Evaluations of link polynomials and recent constructions in Heegaard Floer theory
aa r X i v : . [ m a t h . G T ] J a n EVALUATIONS OF LINK POLYNOMIALS AND RECENTCONSTRUCTIONS IN HEEGAARD FLOER THEORY
LARRY GU AND ANDREW MANION
Abstract.
Using a definition of Euler characteristic for fractionally-graded complexesbased on roots of unity, we show that the Euler characteristics of Dowlin’s “ sl ( n )-like”Heegaard Floer knot invariants HFK n recover both Alexander polynomial evaluations and sl ( n ) polynomial evaluations at certain roots of unity for links in S . We show that theequality of these evaluations can be viewed as the decategorified content of the conjecturedspectral sequences relating sl ( n ) homology and HFK n . Introduction
Ozsv´ath–Szab´o’s theory of Heegaard Floer homology [OSz04b] is a flexible set of con-structions yielding many types of invariants for low-dimensional manifolds. Even for knotsand links in S , the ideas of Heegaard Floer homology can be applied in several ways toproduce a family of related invariants known collectively as knot Floer homology or HFK [OSz04a, Ras03]. The simplest variant, [ HFK as applied to (single-component) knots in S ,assigns to a knot K a bigraded vector space [ HFK ( K ) (say over Q ) whose graded Eulercharacteristic is the Alexander polynomial ∆ K ( t ). Here we focus on more recent variants [ HFK n (which we will call HFK n ) and HFK n , due to Dowlin [Dow18a], with relationshipsto Khovanov homology and sl ( n ) homology more generally.Reduced and unreduced sl ( n ) homology [KR08a], like [ HFK , also assign bigraded vectorspaces to knots K . Their graded Euler characteristics are the reduced and unreduced sl ( n )polynomials of K . The sl ( n ) polynomials and the Alexander polynomial are all specializa-tions of the two-variable HOMFLY-PT polynomial of K , leading to various relationshipsbetween the sl ( n ) and Alexander polynomials at special values.For the homology theories at the categorified level, one can often think of these relation-ships between knot polynomial evaluations as being categorified by certain spectral sequencesthat are known or conjectured to exist. For instance, the appearance of the sl ( n ) polynomialas an evaluation of the HOMFLY-PT polynomial is categorified by Rasmussen’s spectral se-quences [Ras15] from triply graded HOMFLY-PT homology [KR08b] to sl ( n ) homology; theappearance of the Alexander polynomial as a HOMFLY-PT evaluation should be categorifiedby the conjectured spectral sequence from HOMFLY-PT homology to HFK [DGR06].In general, given some construction or conjecture in the realm of sl ( n ) homology or HFK ,it is natural to ask “what does it categorify, if anything?”; in other words, “what is its decate-gorified content?”. Often this is something simpler than what one started with; for example,the identities relating sl ( n ) polynomials and Alexander polynomials with the HOMFLY-PTpolynomial are simpler than the known and conjectured spectral sequences from HOMFLY-PT homology to sl ( n ) homology and HFK . Investigating the decategorified level can be aneasy way to gain valuable information about the structure one expects at the categorifiedlevel.
For this reason, it is natural to ask about the decategorified content of Dowlin’s conjecturedspectral sequences [Dow18a, Conjecture 1.6] from reduced and unreduced sl ( n ) homology to HFK (generalizing Rasmussen’s conjecture for sl (2), proved by Dowlin in [Dow18c]). Thespecific variants of HFK appearing in the conjectured spectral sequences are Dowlin’s singly-graded variants
HFK n and HFK n for links (the first of these agrees with a grading collapseof [ HFK when applied to knots).One complication is that as defined,
HFK n and HFK n are the homology of complexeswhose differentials increase the single grading gr n by n . The usual Euler characteristic for-mula, applied to such a complex, will not always be homotopy invariant. Instead, we dividethe grading on HFK n and HFK n by n , producing n Z -graded complexes whose differentialsincrease the grading by one, and work with a natural generalization of the Euler charac-teristic to this setting (based on roots of unity and admitting an interpretation in terms ofGrothendieck groups of triangulated categories). Theorem 1.1.
Let L be an ℓ -component link in S and let n ≥ . The Euler characteristicof gr n n -graded HFK n ( L ) , in our sense, equals e πi (1 − ℓ ) /n ∆ L ( t ) | t / = − e − πi/n where ∆ L ( t ) is thesymmetric single-variable Alexander polynomial of L (a Laurent polynomial in t / ), andthe Euler characteristic of HFK n ( L ) is zero. For n = 1 , the Euler characteristics of bothHFK ( L ) and HFK ( L ) are for all links L . We introduce grading-modified versions
HFK ′ n ( L ) and HFK ′ n ( L ) of Dowlin’s invariantssuch that the Euler characteristic of HFK ′ n ( L ) equals ∆ L ( t ) | t / = − e πi/n . These n -dependentinvariants are related to bigraded versions HFK ′ ( L ) and HFK ′ ( L ) categorifying ∆ L ( t ) andzero respectively. We propose
HFK ′ n ( L ) and HFK ′ n ( L ) as the E ∞ pages of Dowlin’s conjectured spectralsequences, and support our proposal with Euler characteristic evidence. The E pages ofthese conjectured spectral sequences should be reduced and unreduced sl ( n ) homology withthe bigrading collapsed to a single grading and divided by n as above. By analogy with HFK n , we will refer to the single collapsed grading as gr n and its quotient by n as gr n n . Theorem 1.2.
