A refinement of the Ozsváth-Szabó large integer surgery formula and knot concordance
We compute the knot Floer filtration induced by the –cable of the meridian of a knot in the manifold obtained by large integer surgery along the knot. We give a formula in terms of the original knot Floer complex of the knot in . As an application, we show that the concordance invariant of Hom can equivalently be defined in terms of filtered maps on the Heegaard Floer homology groups induced by the two-handle attachment cobordism of surgery along a knot in .
Let denote the manifold constructed as Dehn surgery along with surgery coefficient . In [HFK] Ozsváth and Szabó construct a chain homotopy equivalence between certain subquotient complexes of the full knot Floer chain complex and Heegaard Floer chain complexes for sufficiently large integers for each spin structure . This equivalence is known as the large integer surgery formula.
The meridian of naturally lies inside of the knot complement and the surgered manifold . The meridian induces a filtration on for each spin structure . In [hedden] Hedden gives a formula for the filtered complex in terms of for sufficiently large . As an application of this formula, Hedden computes the knot Floer homology of Whitehead doubles and the Ozsváth-Szabó concordance invariant of Whitehead doubles. In [hedden-kim-livingston] Hedden, Kim, and Livingston generalize Hedden’s formula by computing the full knot Floer complex in terms of for sufficiently large . As an application to knot concordance, they show that the subgroup of topologically slice knots of the concordance group contains a subgroup.
We refine the theorems of Ozsváth-Szabó, Hedden and Hedden-Kim-Livingston to determine the filtered chain homotopy type of , where denotes the –cable of the meridian of , viewed as a knot in . See Figure 1. For each spin structure , we show that the complex is isomorphic to , but endowed with a different filtration and an overall shift in the homological grading.
Let be a knot in and fix . Then there exists such that for all , the complex is isomorphic to as an unfiltered complex, where denotes a grading shift that depends only on and . Given a generator for , the filtration level of the same generator, viewed as a chain in , is given by:
As a corollary, the –filtered complex is isomorphic to a subquotient complex of , endowed with an step filtration :
This filtration is illustrated in Figure 2 in the case .
Let be a knot, and fix . Then there exists such that for all , the –filtration on induced by is isomorphic to the filtered chain homotopy type of the step filtration on described above.
As an application, we show that the concordance invariant of Hom [Hom-concgroup] can equivalently be defined in terms of filtered maps on the Heegaard Floer homology groups induced by the two-handle attachment cobordism of surgery along a knot in . The rationally null-homologous knot induces a -filtration of and , that is, a sequence of subcomplexes:
Using the knot filtrations, an equivalent definition of can be formulated in terms of the filtration and induced by as a knot inside and .
For sufficiently large surgery coefficient , the concordance invariant is equal to:
This interpretation of the invariant offers a topological perspective that complements the original algebraic definition of . We will also include properties of the invariant as well as computations of for homologically thin knots and –space knots.
The author thanks her advisors, Peter Ozsváth and Zoltán Szabó, for their guidance. Adam Levine for reading the version of this work which appeared in the author’s PhD thesis and for helpful comments. The author would also like to thank Matt Hedden, Jen Hom and Olga Plamenevskaya for helpful conversations.
2. The knot Floer filtration of cables of the meridian in Dehn surgery along a knot
In this section we will refine the theorem of Ozsváth-Szabó to determine the filtered chain homotopy type of the knot Floer complex of (, ).
We begin by recalling the large integer surgery formula from Ozsváth and Szabó [HFK]. Let , , , , be a doubly-pointed Heegaard diagram for , where
the curve is a meridian of the knot
the curve is a longitude for
there is a single intersection point in
the basepoints and lie on either side of
Let be the set of curves in , with replaced by a longitude winding times around . Label the unique intersection point . The Heegaard triple diagram represents a cobordism between and . See Figure 3.
Let denote the subquotient complex of generated by triples with the and filtration levels satisfying the specified constraints.
Theorem 2.1 ([Hfk]).
Let be a knot, and fix . Then there exists such that for all , the chain map
induces an isomorphism of chain complexes.
