequence of rectangles. This follows from the fact that is in bijection with the symmetric group on letters, . If , there exists a finite sequence of transpositions that will turn into . If are the generators in corresponding to and , respectively, then that sequence of transpositions corresponds to a sequence of rectangles connecting to . The following lemma is very similar to Lemma 4.13 in [HO17].

Lemma 3.5.

If are generators of the chain complex and is a rectangle (not necessarily empty) connecting to , then the relative Alexander grading of and is

Proof.

By Definition 3.4,

Where and are the regions of the grid indicated in Fig. 3.4.

Figure 3.4. The regions of the grid referred to in Lemma 3.5

Definition 3.6.

The Alexander filtration of is , where is generated by those elements of whose Alexander grading is less than or equal to .

Proposition 3.7.

is a filtered chain complex. That is, , the boundary map decreases by one the Maslov grading of elements which are homogeneous with respect to the Maslov grading, and the boundary map preserves the relative Alexander filtration.

Proof.

That follows directly from the proof of Proposition 2.10 of [MOST07], since graph grid diagrams differ from link grid diagrams only in the -markings, and the definition of does not involve -markings.

The proof that decreases Maslov grading by one is also the same as in [MOST07]. By their Lemma 2.5, if is an empty rectangle from to , then . Therefore the term in corresponding to will have Maslov grading

To show that preserves the relative Alexander filtration, note that if a rectangle connects to , then . Therefore the term in corresponding to will have Alexander grading

Definition 3.8.

Suppose the -markings in are numbered so that are edge ’s and are vertex ’s. Let be the minimal subcomplex of containing . Then is the filtered chain complex obtained from by setting and letting be the map on the quotient induced by . We consider as a vector space over .

We denote by the homology of the associated graded object of . It is finitely generated as a vector space over , since all of the ’s act trivially on it ([HO17] Proposition 4.29).

3.2. Alexander filtration and the invariant

For a knot , the Alexander filtration of the knot Floer homology chain complex for is an absolute grading preserved under the maps associated to the commutation and (de)stabilization grid moves. For balanced spatial graphs, as discussed elsewhere in this chapter, only the relative Alexander filtration of the graph Floer homology chain complex is preserved under the maps associated to the commutation’ and (de)stabilization’ grid moves. Therefore, in order to define a invariant for balanced spatial graphs, we need to fix an absolute Alexander grading and filtration of the graph Floer homology chain complex that will be preserved under the maps associated to all of the graph grid moves.

To do this, we show that the homology of the associated graded complex is non-trivial. To show this, we appeal to the following lemma.

Lemma 3.9.

Let be a filtered chain complex with filtration of such that and . If for each homological grading , the chain group is finitely generated, then for some .

Proof.

Since and , there exists some for which . Therefore there is some non-zero which is homogeneous with respect to the homological grading , with , and whose homology class is nonzero. We can then choose the minimal filtration level so that .

Let . Then . If is not a boundary in the chain complex , then and we are done.

If is a boundary in , then there is some with . Set . Since is a cycle, . Therefore is a cycle and since and differ by a boundary, in .

We can repeat this process, choosing the minimal filtration level so that , yielding a cycle with in . Iterating this process will produce infinitely many representatives of , each in different filtration levels. This contradicts our hypothesis that for each homological grading , the chain group is finitely generated. ∎

Note that the grid chain complex satisfies the condition in Lemma 3.9 that for each Maslov grading level , the chain group is finitely generated. This is because all elements of are of the form for some generator and with for all , so

Since there are finitely many generators, since is finite, and since there are only finitely many ways to write a given number as the sum of finitely many positive integers, the condition is satisfied.

Definition 3.10.

For a grid diagram representing a balanced spatial graph , define the symmetrized Alexander filtration to be the absolute Alexander filtration obtained by fixing the relative Alexander grading so that , where and .

Now that we have symmetrized the Alexander filtration of , we can lift that filtration to a symmetrized filtration of .

Definition 3.11.

Define the symmetrized Alexander filtration of to be , obtained by fixing the relative Alexander grading of so that each generator is in the same filtration level of as it is in .

Remark 3.12.

This is not necessarily the only way to symmetrize the Alexander filtration. If we knew that the bigraded Euler characteristic of (which is an Alexander polynomial, see [HO17]) were non-zero, then we could fix an absolute Alexander grading so that the maximal and minimal terms with non-zero coefficients in the Alexander polynomial were centered around zero. It would be interesting to answer the question of whether these two ways of fixing the Alexander grading are equivalent.

Definition 3.13.

For a graph grid diagram representing a balanced spatial graph , define the invariant of to be

where is the map induced by inclusion.

In proving the next theorem, we will appeal to this lemma.

Lemma 3.14 ([McC01], Theorem 3.2).

If is a filtered chain map which induces an isomorphism on the homology of the associated graded objects of and , then is a filtered quasi-isomorphism.

Theorem 3.15.

For grid diagrams representing , there exist filtered quasi-isomorphisms and which preserve the symmetrized filtration .

Proof.

