An Involutive Upsilon Knot Invariant
Using the theory of involutive Heegaard Floer knot theory developed by Hendricks-Manolescu, we define two involutive analogs of the Upsilon knot concordance invariant of Ozsváth-Stipsicz-Szabó. These involutive invariants are piecewise linear functions defined on the interval [0,2]. Each is a concordance invariant and provides bounds on the three-genus of a knot.
Heegaard Floer knot theory  associates to a knot a chain complex . To be more precise,
where is a graded –chain complex with boundary map of degree and and are increasing filtrations on . Furthermore, is a free, finitely generated –module; the action by commutes with , lowers gradings by and lowers both filtration levels by 1. The construction of depends on a series of choices, but any two complexes associated to are bifiltered chain homotopy equivalent.
Two further structural properties of have been discovered: Sarkar  described a naturally defined self-chain homotopy equivalence , now called the Sarkar map, and Hendricks and Manolescu  used the existence of a skew-bifiltered chain homotopy equivalence, first constructed in ,
to define a family of new invariants called the involutive homology groups, , where or . Here, by skew-bifiltered we mean that switches algebraic and Alexander filtration levels. That is well-defined depends on the naturality of , which follows from results of Juhasz-Thurston . Hendricks and Manolescu also proved that is bifiltered chain homotopic to .
The group is the homology of the mapping cone of . In Section 2 we will review the construction of this mapping cone, . We will also describe a bifiltration on . Section 3 describes the computation of for the torus knot and presents a generalization that applies to all torus knots or, more generally, to –space knots and their mirror images. In Section 4 we describe how the Upsilon function associated to , defined in , can be extended to give a pair of what we call the involutive Upsilon functions. Section 5 focuses on a single example, the torus knot . In Section 6 we prove the concordance invariance of the involutive Upsilon functions. We show in Section 7 that the value of each Upsilon at is determined by previously defined invariants, and . Section 8 briefly discusses the three-genus of knots.
Throughout this paper, all complexes are –graded chain complexes over the field , henceforth denoted . All differentials will lower homological degree by 1. The word grading is synonymous with homological degree.
Recall that if is a partially ordered set, then an –filtered complex is a complex together with a collection of subcomplexes , indexed by , with whenever . We will always assume that
A homotopy equivalence between –filtered complexes is said to be an –filtered homotopy equivalence if all relevant maps (the chain map , its chain homotopy inverse , and the chain homotopy) are filtration preserving.
By default, filtered complex means –filtered and bifiltered means –filtered. If is a filtered complex, we may define the filtration degree of a nonzero by
By convention, we set . Thus, given the assumptions (1), is a well-defined set function . The function satisfies
Conversely, given a set function with these properties, we can recover the subcomplex as the linear span of all elements such that . In fact, we can allow more general functions . If is a set function satisfying the properties (F1), (F2), and (F3), then we can define an –filtration of , which is the associated ordered family of subcomplexes , indexed by . In all examples of interest to us, the image of under will be a discrete subset which, as an ordered set, is isomorphic to . Thus, every –filtered complex considered here can also regarded as a –filtered complex, though the precise description would require choosing an isomorphism .
Henceforth, a filtered complex will mean a pair , where is a chain complex with differential , and is a function satisfying properties (F1), (F2), and (F3) above. Similarly, a bifiltered complex is a triple such that and are filtered complexes. The reader can verify that a bifiltered complex in this sense corresponds to a filtration indexed by .
Acknowledgments We thank Kristen Hendricks and Jen Hom for helpful comments.
2. Involutive homology
We begin by reviewing the definition of the mapping cone complex in the context of the chain map ; we denote this complex . The underlying graded vector space is , where is the same complex as with gradings shifted up by 1. The boundary map is given by
In other words, the boundary of is . It is easily checked that .
We denote by .
The homology of , , is isomorphic as an –module to , where has grading 1 and has grading . This splitting is natural.
For the mapping cone of any map of complexes , there is a long exact sequence
The map is trivial on homology and with the generator 1 of grading 0. ∎
, where is the odd tower isomorphic to with all gradings odd, and is similarly the even tower.