Here, as usual, the labeling of the spin structures is determined by the condition that can be extended over the cobordism from to associated to the two-handle addition along with framing , yielding a spin structure satisfying
Above, denotes a surface in obtained from closing off a Seifert surface for in to produce a surface of square .
We refine the theorem of Ozsváth-Szabó to determine the filtered chain homotopy type of the knot Floer complex of (, ). Consider the meridian of a knot . The meridian naturally lies inside of the knot complement and the surgered manifold . For , denotes the –cable of , and also lies inside and the surgered manifold . The knot is homologically equivalent to in . When , . See Figure 1 for a picture of the two-component link .
For all there is a natural -step algebraic filtration on the subquotient complex of :
This filtration is illustrated in the case in Figure 2.
Theorem 2.3 says that this algebraic filtration corresponds to a relative -filtration on induced by . This generalizes work of Hedden [hedden] who studied the case of the filtered complex .
Let be a knot, and fix . Then there exists such that for all , the following holds: The filtered chain homotopy type of the step filtration on described above is filtered chain homotopy equivalent to that of the filtration on induced by .
The key observation will be that the triple diagram used to define not only specifies a Heegaard diagram for the knot , but also a Heegaard diagram for the knot with the addition of a basepoint . Place an extra basepoint so that it is regions away from the basepoint in the Heegaard triple diagram representing the cobordism between and as in Figure 4. (This can be accomplished if is sufficiently large, e.g. if ). The knot represented by the doubly-pointed Heegaard diagram is in .
An intersection point is said to be supported in the winding region if the component of in lies in the local picture of Figure 4. Intersection points in the winding region are in to 1 correspondence with intersection points in .
Fix a Spin structure where . For (the surgery coefficient) sufficiently large, any generator representing structure is supported in the winding region. In this case, there is a uniquely determined and a canonical small triangle .
Suppose is the canonical small triangle and is a generator representing structure . If (so ), then the component of is (and lies units to the left of ) in Figure 4. In this case, maps to . On the other hand, if is a generator with and , then the component of is (and lies steps to the right of ) in Figure 4. In this case, maps to the subcomplex .
The following lemma (which generalizes Lemma 4.2 of [hedden]) will be used to finish the proof.
Let be a generator supported in the winding region, and let denote the component of the corresponding intersection point in , where the are labeled as in Figure 4. Then
Here, (respectively, ) denotes the top (respectively, bottom) filtration level of . denotes the filtration level that is lower than . In addition , so this is an -step filtration.
The -filtration is defined by the relative Alexander grading induced by on . That is,
where is a Whitney disk connecting .
Let be generators supported in the winding region, and let , denote the components of the corresponding intersection points . Assume without loss of generality that (so that lies to the right of ).
We will define a set of arcs on as follows. Let denote the arc on connecting to . Let denote on the arc on connecting to , for .
We will construct a Whitney disk with the following properties:
If and , (that is, , both lie on the left of ), then doesn’t contain any arc . Therefore,
If and , (that is, , both lie steps to the right of ), then doesn’t contain any arc . Therefore,
If and , (that is, lies to the left of and lies steps to the right of ), then contains the arcs , each with multiplicity one. Therefore,
If and , (that is, lies to the left of and lies steps to the right of ), then contains the arcs , each with multiplicity one. Moreover, doesn’t contain the arcs for . Therefore,
If and , (that is, lies steps to the right of and lies steps to the right of ), then contains the arcs , each with multiplicity one. Moreover, doesn’t contain the arcs for . Therefore,
If and , (that is, lies steps to the right of and lies steps to the right of ), then contains the arcs , each with multiplicity one. Therefore,
Assuming the existence of such , the lemma follows immediately.
In [hedden, Lemma 4.2] Hedden constructs a Whitney disk . The above enumerated properties of will be immediate from the construction. We restate his construction here. Note first since lie in the winding region, they correspond uniquely to intersection points , . These intersection points , can be connected by a Whitney disk with and for some . This means that contains with multiplicity , which further implies that the distance between and is , that is, . The domain of can then be obtained from the domain of by a simple modification in the winding region as described in [hedden]. This modification is shown in Figure 5. It replaces the boundary component by a simple closed curve from an arc connecting and along followed by an arc connecting to along , and which wraps times around the neck of the winding region. ∎
This completes the description of the knot Floer complex in terms of the complex . ∎
Theorem 2.3 described the -filtered chain homotopy type of knot Floer chain complex for large with respect to and . In Theorem 1.1, we describe the -filtered chain homotopy type of . This generalizes Theorem 4.2 of Hedden-Kim-Livingston [hedden-kim-livingston] which studies the case.