Let L be a link in S . The Euler characteristic of the gr n n -graded reduced sl ( n ) homology of L , in our sense, equals the reduced sl ( n ) polynomial of L evaluated at q = e πi/n . The Euler characteristic of the gr n n -graded unreduced sl ( n ) homology of L equalsthe unreduced sl ( n ) polynomial of L evaluated at q = e πi/n . For n ≥ L ( t ) | t / = − e πi/n and zero respectively; indeed, in the reduced case both evaluations are equal to P L ( − , e πi/n )where P L ( a, q ) is the reduced HOMFLY-PT polynomial of L , and similarly in the unreducedcase. When n = 1 both evaluations are equal to 1. As we discuss below, we can view Dowlin’sconjectured spectral sequences as categorifications of these equalities.We situate these results in the context of Rasmussen’s spectral sequences from HOMFLY-PT homology to sl ( n ) homology and the conjectured spectral sequences from HOMFLY-PT From the representation-theoretic perspective,
HFK ′ ( L ) categorifies the U q ( gl (1 | , HFK ′ ( L ) categorifies the U q ( gl (1 | This usage of gr n conflicts with the notation in [Ras15], where gr n is used for what we call the quantumgrading on sl ( n ) homology (itself related in an n -dependent way to the quantum and horizontal gradings onHOMFLY-PT homology; see [Ras15]). VALUATIONS OF LINK POLYNOMIALS 3 homology to
HFK , which fit with Dowlin’s conjectured spectral sequences into a square asshown in [Dow18a, Figure 1]. We review the decategorified content of the known and con-jectured spectral sequences starting at HOMFLY-PT homology, which we generalize to linksin terms of our shifted gradings, and we add to Dowlin’s square by labeling the edges withtheir decategorified content (see Figure 2). Examining the decategorified content along thepossible paths in the square, in terms of link polynomial evaluations, reveals a compatibilitythat could be a sign of a more structured relationship between these spectral sequences atthe categorified level.
Example 1.3.
Dowlin computes
HFK n of the unknot in [Dow18a, Example 2.10]; the resultis Q [ U ] / ( U n ), and the generator 1 ∈ Q [ U ] has gr n equal to 1 − n (so it has gr n n equal to n − q n − q − n q − q − , agreeing withthe sl ( n ) polynomial of the unknot.We propose that q n − q − n q − q − is the gr n -graded Poincar´e polynomial of this homology group,where the coefficient of q i in the polynomial is the dimension of the homology in gr n = i . Infact, HFK n of the unknot is isomorphic to unreduced sl ( n ) homology of the unknot, whichis naturally bigraded; in this example, the homological component of the bigrading is zeroand the intrinsic component agrees with gr n . With respect to this bigrading, which makessense on HFK n of the unknot but not on HFK n in general, it is indeed true that the gradedEuler characteristic of HFK n of the unknot is q n − q − n q − q − .However, here we are considering gr n (divided by n ), which makes sense on HFK n of arbi-trary links, as a homological grading. Since it is only a single grading, its Euler characteristicin our sense will be a single complex number (not necessarily an integer because the gradingis fractional). A generator of the homology in gr n = k will contribute a term e kπi/n to theEuler characteristic by our definitions; the Euler characteristic for the unknot homology isthus e πi (1 /n − + e πi (3 /n − + · · · + e πi (1 − /n ) + e πi (1 − /n ) . For n = 1 we get e πi (0) = 1; for n ≥ Remark 1.4.
Here we see chain complexes with gradings by n Z (and d = 0) categorifyingevaluations of sl ( n ) polynomials at 2 n th roots of unity. For categorification of these polyno-mials at roots of unity e πi/p for p prime, complexes with d = 0 are no longer suitable, andone often works with p -complexes satisfying d p = 0 (see e.g. [Kho16, Qi14]). Combiningthese ideas, one could look for p -complexes with gradings by n Z categorifying evaluationsof sl ( n ) polynomials at pn th roots of unity, although we are not aware of such complexes inthe literature. Remark 1.5.
If one is only interested in categorifying e.g. P n,L ( e πi/n ) using the ideas ofthis paper where P n,L ( q ) is the sl ( n ) polynomial, one does not need to use HFK n ; it sufficesto take a grading-collapse of reduced sl ( n ) homology. However, HFK n is a more natural orminimal categorification of this evaluation; analogously, to categorify the Alexander polyno-mial one can take a grading collapse of HOMFLY-PT homology, but HFK is a more minimalway to do it.
Remark 1.6.
Let K be a knot. For n = 2 where a spectral sequence from Khovanovhomology to HFK has been constructed by Dowlin [Dow18c], the equality P ,K ( i ) = ∆ K ( − LARRY GU AND ANDREW MANION is familiar (both evaluations give the knot determinant) and is a sign of a deeper relationshipbetween the representation theory of U q ( gl (1 | U q ( gl (2)) at q = i ; see [KS91, Section1]. We do not know whether there is any similar story one can tell about the analogousequalities for n >
2, although the n = 2 case is special at least in that both gl (1 |
1) and gl (2)are defined using 2 × Organization.
In Section 2 we define Euler characteristics for fractionally graded complexesand discuss spectral sequences. In Section 3 we review what we need about HOMFLY-PTpolynomials, sl ( n ) polynomials, and Alexander polynomials as well as HOMFLY-PT homol-ogy and sl ( n ) homology (focusing on the gradings). In Section 4 we do the same for HFK while introducing bigrading-shifted versions of
HFK theories adapted to the three variantsof HOMFLY-PT homology. In Section 5 we recall the definitions of Dowlin’s
HFK n invari-ants; in Section 6 we compute their fractionally-graded Euler characteristics and introducegrading-shifted variants of HFK n . In Section 7 we state a version of Dowlin’s spectral se-quence conjectures involving grading-shifted HFK n , compute its decategorified content, andplace it in the context of spectral sequences from HOMFLY-PT homology to sl ( n ) homologyand HFK . Acknowledgments.
We would like to thank Aaron Lauda for useful conversations andsuggestions. A.M. was partially supported by NSF grant DMS-1902092 and Army ResearchOffice W911NF2010075. 2.
Algebraic preliminaries
Euler characteristics of fractionally-graded complexes.
Following [Dow18a], wewill work over Q . Definition 2.1.
For n ≥
1, a n Z -graded complex C of Q -vector spaces (or just a n Z -gradedcomplex for short) is a n Z -graded Q -vector space C = M α ∈ n Z C α equipped with a Q -linear endomorphism d of degree +1 satisfying d = 0. Remark 2.2. A n Z -graded complex is the same data as n ordinary complexes, one for eachelement of ( n Z ) / Z . However, the examples of interest here more naturally give a n Z -gradedcomplex than n ordinary complexes.The category of n Z -graded complexes and homotopy classes of degree-zero chain maps istriangulated; the translation functor is degree shift downward by one. Furthermore, degreeshift downward by n equips this triangulated category with an n th root of its translationfunctor. We let C [ α ] denote C with its degrees shifted downward by α , so that ( C [ α ]) α ′ = C α ′ + α ; the usual notation [1] for the translation functor of a triangulated category agreeswith our notation. Definition 2.3.