For graph grid diagrams and both representing a balanced spatial graph , we know by Theorem 2.2 [HO17] that there is a finite sequence of cyclic permutation, commutation’, stabilization’, and destabilization’ moves which turns into . Thus, once we show that each of these grid moves is associated to a quasi-isomorphism of filtered chain complexes, we can take the composition of the maps associated to each of the grid moves in the sequence, resulting in a filtered quasi-isomorphism from to . The proof that each of the grid moves is associated to a quasi-isomorphism of filtered chain complexes consists of three steps:

  1. We need to show that if and are graph grid diagrams which are related by a cyclic permutation, commutation’, stabilization’, or destabilization’ grid move, there exists a chain map and an integer such that for all , we have , and such that induces an isomorphism . Note that here, we are working with the original Alexander filtration rather than the symmetrized version. This will be proved in Section 3.3, Section 3.4, and Section 3.5.

  2. We need to show that each of the maps from Step (1) induces a quasi-isomorphism on the symmetrized Alexander filtration. That is, we need to show that is an isomorphism. Since we know from Step (1) that induces an isomorphism on the homology of the associated graded objects, it is sufficient to show that the span is the same for both and . We will show that and .

    Assume for the sake of contradiction that . Then there exists some such that is non-trivial. Then, since

    is an isomorphism, there exists some non-trivial in. This contradicts our assumption that , so we have that . Similar arguments show that , and that . Therefore we have shown that .

  3. We need to know that the existence of a quasi-isomorphism on the associated graded object of a filtered chain complex implies the existence of a filtered quasi-isomorphism on the filtered chain complex. This is exactly what Lemma 3.14 [McC01] says.

Lemma 3.16.

Suppose that there exist filtered quasi-isomorphisms and . Then .

Proof.

Suppose that . Then we have the following commutative diagram:

{tikzcd}

H_*(^F_a(g)) \arrowri_* \arrowdF^a_* & H_*(^CF(g)) \arrowdF_*

H_*(^F_a(¯g)) \arrowrj_* & H_*(^CF(¯g))

Thus there is some which maps via to a non-zero element of to which sends . Therefore is non-trivial, so .

The same argument using says that , so putting the two inequalities together gives the result that . ∎

With the previous lemma, we have shown the following corollary to Theorem 3.15.

Corollary 3.17.

If and are graph grid diagrams representing a balanced spatial graph , then .

Now we have a well-defined invariant for balanced spatial graphs.

Definition 3.18.

For a balanced spatial graph , if is any graph grid diagram representing , then

3.3. Cyclic Permutation

Suppose that and are graph grid diagrams which differ by a cyclic permutation move. Since the chain complex and are defined from toroidal grid diagrams, the chain map associated to the cyclic permutation grid move is the identity map, so it is a quasi-isomorphism. However, we still need to show that the map preserves the Alexander filtration, which was defined using planar grid diagrams.

From Lemma 3.5 and Corollary 4.14 in [HO17], we know that the relative Alexander grading is well-defined on the toroidal grid diagram. Define new gradings and by shifting the Alexander gradings on and , respectively, so that in each one, , the generator whose points are at the lower left corner of each of the grid squares containing and -marking, has grading zero. Now the identity map preserves this shifted grading. If and were the shifts from to and from to , respectively, then we see that the identity map sends elements of with Alexander grading to elements of with Alexander grading . Therefore , the map induced by the identity, is an isomorphism.

3.4. Commutation’

Let and be graph grid diagrams which differ by a commutation’ move. We can depict both grids in a single diagram, as shown in Fig. 3.5. In this example is the graph grid diagram obtained from Fig. 3.5 by deleting the line labeled , and is the graph grid diagram obtained from it by deleting . The proof of commutation’ invariance closely follows that in [HO17].

Figure 3.5. A grid showing both and . The grid diagram with but not is , and the diagram with but not is .

Recall that the differential map counts empty rectangles connecting generators in . In this section, we will consider maps that count empty pentagons and hexagons in the combined grid showing both and . An embedded pentagon in the combined grid diagram connects to if and agree in all but two points, and if the boundary of is made up of arcs of five grid lines, whose intersection points are, in counterclockwise order, , where , , and . See Fig. 3.6 for an example. Such a pentagon is empty if its interior does not contain any points of or . The set of empty pentagons connecting to is denoted .

Definition 3.19.

For , let

and note that .

Figure 3.6. A pentagon connecting the black generator to the white generator, counted in .
Lemma 3.20.

The map is a chain map which preserves Maslov grading and respects the Alexander filtration, which is to say that for some , where is the unsymmetrized Alexander filtration of . Moreover, it induces an isomorphism on the homology of the associated graded object, so

is an isomorphism for all .

Proof.

This proof has three parts:

  1. preserves Maslov grading. This follows immediately from Lemma 5.2 in [HO17] because the difference between their map between associated graded chain complexes and our filtered map between filtered chain complexes is that in the filtered setting pentagons may contain -markings, but Maslov grading does not involve the -markings on the grid in any way.

  2. The map preserves the Alexander filtration in the sense given in the statement of the lemma and induces an isomorphism on the homology of the associated graded object. In the proof of Lemma 5.2 in [HO17], Harvey and O’Donnol show that their map shifts the Alexander grading by some fixed element , which is the class in of the sums of the meridians of the graph arcs connecting the and -markings in the upper region and the lower region between and in the combined grid. By collapsing their Alexander grading using the obvious map from to , we obtain from their the induced map of our on the associated graded objects