Proof of Theorem 1.1.
The isomorphism of chain complexes induced by the map (defined in [HFK])
respects the -module structure of both complexes, and hence determines one of the -filtrations (called the -filtration) of .
The knot induces an additional -filtration (the Alexander filtration) on and on . The additional -filtration on can be determined in exactly the same way as it was determined for the case of . Lemma 2.4 identifies the -filtration induced on any given slice in with a -step filtration as above. This yields the statement of the theorem.
Alternatively, the additional (Alexander) -filtration on can be obtained from the Alexander filtration on by the fact that the variable decreases Alexander grading by one, i.e. we have the relation . ∎
Let be a knot in and fix . Then there exists such that for all the following holds: Up to a grading shift, the filtration level of is described in terms of the original filtered knot Floer homology as
That is, each Alexander filtration level of is a “hook” shaped region in .
This follows from Theorem 1.1. ∎
Let with and let . For sufficiently large surgery coefficient , the Alexander filtration induced by on coincides with the algebraic -filtration on under the correspondence given by .
Since has degree equal to the Seifert genus of the knot, is supported along a thick diagonal of width . By the hypothesis, we have
Therefore the corner of the hook region of each constant Alexander filtration level of lies above the thick diagonal along which is supported. See Figure 6. For spin structures where , this means that the Alexander filtration induced by on coincides with the algebraic -filtration on under the correspondence given by . ∎
Because the algebraic -filtration is used to define concordance invariants (such as , which can be interpreted as an integer lift of the Hom invariant [epsilon]), the filtration induced by on can be used to study the concordance class of a knot . We will see that we can extract concordance invariants of from .
3. A knot concordance invariant
As an application for the results in the previous section on the –filtration induced on by the –cable of the meridian , our main result in this section (Theorem 3.5) shows that the concordance invariant of Hom [Hom-concgroup], which has an algebraic definition in terms of maps on subquotient complexes of , can be equivalently defined by studying filtered maps on the (hat version of the) Heegaard Floer homology groups induced by the two-handle attachment cobordism of large integer surgery along a knot in and the filtration induced by the knot inside of the surgered manifold.
Our result is analogous to the statement that the concordance invariants of Ozsváth-Szabó [OzSzRational] and of Hom [epsilon] can be defined algebraically or in terms of maps on the (hat version of the) Heegaard Floer homology groups induced by the two-handle attachment cobordism of large integer surgery along a knot in . Definition 3.1 gives an algebraic definition of in terms of certain chain maps on the subquotient complexes of the knot Floer chain complex . Due to the Ozsváth-Szabó large integer surgery formula [HFK], can equivalently be defined in terms of maps on the Heegaard Floer chain complexes induced by the two-handle attachment cobordism of (large integer) surgery.
We begin by recalling the definition of the concordance invariants . Let be a sufficiently large integer relative to the genus of a knot . Consider the map
induced by the two-handle cobordism . Here, denotes the restriction to of the Spin structure over with the property that
where and denotes the capped off Seifert surface in the four-manifold. We also consider the map
induced by the two-handle cobordism .
The maps and can be defined algebraically by studying certain natural maps on subquotient complexes of , as in [HFK]. The map is induced by the chain map
consisting of quotienting by followed by the inclusion. Similarly, the map is induced by the chain map
consisting of quotienting by followed by the inclusion.
Definition 3.1 ([epsilon], [Hom-concgroup]).
Let be the Ozsváth-Szabó concordance invariant. The invariant is defined as follows:
if is trivial (in which case is necessarily non-trivial).
if is trivial (in which case is necessarily non-trivial).
if and are both non-trivial.
In [Hom-concgroup], Hom defines a concordance invariant for knots with that is a refinement of .
Definition 3.2 ([Hom-concgroup]).
If ( is trivial), define