Let C be an (essentially small) triangulated category equipped with an n th root (cid:2) n (cid:3) of its translation functor [1]. Let ζ n = e πi/n . We define the Grothendieck group K ( C ) to be the quotient of the free Z [ ζ n ]-module spanned by isomorphism classes of objects VALUATIONS OF LINK POLYNOMIALS 5 of C by the relations X − Y + Z = 0 for every distinguished triangle X → Y → Z → X [1]in C , as well as (cid:20) X (cid:20) n (cid:21)(cid:21) = ζ − n [ X ]for all objects X of C .We can apply Definition 2.3 to the homotopy category H of finite-dimensional n Z -gradedcomplexes (a full triangulated subcategory of the homotopy category of all n Z -graded com-plexes, preserved by the n th root of the translation functor). The result is a free Z [ ζ n ]-module K ( H ) of rank 1 spanned by [ Q ], where [ Q ] denotes the class of the complex that has Q indegree zero and zero in all other degrees. Definition 2.4.
Let C be a finite-dimensional n Z -graded complex. The Euler characteristic χ ( C ) of C is the unique element of Z [ ζ n ] such that [ C ] = χ ( C )[ Q ] in K ( H ). Explicitly, χ ( C ) = X α ∈ n Z e πiα dim Q C α . Just as for ordinary Euler characteristics, we have χ ( C ) = χ ( H ∗ ( C )). We also recall theusual graded Euler characteristics for bigraded and triply-graded complexes. Definition 2.5.
Let C = ( { C I,J : I, J ∈ Z } , d ) be a bigraded chain complex which is finite-dimensional in each I -degree, such that d has degree (0 , −
1) or (0 , C is defined to be χ u ( C ) = X I,J ∈ Z ( − J u I dim Q ( C I,J ) , a formal Laurent series in a variable u . If the I -grading is valued in Z rather than Z , thesame definition gives a formal Laurent series in u / .Similarly, let C = ( { C I,J,K : I, J, K ∈ Z } , d ) be a triply graded chain complex which isfinite-dimensional in each ( I, J )-bidegree, such that d has degree (0 , , −
1) or (0 , , C is defined to be χ u,v ( C ) = X I,J,K ∈ Z ( − K u I v J dim Q ( C I,J,K ) , a formal Laurent series in variables u and v . Remark 2.6.
Rather than u and v , we will often use variable names corresponding to thegradings in question.2.2. Spectral sequences.Definition 2.7.
A spectral sequence of n Z -graded complexes is a sequence of n Z -gradedcomplexes ( E r , d r ) r ≥ together with isomorphisms H ∗ ( E r , d r ) ∼ = E r +1 for r ≥ d r = 0 for large enough r , so that for some n Z -graded vector space E ∞ we have ( E r , d r ) = ( E ∞ ,
0) for large enough r . Remark 2.8.
It is built into the above definition that the differential d r on each page of thespectral sequence has degree +1 with respect to the n Z grading. LARRY GU AND ANDREW MANION
Figure 1.
Links appearing in the HOMFLY-PT skein relation.Since χ ( C ) = χ ( H ∗ ( C )) for finite-dimensional n Z -graded chain complexes, if some page E r of a spectral sequence as in Definition 2.7 is finite-dimensional then for any r ′ ≥ r wehave χ ( E r ) = χ ( E r ′ ); in particular, χ ( E r ) = χ ( E ∞ ). Remark 2.9.
Suppose one has a spectral sequence of bigraded complexes as in Defini-tion 2.5, such that each d r has bidegree (0 , −
1) or (0 , E r isfinite-dimensional in each j -degree; the same is then true for each page E r ′ for r ≥ r , and itfollows for the same reason as above that χ u ( E r ) = χ u ( E r ′ ). In particular, χ u ( E r ) = χ u ( E ∞ ).A similar equality for χ u,v holds in the triply graded case, assuming each d r has bidegree(0 , , −
1) or (0 , , Link polynomials and Khovanov–Rozansky homology
All links below are assumed to be oriented.3.1.
Link polynomials.
The HOMFLY-PT polynomial P L ( a, q ) of a link L in S [FYH + aP L + ( a, q ) − a − P L − ( a, q ) = ( q − q − ) P L ( a, q )(where L + , L − , and L are related near a crossing as in Figure 1) together with the HOMFLY-PT polynomial of the unknot as a normalization. We consider three variants: • The reduced HOMFLY-PT polynomial P L ( a, q ) has P unknot ( a, q ) = 1. • The middle HOMFLY-PT polynomial P − L ( a, q ) has P − unknot ( a, q ) = − q − q − . • The unreduced HOMFLY-PT polynomial P L ( a, q ) has P unknot ( a, q ) = a − a − q − q − . Remark 3.1.
In these variables the middle and unreduced HOMFLY-PT “polynomials” arerational functions in general, although they are Laurent polynomials in a and z = q − q − .We also consider three variants of the sl ( n ) polynomial: • The reduced sl ( n ) polynomial is P n,L ( q ) := P L ( q n , q ). • The unreduced sl ( n ) polynomial is P n,L ( q ) := P L ( q n , q ).Finally, if we let ∆ L ( t ) denote the (symmetric single-variable) Alexander polynomial of L ,a Laurent polynomial in t / , then we have • P L (1 , t / ) = ∆ L ( t ), • P − L (1 , t / ) = ∆ L ( t ) t − / − t / , • P L (1 , t / ) = 0.More relevant for us will be the following identities, which are consequences of the sym-metries P L ( a, q ) = P ( − a, − q ), P − L ( a, q ) = − P − L ( − a, − q ), and P L ( − a, − q ) = P L ( a, q ) of theHOMFLY-PT polynomials: • P L ( − , − t / ) = ∆ L ( t ), • P − L ( − , − t / ) = ∆ L ( t ) t / − t − / , VALUATIONS OF LINK POLYNOMIALS 7 • P L ( − , − t − / ) = 0.3.2. Khovanov–Rozansky homology.