2.1. The folded bifiltration
The map is skew; it does not preserve the algebraic-Alexander bifiltration. However, there is a pair of filtrations on that are preserved.
Suppose is a bifiltered complex. Define a new bifiltered complex , where and . We call the resulting bifiltration the folded, or Min-Max bifiltration on .
We may regard as a bifiltered complex with respect to the Min-Max filtration. Note that swaps the and filtrations, and hence it preserves the Min-Max filtrations. Thus, is a bifiltered complex.
The left diagram in Figure 1 illustrates a model complex for ; in general, the structure of is determined by the results of , which study a more general family of knots, called –space knots. In the diagram, the vertex at represents a generator at grading 0. The full complex is constructed from the illustrated model complex by tensoring with ; the translates, if illustrated, would be represented by copies of the finite complex that is drawn, shifted units along the main diagonal. It is evident that the only self-chain homotopy equivalence that is also skew is represented by reflection through the diagonal. On the right in Figure 1 the complex is illustrated, where now the bifiltration is given by Min-Max. Figure 2 illustrates the involutive complex for as well as an equivalent reduced complex obtained by bifiltered Gaussian elimination. In the next section we describe the steps in constructing this reduction.
Suppose is a partially ordered set, and let be a filtered complex. We say that is reduced if each subquotient has zero differential. Under mild finiteness assumptions on (which are satsified by and ), any –filtered complex is isomorphic to where is reduced and is homotopically trivial in the filtered sense. An algorithm for reducing bifiltered complexes is presented in . In this section we will summarize the procedure in the case that the starting complex is the involutive complex associated to a staircase, such as the one illustrated on the left of Figure 1. In this example there are 9 vertices and the steps are . The goal is to perform a bifiltered change of basis so that, after removing acyclic summands, the remaining diagram has no arrows within any given square. We first introduce some terminology.
A staircase complex is symmetric if , where denotes the staircase with bifiltrations swapped. Any symmetric staircase has steps of size with . A symmetric staircase is inward-pointing if the generator corresponding to the central dot is a cycle, and is outward-pointing otherwise. We call a staircase positive if the first (top) step is to the right, not down, and negative otherwise.
Note that every symmetric staircase has an even number of edges. A positive symmetric staircase is inward-pointing exactly when its length is 0 (mod 4), and a negative symmetric staircase is inward-pointing exactly when its length is 2 (mod 4).
Note: Positive torus knots, with , have positive symmetric staircase complexes, while their mirror images, , have negative staircase complexes. More generally, all –space knots have positive staircase complexes.
The appropriate change of basis is best described by a schematic, as in Figure 3.
The diagram on the left represents the complex . The box labeled is the portion of the staircase complex that arises from the portion of the staircase with , and is the portion with . The boxes and represent the same complexes, with grading shifted up by one.
A change of basis in which generators from are added to corresponding generators of , and generators of are added to corresponding generators of , changes the diagram so that it appears as on the right in Figure 3. The complexes and are each staircase complexes; the one with an even number of vertices is acyclic and the other has homology of rank one. (In the current example, and both have four vertices, so it is that is not acyclic.) The complex involving and is illustrated schematically on the left in Figure 4. Changing basis, adding some of the generators from the top row to those of the bottom row, as indicated in the diagram, offers the reduction as illustrated on the right in Figure 4.
We now see that is bifiltered chain homotopy equivalent to the complex illustrated in Figure 2. In the diagram, the staircase with four vertices is acyclic, and thus it doesn’t contribute to later computations. Henceforth, we will not include such acyclic pieces in our diagrams.
In considering a general symmetric staircase complex, there are two parity issues: the length of the staircase (mod 4) and whether the staircase is positive or negative. The following theorem summarizes the result in all cases. The proof in each case follows the exact same lines as the computations above.
Let be a staircase complex , let and let . The involutive complex associated to is the direct sum of two complexes, one represented by a single vertex on the diagonal and the other a staircase complex .