Gradings and Euler characteristics.
We briefly establish notation for the sl ( n ) homol-ogy and HOMFLY-PT homology of Khovanov–Rozansky [KR08a, KR08b]; see also [Ras15].Let H ( L ), H − ( L ), and H ( L ) be the reduced, middle, and unreduced HOMFLY-PT homol-ogy of a link L in S . In the notation of [Ras15], these variants of HOMFLY-PT homology(denoted there by H ( L ), H ( L ), and e H ( L ) respectively) have a Z -grading gr q (or just q ),a Z -grading gr h , and a Z -grading gr v . Rasmussen also writes i = gr q , j = 2 gr h , and k = 2 gr v ; the value of j − k is always even. We let • gr A = 2 gr h = j , • gr Q = gr q = i , • gr H = gr v − gr h = k − j ,each of which is a grading by Z on the above three variants of HOMFLY-PT homology. Remark 3.2.
While h in gr h stands for horizontal, H in gr H stands for homological.Each variant of HOMFLY-PT homology is finite-dimensional in each (gr A , gr Q )-bidegree,so the following proposition makes sense. Proposition 3.3 (cf. Theorem 2.11, Section 2.8 of [Ras15]) . For a link L in S , we have: • χ a,q ( H ( L )) = P L ( a, q ) , • χ a,q ( H − ( L )) = P − L ( a, q ) , • χ a,q ( H ( L )) = P L ( a, q ) . Now let H n ( L ) and H n ( L ) be the reduced and unreduced sl ( n ) homology of a link L in S ; the reduced homology H n ( L ) also depends on a choice of component of L . Both variantsof sl ( n ) homology have Z -gradings gr Q,n and gr H ; in [Ras15, Section 2.9] these gradings arecalled gr n and gr v respectively, while in [Ras15, Section 5] they are called gr ′ n and gr − (see[Ras15, Proposition 5.14]). Proposition 3.4 (cf. Theorem 2.16 of [Ras15]) . Write χ q for the gr Q,n -graded Euler char-acteristic, with gr H treated as the homological grading. For a link L in S , we have: • χ q ( H n ( L )) = P n,L ( q ) , • χ q ( H n ( L )) = P n,L ( q ) . Rasmussen’s spectral sequences.
In [Ras15], Rasmussen constructs spectral sequenceswith E page H ( L ) (respectively H ( L )) and E ∞ page H n ( L ) (respectively H n ( L )) for n ≥ Q,n = 0and gr H = 1 where gr Q,n on HOMFLY-PT homology is defined by gr
Q,n = gr Q + n gr A . Thus,each page gets a bigrading as the homology of the previous page, and the induced bigradingon the E ∞ page agrees with (gr Q,n , gr H ) on sl ( n ) homology.These spectral sequences give equalities of Euler characteristics χ q ( H ( L )) = χ q ( H n ( L ))and χ q ( H ( L )) = χ q ( H n ( L )), where we are viewing H ( L ) and H ( L ) as bigraded by (gr Q,n , gr H )and χ q denotes the gr Q,n -graded Euler characteristic. As in the proof of [Ras15, Lemma 5.4],we have χ q ( H ( L )) = P L ( q n , q ); similarly, we have χ q ( H ( L )) = P L ( q n , q ). Thus, applyingEuler characteristics to these spectral sequences recovers the usual identity of the sl ( n ) poly-nomial with an evaluation of the HOMFLY-PT polynomial. We view these identities as LARRY GU AND ANDREW MANION the “decategorified content” of Rasmussen’s spectral sequences; in other words, we view thespectral sequences as categorifications of these identities.4.
Knot Floer homology
The master complex.
Let L be a link in S ; let H be a multi-pointed Heegaarddiagram for L (versions of the below theories can be defined for links in more general 3-manifolds but we restrict attention to links in S here). Write { z , w , . . . , z m , w m } for the set of basepoints in H . We assume that H is equipped with the appropriate analyticdata such that the knot Floer homology “master complex ” CFK
U,V ( L ), a finitely generatedbigraded free module over Q [ U , V . . . , U m , V m ] with an endomorphism ∂ U,V satisfying ∂ U,V = m X i =1 ( U a ( i ) − U b ( i ) ) V i , is defined (here a ( i ) denotes the index of the unique w basepoint in the same componentas z i of the Heegaard surface with alpha curves removed, and similarly for b ( i ) and betacurves). See [Zem19, Dow18a] for more details on the master complex.The two gradings on CFK
U,V ( L ) are called the Alexander grading gr T (a grading by Z in general) and the Maslov grading gr M (a grading by Z ); our conventions for these gradingsfollow [OSz04a, OSz08]. The variables U i have gr T = − M = −
2, the variables V i have gr T = 1 and gr M = 0, and ∂ U,V has gr T = 0 and gr M = − H whose number m of z and w basepoints is equal to the number ℓ of components of L .Then the homology of CFK
U,V ( L ) ⊗ Q [ U ,V ,...,U ℓ ,V ℓ ] Q [ U , V , . . . , U ℓ , V ℓ ]( U , V − , . . . , V ℓ − , which is a complex which ∂ = 0 with a single grading by gr M , computes d HF ( S ) ∼ = Q (seethe discussion after Theorem 4.4 of [OSz08]), while the homology of CFK
U,V ( L ) ⊗ Q [ U ,V ,...,U ℓ ,V ℓ ] Q [ U , V , . . . , U ℓ , V ℓ ]( U , . . . , U ℓ , V − , . . . , V ℓ − d HF ( ℓ − ( S × S )) ∼ = ∧ ∗ V where V is a vector space of dimension ℓ −
1. Asmentioned in [OSz08, proof of Theorem 1.1], in the grading conventions of that paper thetop-dimensional generator of ∧ ∗ V corresponds to the generator of d HF ( S ) ∼ = Q . By thediscussion after [OSz08, Theorem 1.2], the absolute Maslov grading on CFK is fixed so thatthe top-dimensional generator of ∧ ∗ V has Maslov degree zero, so we can equivalently saythat the generator of d HF ( S ) ∼ = Q has degree zero. The absolute Alexander grading is fixedby symmetry. While it would be more accurate to write
CFK
U,V ( H ), it will help avoid confusion with reduced andunreduced versions of knot Floer homology below to write CFK
U,V ( L ), with the H dependence left implicit. VALUATIONS OF LINK POLYNOMIALS 9
Other bigraded variants of
HFK . The following complexes are all derived from themaster complex and satisfy ∂ = 0. We focus on bigraded versions of HFK ; there are alsomulti-graded versions as in [OSz08]. Let L be an ℓ -component link in S ; when we mentionknot Floer complexes for links, the dependence on a choice of Heegaard diagram for L (saywith basepoints { z , w , . . . , z m , w m } ) is implicit. Definition 4.1.
The bigraded complex ] CFK ( L ) is CFK
U,V ( L ) ⊗ Q [ U ,V ,...,U m ,V m ] Q . Definition 4.2.
Assume that, in our Heegaard diagram H representing L , we are given somechoice of basepoints ( z i j , w i j ) on each component L j of L . The bigraded complex [ CFK ( L ) is CFK
U,V ( L ) ⊗ Q [ U ,V ,...,U m ,V m ] Q [ U , V , . . . , U m , V m ]( V , . . . , V m , U i , . . . , U i ℓ ) . Definition 4.3.
Assume that L is equipped with a distinguished component and that thebasepoints z m , w m of the Heegaard diagram H representing L lie on the distinguished com-ponent of L . The bigraded complex CFK ( L ) is CFK
U,V ( L ) ⊗ Q [ U ,V ,...,U m ,V m ] Q [ U , V , . . . , U m , V m ]( V , . . . , V m , U m ) . Definition 4.4.
The bigraded complex
CFK − ( L ) is CFK
U,V ( L ) ⊗ Q [ U ,V ,...,U m ,V m ] Q [ U , V , . . . , U m , V m ]( V , . . . , V m ) . Definition 4.5. [cf. Section 2.6 of [Dow18a]] Let L ′ be the disjoint union of L with asplit unknot; we choose the unknot component as a distinguished component for L ′ , and weassume that the only basepoints of the diagram H ′ we choose to represent L ′ that lie on thedistinguished component of L ′ are the final pair ( z m ′ , w m ′ ) of basepoints. We define CFK ( L ) := t − / CFK ( L ′ )where t − / denotes a downward shift by in the Alexander grading gr T . Remark 4.6.
The use of a split unknot to define unreduced
HFK appears in Baldwin–Levine–Sarkar [BLS17], although these authors use the term “unreduced
HFK ” for a slightlydifferent theory.The homology of each of these complexes is an invariant of L (equipped with a distin-guished component in Definition 4.3) and will be denoted by ] HFK ( L ), [ HFK ( L ), etc. Eachof the above bigraded versions of HFK is finite-dimensional in each Alexander degree.
Remark 4.7.
The complex
CFK ( L ) and its homology appear to be less common in theliterature; we use ( · ) to match the notation of HOMFLY-PT homology and sl ( n ) homology,although it is possible that our use of the notation HFK ( L ) conflicts with uses of this notationelsewhere. Graded Euler characteristics.Proposition 4.8.
For a link L in S , we have: • χ t ( [ HFK ( L )) = ( t − / − t / ) ℓ − ∆ L ( t ) = ( − ℓ − t ℓ − (1 − t − ) ℓ − ∆ L ( t ) , • χ t ( HFK ( L )) = ( − ℓ − t ℓ − ∆ L ( t ) , • χ t ( HFK − ( L )) = ( − ℓ − t ℓ − ∆ L ( t )1 − t − , • χ t ( HFK ( L )) = 0 .Proof. The first claim follows from [OSz04a, equation (1)] and [OSz08, Theorem 1.1]. Thesecond claim follows from the first because the chain groups in
CFK ( L ) are free modules overpolynomial rings in ℓ − [ CFK ( L );the graded Euler characteristic of a polynomial ring in one of these variables (with degrees A = − M = −
2) is − t − . The third claim follows similarly; the fourth claim followsfrom the second claim along with the fact that ∆ L ( t ) vanishes on split links. (cid:3) Remark 4.9.
For an ℓ -component link L in S , the single-variable and multi-variableAlexander polynomials of L satisfy the relation ∆ L ( t ) . = ( ∆ multi L ( t ) ℓ = 1∆ multi L ( t, . . . , t )(1 − t ) ℓ > . = means equality up to multiplication by a unit in Z [ t, t − ] (see [Kaw96, Proposition7.3.10(1)]). This relation explains why, unlike in [OSz08, equations (1) and (2)], we do notneed to treat ℓ = 1 and ℓ > Definition 4.10.
To more closely match the three variants of HOMFLY-PT homology,we introduce grading-shifted variants
HFK ′ ( L ), ( HFK − ) ′ ( L ), and HFK ′ ( L ) of knot Floerhomology. We first replace the Alexander and Maslov degrees of HFK ( L ), HFK − ( L ), and HFK ( L ) by their negatives, so that the variable U i now has gr A = 1 and gr M = 2 and thedifferentials on CFK complexes now have Maslov degree +1. In the Euler characteristiccomputations of Proposition 4.8, t gets replaced by t − ; note that ∆ L ( t − ) = ( − ℓ − ∆ L ( t ).We then make the following shifts: • For
HFK ′ ( L ), we shift the Alexander grading on grading-reversed HFK ( L ) upwardby ℓ − . We have χ t ( HFK ′ ( t )) = ∆ L ( t ) . • For (
HFK − ) ′ ( L ), we shift the Alexander grading on grading-reversed HFK − ( L ) up-ward by ℓ ; we also shift the Maslov grading upward by 1. We have χ t (( HFK − ) ′ ( L )) = ∆ L ( t ) t / − t − / . • For
HFK ′ ( L ), we shift the Alexander grading on grading-reversed HFK ( L ) upwardby ℓ − (recall that HFK ( L ) already had an Alexander grading shift in Definition 4.5).We have χ t ( HFK ′ ( L )) = 0.4.4. Conjectured spectral sequences from HOMFLY-PT homology to
HFK . In[DGR06], Dunfield–Gukov–Rasmussen conjectured the existence of spectral sequences from H ( K ) to [ HFK ( K ) for knots K in S . Manolescu [Man14] gives a similar conjecture for HFK − and the middle HOMFLY-PT homology. Dowlin [Dow18a] conjectures spectral sequences Note that [OSz08, equation (1)], when proved in [OSz08, Proposition 9.1], is stated with a ± sign. VALUATIONS OF LINK POLYNOMIALS 11 from H ( L ) to [ HFK ( L ) and from H ( L ) to HFK ( L ) for all links in S . We believe a spectralsequence involving HFK ( L ) is more plausible in the reduced case for links, so we will statethe following version of these spectral sequence conjectures. Conjecture 4.11.