If the staircase is positive and is even, then has grading level 1 and is at filtration level ; is the staircase with homology at grading 0, beginning at filtration level and ending at .
If the staircase is positive and is odd, then has grading level 1 and is at filtration level ; is the staircase with homology at grading 0, beginning at filtration level .
If the staircase is negative and is even, then has grading level 0 and is at filtration level ; is the staircase with homology at grading 1, beginning at the filtration level and ending at .
If the staircase is negative and is odd, then has grading level 0 and is at filtration level ; is the staircase with homology at grading 1, beginning at filtration level .
In summary we have the following general result.
The complexes and are bifiltered by and . The homology group is isomorphic to a tower of even homological grading. The homology group is isomorphic to a direct sum of towers and of odd and even homological grading, respectively.
For each , let be the function on either or given by
The image of as ranges over all (nonzero) elements of is some discrete subset ; for each we have a subcomplex which is the –span of all elements for which . We refer to this as the slope filtration of (or of ). Note that for all and , is an –module. Similar statements hold for . Note also that induces filtrations on and .
Recall that we have a short exact sequence of bifiltered complexes
The connecting homomorphism for the associated long exact sequence is induced by the map , and since induces the identity map in homology, the connecting map is zero. Thus the long exact sequence in homology splits into a collection of short exact sequences
for all . Since is supported in even homological degrees, it follows that is an isomorphism and is an isomorphism .
Let , , and denote generators. It follows that and . We now define three functions of :
Here the superscript on indicates that we are using the folded bifiltration on . Using these, we define two involutive Upsilon functions and a folded Upsilon function.
Our conventions for “upper bar” and “lower bar” where chosen so that the following inequalities hold:
If and are filtered vector spaces, then any filtration preserving map satisfies by definition. Thus, the inequalities follow immediately from the fact that and , discussed in the remarks preceding the proposition. This proves , as claimed. ∎
In Figure 9 we have redrawn the fully reduced complex .
The figure includes two dashed lines of slope corresponding to filtration levels when . The lower line is the boundary of the region
The upper line is the boundary of the region
Continuing to work with , the least value of so that the region contains a generator of homology at grading 0 is . The least value of so that the region contains a generator of homology at grading 1 is . Thus, and .
For general we have
6. Concordance invariance and knot inverses.
If and are concordant knots, then and .
A result of Hendricks and Hom  states that if is a slice knot, then splits as the direct sum of involutive complexes:
where and is acyclic. (This result generalizes an analogous result of Hom  that holds for noninvolutive complexes. The proof depended on results of Zemke  concerning the involutive homology of connected sums of knots.) Thus, we can write
as involutive complexes, where and is acyclic. It follows that
and according to a connected sum formula for involutive homology given in , this is again a direct sum of involutive complexes (but the involution on is not necessarily the tensor product of the involutions; see  for details). Since the second summand is acyclic, we write
Next, we write
which can be rewritten as
since is slice. As before, is a direct sum of involutive complexes,
In summary we have the following decompositions of involutive complexes:
The acyclic summands do not affect the value of either or , and thus the proof is complete. ∎
7. The knot concordance invariants and
In , two knot concordance invariants and are defined. These can be interpreted in terms of .
Both and are defined in terms of the involutive correction terms for large surgery on : and . These are computed in terms of the maximal gradings of even and odd non-torsion elements in the homology of . More precisely, they are minus one half these gradings.
Suppose the maximal grading of a (non-torsion) class in of even grading is . Then it follows from Formula 1 of  that . As a consequence, the involutive complex contains a non-torsion class of grading 0 if and only if . Thus . It now follows that , as desired. A similar argument works for the lower invariants.
The three-genus bounds that arises from and are almost immediate, following in the same way as the lower bounds on coming from : for all at which is nonsingular. The proof of this inequality uses only the fact that is chain homotopy equivalent to complex for which all filtration levels satisfy (see , or the expository account in ). The same constraint holds for the involutive complex with the Max-Min filtration, so the same proof applies.
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