Let L be a link in S . Ignoring gradings at first, there are spectralsequences with: • E page H ( L ) and E ∞ page HFK ( L ) ; • E page H − ( L ) and E ∞ page HFK − ( L ) ; • E page H ( L ) and E ∞ page HFK ( L ) .Moreover, such sequences are given by the construction of Manolescu [Man14, Theorem 1.1] ,which is known to give E ∞ pages recovering HFK . Remark 4.12.
In [Dow18b], Dowlin identifies the E page of the spectral sequence from[Man14, Theorem 1.1] with the appropriate sum of HOMFLY-PT complexes for singularresolutions of L ; it remains to identify the E page with HOMFLY-PT homology for linkswith nonsingular crossings.Manolescu [Man14, Section 4] discusses the grading properties of his conjectured spectralsequences from HOMFLY-PT homology to HFK in detail; we will rephrase some of hisdiscussion in terms of the grading-shifted variants of
HFK from Definition 4.10. We definea Z -grading gr T and a Z -grading gr M on each of the variants of HOMFLY-PT homologyby • gr T = gr Q , • gr M = gr A + gr Q + gr H (Manolescu includes constant grading-shift terms in the above formulas but here we incor-porate the grading shifts into HFK ; he also has negative signs since, unlike us, he has notmultiplied the Alexander and Maslov gradings on
HFK by − d r on the E r page of Manolescu’s conjectured sequences has gr A = 2 − r , gr Q = 0, and gr H = 2 r − d r has gr T = 0 and gr M = 1. Writing H ( L ) = ⊕ i,j,k ∈ Z H i,j,k ( L ) as in [Ras15] (andsimilarly for the other versions), we equivalently have gr T = i/ M = i + j/ k/ Conjecture 4.13.
Let L be a link in S . There are spectral sequences with each page bigradedby (gr T , gr M ) , such that the differential on each page has (gr T , gr M ) = (0 , and each pageis the bigraded homology of the previous page, and with • E page H ( L ) and E ∞ page HFK ′ ( L ) ; • E page H − ( L ) and E ∞ page ( HFK − ) ′ ( L ) ; • E page H ( L ) and E ∞ page HFK ′ ( L ) as bigraded vector spaces. These spectral sequences would give equalities of Euler characteristics χ t ( H ( L )) = χ t ( HFK ′ ( L )) , χ t ( H − ( L )) = χ t (( HFK − ) ′ ( L )) , χ t ( H ( L )) = χ t ( HFK ′ ( L )) where χ T denotes the gr T -graded Euler characteristic. We have χ t ( H ( L )) = X I ∈ Z ,J ∈ Z ( − J t I dim Q (cid:0) H ( L ) gr T = I, gr M = J (cid:1) = X i,j,k ∈ Z ( − i + j/ k/ t i/ dim Q (cid:16) H i,j,k ( L ) (cid:17) = X i,j,k ∈ Z a j q i ( − ( k − j ) / dim Q (cid:16) H i,j,k ( L ) (cid:17)! (cid:12)(cid:12)(cid:12)(cid:12) a = − , q = − t / = P L ( − , − t / ) . Similarly, χ t ( H − ( L )) = P − L ( − , − t / ) , χ t ( H ( L )) = P L ( − , − t / ) = 0 . Thus, these conjectured spectral sequences can be viewed as categorifications of the threeequalities involving Alexander polynomials and HOMFLY-PT polynomial evaluations (with a = −
1) at the end of Section 3.1.5.
Dowlin’s
HFK n invariants We now consider two versions of
HFK defined by Dowlin [Dow18a], applied to links in S rather than more general 3-manifolds. Rather than bigradings, these versions will havesingle gradings by n Z in our conventions.Let L be a link in S , represented by a Heegaard diagram H as in Section 4.1 withbasepoints { z , w , . . . , z m , w m } . Following Dowlin, for n ≥ n ofthe bigrading on CFK
U,V ( L ) defined bygr n = − n gr M +2( n −
1) gr T . We divide gr n by n to get gr n n = − gr M +2 (cid:18) − n (cid:19) gr T which is valued in n Z even for half-integral values of gr T . The variables U i have gr n n = n ,the variables V i have gr n n = 2 − n , and ∂ U,V has gr n n = 1. Definition 5.1 (cf. Definition 2.19 of [Dow18a]) . Assume that L is equipped with a distin-guished component and that the basepoints z m , w m of the Heegaard diagram H representing L lie on the distinguished component. The n Z -graded complex CFK n ( L ) is CFK
U,V ( L ) ⊗ Q [ U ,...,U m ,V ,...,V m ] Q [ U , . . . , U m , V , . . . , V m ] (cid:16) V i − U na ( i ) − U nb ( i ) U a ( i ) − U b ( i ) : 1 ≤ i ≤ m − (cid:17) + ( U m , V m )where a ( i ) and b ( i ) are defined as in Section 4.1. The grading is given by gr n n ; note that U na ( i ) − U nb ( i ) U a ( i ) − U b ( i ) equals the telescoping sum U n − a ( i ) + U n − a ( i ) U b ( i ) + · · · + U n − b ( i ) , which (like V i ) has gr n n = 2 − n . Definition 5.2 (cf. Definition 2.5 of [Dow18a]) . Let L ′ be the disjoint union of L with asplit unknot, and choose the unknot component to be distinguished. As in Definition 4.5, VALUATIONS OF LINK POLYNOMIALS 13 we assume that the only basepoints of the diagram H ′ we choose to represent L ′ that lie onthe distinguished component of L ′ are the final pair ( z m ′ , w m ′ ) of basepoints. We define CFK n ( L ) := CFK n ( L ′ )[1 − /n ];note that a downward shift by in gr T as in [Dow18a, Section 2.2] produces a downwardshift by 1 − n in gr n n = − gr M +2(1 − /n ) gr T .We write ∂ n for the differential on either variant of CFK n ; it satisfies ∂ n = 0. When n = 1, the complex CFK ( L ) computes d HF ( S ) (see Section 4.1), so its homology is Q ingr M = 0 (and thus gr n n = 0) and zero in other degrees; see [Dow18a, Lemma 5.2]. It followsthat HFK ( L ) is also Q in degree 0 and zero in other degrees.Since the tensor product (after annihilating the final pair of variables) sets each V i vari-able equal to a polynomial in the U i variables while imposing no further relations on the U i variables, the complexes CFK n ( L ) and CFK n ( L ) are free over Q [ U , . . . , U m − ] and Q [ U , . . . , U m ′ − ] respectively. Their homology groups HFK n ( L ) (respectively HFK n ( L )) de-pend only on L with its distinguished component (respectively, L ) and are finite-dimensionalover Q as shown in [Dow18a]. Remark 5.3.
In [Dow18a], Dowlin uses the notation [ HFK n ( L ) to refer to what we call HFK n ( L ); however, in [Dow18c], [ HFK n (at least for n = 2) is given a different definitionwhich is closer to Definition 4.2 for [ HFK ( L ).6. The Euler characteristic of
HFK n Let L be an ℓ -component link in S equipped with a distinguished component; in thissection we compute the Euler characteristics of HFK n ( L ) and HFK n ( L ).For simplicity, assume we are working with a Heegaard diagram H for L that has exactly 2 ℓ basepoints. Let R = Q [ U , ..., U ℓ − ], a n Z -graded ring where U i has degree n as in Section 5,and let K be the n Z -graded Koszul complex K = ℓ − O i =1 (cid:18) R (cid:20) − n (cid:21) U i −→ R (cid:19) where the tensor products are over R . Lemma 6.1.
Let n ≥ . There exists a spectral sequence with each page graded by n Z , withdifferentials of degree +1 such that each page is the n Z -graded homology of the previous page,and with E page HFK n ( L ) ⊗ R K and E ∞ page [ HFK ( L ) as n Z -graded vector spaces.Proof. The complex
CFK n ( L ) ⊗ R K can be viewed as a cube of dimension ℓ − CFK n ( L ). We equip CFK n ( L ) ⊗ R K with a filtration such thatevery oriented edge of this cube increases the filtration level by 1. Then the differential d on CFK n ( L ) ⊗ R K can be decomposed as d = d + d , where d is the differential on each copyof CFK n ( L ) and d comes from the differential on K .From this filtration, we get a spectral sequence whose E page is ( HFK n ( L ) ⊗ R K, ( d ) ∗ ).The spectral sequence converges because there are only finitely many nontrivial filtration lev-els, and the E ∞ page is [ HFK ( L ); indeed, the total complex CFK n ( L ) ⊗ R K has a contractiblesubcomplex such that the quotient by this subcomplex is [ CFK ( L ). (cid:3) Since
HFK n ( L ) is finitely generated over Q , the same holds for HFK n ( L ) ⊗ R K . Thus, thespectral sequence of Lemma 6.1 gives an equality between the gr n n -graded Euler characteristicsof HFK n ( L ) ⊗ R K and [ HFK ( L ). The Euler characteristics of HFK n ( L ) ⊗ R K and HFK n ( L )are related by χ ( HFK n ( L ) ⊗ R K ) = (cid:0) − e πi/n (cid:1) ℓ − χ ( HFK n ( L )) , so since n ≥ χ ( HFK n ( L )) = (cid:0) − e πi/n (cid:1) − ℓ χ ( [ HFK ( L )) . Using Proposition 4.8, we can compute the gr n n -graded Euler characteristic of [ HFK ( L ) asfollows: χ ( [ HFK ( L )) = X α ∈ n Z e πiα dim Q (cid:16) [ HFK ( L ) gr nn = α (cid:17) = X I ∈ Z ,J ∈ Z e πi ( − J +2(1 − /n ) I ) dim Q (cid:16) [ HFK ( L ) gr T = I, gr M = J (cid:17) = X I ∈ Z ,J ∈ Z ( − J t I dim Q (cid:16) [ HFK ( L ) gr T = I, gr M = J (cid:17) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t / = e πi (1 − /n ) = − e − πi/n = (cid:16) ( − ℓ − t ℓ − (1 − t − ) ℓ − ∆ L ( t ) (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) t / = − e − πi/n = e πi (1 − ℓ ) /n (cid:0) − e πi/n (cid:1) ℓ − ∆ L ( t ) | t / = − e − πi/n . Corollary 6.2.
For n ≥ , the gr n n -graded Euler characteristic of HFK n ( L ) is e πi (1 − ℓ ) /n ∆ L ( t ) | t / = − e − πi/n . Since
HFK n ( L ) is defined as a grading shift of HFK n of the disjoint union of L with asplit unknot, and the Alexander polynomial vanishes on split links, we see that if n ≥
2, the gr n n -graded Euler characteristic of HFK n ( L ) is zero for all links L . When n = 1, we have χ ( HFK ( L )) = χ ( HFK ( L )) = 1for all links L . Definition 6.3.
As in Definition 4.10, we define grading-shifted variants
HFK ′ n ( L ) and HFK ′ n ( L ) of HFK n ( L ) and HFK n ( L ). Starting with bigradings on CFK n ( L ) and CFK n ( L )corresponding to the bigradings on CFK ′ ( L ) and CFK ′ ( L ), the differentials ∂ n have degree+1 with respect to gr n n := gr M − (cid:18) − n (cid:19) gr T (note that since we still want +1 differentials on n Z -graded complexes, this is the negativeof the earlier definition of gr n n in terms of gr T and gr M ). We define CFK ′ n ( L ) to be CFK ′ ( L )with grading given by gr n n and differential given by ∂ n ; we define CFK ′ n ( L ) similarly. Remark 6.4.
Starting from
HFK ( L ), we negated both gradings and shifted the Alexandergrading upward by ℓ − to get HFK ′ ( L ), then applied the collapse gr M − − /n ) gr T to VALUATIONS OF LINK POLYNOMIALS 15 get the grading on
HFK ′ n ( L ). Equivalently, we could first apply the collapse − gr M +2(1 − /n ) gr T on HFK ( L ) to get the grading on HFK n ( L ), then shift this n Z grading upward by(1 − ℓ )(1 − /n ). In other words, HFK ′ n ( L ) is HFK n ( L ) with its n Z -grading shifted upwardby (1 − ℓ )(1 − /n ); similarly, HFK ′ n ( L ) is HFK n ( L ) with its n Z -grading shifted upward by(1 − ℓ )(1 − /n ). Corollary 6.5.
For n ≥ , the gr n n -graded Euler characteristic of HFK ′ n ( L ) is ∆ L ( t ) | t / = − e πi/n , and the gr n n -graded Euler characteristic of HFK ′ n ( L ) is zero.Proof. For the reduced case, we have e πi (1 − ℓ )(1 − /n ) e πi (1 − ℓ ) /n ∆ L ( t ) | t / = − e − πi/n = ( − − ℓ ∆ L ( t ) | t / = − e − πi/n = ∆ L ( t ) | t / = − e πi/n (using that ∆ L ( t − ) = ( − ℓ − ∆ L ( t )); for the unreduced case, we have e πi (1 − ℓ )(1 − /n ) · (cid:3) The proof of [Dow18a, Lemma 2.23] gives us n Z -graded spectral sequences from HFK ′ ( L )to HFK ′ n ( L ) and from HFK ′ ( L ) to HFK ′ n ( L ), where gr n n on HFK ′ ( L ) and HFK ′ ( L ) is definedto be gr M − − /n ) gr T .When n = 1, the shifted homology groups HFK ′ ( L ) and HFK ′ ( L ) agree with HFK ( L )and HFK ( L ) respectively, so their Euler characteristics are both 1. Remark 6.6.
The arguments in this section can be made simpler in the case of knots ( ℓ = 1),where by [Dow18a, Lemma 2.20], HFK n ( L ) is isomorphic to gr n n -graded HFK ( L ) = [ HFK ( L ).In particular, Lemma 6.1 is unnecessary in this case.7. Euler characteristics and spectral sequences
Dowlin [Dow18a, Conjecture 1.6] conjectures the existence of spectral sequences from H n ( L ) to HFK n ( L ) and from H n ( L ) to HFK n ( L ). These sequences are conjectured torespect the n Z -gradings, where the n Z -grading gr n n on reduced and unreduced sl ( n ) homologyis defined by gr n n = 1 n gr Q,n + gr H . Dowlin works with n times this grading, which we would write as gr Q,n + n gr H . We statethe following version of Dowlin’s conjectures in terms of the grading-shifted theories
HFK ′ n and HFK ′ n . Conjecture 7.1.
Let L be a link in S . There exist spectral sequences with each page gradedby n Z , with differentials of degree +1 such that each page is the n Z -graded homology of theprevious page, and with • E page H n ( L ) and E ∞ page HFK ′ n ( L ) ; • E page H n ( L ) and E ∞ page HFK ′ n ( L ) as n Z -graded vector spaces. In Dowlin’s notation this grading is called gr n + n gr v (where gr n corresponds to our gr Q,n ). As specifiedin [Dow18a, Section 4.2], the grading gr v here is k = 2 gr v in the notation of [Ras15], where gr v correspondsto our gr H . This accounts for the factor of in Dowlin’s formula. These spectral sequences would give equalities of n Z -graded Euler characteristics χ ( H n ( L )) = χ ( HFK ′ n ( L )) , χ ( H n ( L )) = χ ( HFK ′ n ( L )) . We have χ ( H n ( L )) = X α ∈ n Z e πiα dim Q (cid:16) H n ( L ) n gr Q,n + gr H = α (cid:17) = X I,J ∈ Z e πi ( I/n + J ) dim Q (cid:16) H n ( L ) gr Q,n = I, gr H = J (cid:17) = X I,J ∈ Z ( − J q I dim Q (cid:16) H n ( L ) gr Q,n = I, gr H = J (cid:17)! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q = e πi/n = P n,L ( e πi/n );similarly, χ ( H n ( L )) = P n,L ( e πi/n ), which is 0 for n ≥ n = 1. Thus, for n ≥ P n,L ( e πi/n ) = ∆ L ( t ) | t / = − e πi/n , P n,L ( e πi/n ) = 0 . For n = 1, the equalities are P ,L ( −
1) = 1 and P ,L ( −
1) = 1 (note that P ,L ( q ) = P ,L ( q ) = 1in general). Remark 7.2.
Let n ≥ HFK to HFK n ,can be organized as shown in Figure 2, following [Dow18a, Figure 1]. The arrows in thisfigure represent spectral sequences (solid for known, dotted for conjectural); we augmentDowlin’s figure by labeled the arrows with their decategorified content. It is interesting to look at the square formed by the reduced theories; traveling alongthe left edge and then the bottom edge amounts to starting with P L ( a, q ), evaluating at a = q n , and then evaluating the result at q = e πi/n to get P L ( − , e πi/n ). On the other hand,traveling along the top edge and then the right edge amounts to starting with P L ( a, q ),evaluating at a = − q = − t / , and then evaluating the result at t / = − e πi/n to get P L ( − , e πi/n ). This compatibility at the level of Euler characteristics could be a sign of amore elaborate compatibility relationship between the conjectured spectral sequences at thecategorified level. References [BLS17] J. A. Baldwin, A. S. Levine, and S. Sarkar. Khovanov homology and knot Floer homology forpointed links.
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Email address : [email protected] Department of Mathematics, University of Southern California, Los Angeles, CA
Email address : [email protected]@usc.edu