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Direct computation of knot Floer homology and the Upsilon invariant
Taketo Sano, joint work with Kouki Sato
The University of Tokyo
2019-12-20
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Overview
knot homology theory
HFK(K)
knot Floer homology
knot concordance invariant
Tau invariant
τ(K)
Vk sequence
{Vk}
Upsilon invariant
ΥK(t)
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I will describe algorithms for computing theG0 invariant, which is introduced by Kouki Sato. As an application, we obtain algorithms for computing knot concordance invariants such as:
1. τ invariant,a homomorphism fromConc(S3) toZ,
2. Vk sequence,determines thed-invariants ofSp/q3 (K), p/q>0, 3. Υ invariant,a homomorphism fromConc(S3) toPL([0,2],R).
Main Result
We have determinedΥfor almost all (except for 5) knots with crossing number up to 11, including 39 knots whose values have been unknown.
∗The paper is currently in progress.
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Combinatorial knot Floer homology
Knot Floer homologyHFK is a knot homology theory, originated from the Heegaard Floer homology. AlthoughHFK involves heavy analytic machineries, a purely combinatorial description was lately found. It uses thegrid diagramfor the construction, hence also called thegrid homology.
Figure 1:Grid diagram and the corresponding knot
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Construction of the chain complex (sketch)
LetG be a grid diagram of a knot, and N be its grid number. The complexC−(G) is a finitely generated free module over
F2[U1,· · · ,UN], where:
I the generators {x} are given by permutations of lengthN.
(Eachx can be drawn as N-tuple of points on the lattice), I the homological degree of x is given by theMaslov function
MG(x), with each factorUi contributing to degree−2, and I the differential ∂:Ck−(G)→Ck−1− (G) is given by
∂x=X
y
X
r∈Recto(x,y)
U1ε1· · ·UNεNy
wherer runs over the empty rectangles connectingxto y, and the exponents ε1, . . . , εN ∈ {0,1} are given by counting the number of intersections of r and’s.
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∂x=X
y
X
r∈Recto(x,y)
U1ε1· · ·UNεNy
=x =y
r
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Computing H
∗−(G ) algorithmically
Since the ground ringF2[U1,· · ·,UN] isnot a field (nor a PID), we cannot use computational methods to calculate the homology.
However, since the generators are finite and degUi =−2, we may regard eachk-th chain module Ck−(G) as a finitely generated free
::::::::::
F2-module with generators of the form:
U1a1· · ·UNaNx where degx−2P
iai =k.
With theseinflated generators, it becomes possible to compute the homology groupHk−(G) for each k ∈Z.
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Example (G = 31,N = 5)
k -6 -5 -4 -3 -2 -1 0
#{x} 2 10 27 40 30 10 1
rankCk− 622 360 192 90 35 10 1
rankHk− 1 0 1 0 1 0 1
Example (G = 61,N = 8)
k -11 ... -3 -2 -1 0 1
#{x} 1 ... 8,379 4,949 1,873 402 36
rankCk− 5,321,071 ... 24,659 8,165 2,161 402 36
rankHk− 0 ... 0 1 0 1 0
# of inflated generators explodes ask decreases!
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Proposition
H∗−(G)∼=F2[U1].
The homology does not provide any information specific to the knot.
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Bifiltration on C
∗−(G )
C∗−(G) admits abifiltration. Namely, every inflated generator U1a1· · ·UNaNx∈C∗−(G),
is assigned a bidegree (i,j)∈Z2 as
I i =−a1, the exponent ofU1 in its coefficient, and I j =AG(x)−P
`a`, whereAG is the Alexander function.
The first degreei is called the algebraic degree, and the second j is called theAlexander degree. It can be proved that∂ is
:::::::::::::
non-increasing for bothi and j.
Important knot concordance invariants such asτ,Vk and Υ can be obtained from this bifiltration. In year 2019, K. Sato introduced a new invariantG0(K) that unifies these invariants.
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Closed region and Z
2-filtration
We call a subsetR ⊂Z2 an closed region iff:
If (i,j)∈R and (i0 ≤i, j0 ≤j) then (i0,j0)∈R.
For any closed regionR, there is a corresponding subcomplex FRC− :=SpanF2{ z ∈C−|(i(z),j(z))∈R }
wherez is a monomial of the formz =U1a1· · ·UNaNx. For another closed regionR0 ⊂R, we haveFR0C−⊂FRC−.
The differential∂ is closed in FRC−. Thus C∗−(G) admits a
:::::::::::
Z2-filtration with respect to the partial order≤.
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C0−(G)
i j
R R0
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The invariant G
0(K )
Recall thatH0−(G)∼=F2. The setG0(0)(K) is defined as the set of minimal closed regions, each containing a homological generator of H0−(G), namely,
R ∈ G0(0)(K) ⇔
(∃z ∈FRC0−(G) s.t. 06= [z]∈H0−(G), R is minimal w.r.t. the above property.
G0(K) is defined similarly, by regarding homological generators of homological degree≤0.
Theorem (Sato ’19, Section 5.1) G0(K) is a knot concordance invariant.
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Theorem (Sato ’19, Prop. 5.17) G0(K) determines τ,Vk and Υ.
i j
τ
i j
k
Vk
Figure 2:Determiningτ,Vk and Υ fromG0(K).
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Main result
Theorem (S.-Sato)
There is an algorithm for computingG0(K).
Corollary
There is an algorithm for computingτ,Vk andΥ.
As an application, we determined Υ for almost all knots of crossing number up to 11, including 39 knots whose Υ have been unknown.
The following five are the uncomputed ones, due to computational cost.
10152, 11n31, 11n47, 11n77, 11n9.
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Computation of G
0(K )
First we compute one homological generatorz ∈C0−(G), i.e. a cycle whose homology class generatesH0−(G)∼=F2.
For any closed regionR, it contains a homological iff:
∃c ∈C1 s.t. z −∂c ∈FRC0.
LetQRC∗ =C∗/FRC∗, then the above condition is equivalent to:
∃c ∈QRC1 s.t. ∂c =z ∈QRC0
EachQRCi is a finite dimensional F2-vector space. We represent∂ by a matrixA, then the c above corresponds to the solutionx of the linear system:
Ax =vec(z).
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We minimize the region containingz by “sweeping” its
components into a smaller region. The invariantG0 is the set of all such minimal regions.
z
z0
c
∂
sweep
R R0
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Finite candidate regions
From Sato’s theorem:
−g4(K)[T2,3]ν+ ≤[K]ν+ ≤g4(K)[T2,3]ν+,
we can tell that for anyR ∈ G0(K), any of its corner (i,j) satisfies
|i+j| ≤g4(K) (ij ≥0),
|i−j| ≤g3(K) (ij <0).
Thus the set of all closed regions satisfying these conditions is finite, which we can take as a set of candidate regions including G0(K).
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i
j
g3
−g3
−g4
g4
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The algorithm (sketch)
Step 1. Compute one homological generator z ∈C0(G).
Step 2. Setup the candidate regions.
Step 3. Choose one candidateR. Take the basis of QRC0 =C0/FRC0
by modding out the generators of C0 that lie inR.
Step 4. Compute the matrix Arepresenting the differential
∂R :QRC1→QRC0.
Step 5. Check whether Ax =vec(zR) has a solution. If it does, mark R as realizable. If it doesn’t, discard all regions that are included in R. Goto Step 3 if unchecked candidates exist.
Step 6. Collect the realizableR’s that are minimal w.r.t. the inclusion
⊂, and we obtain G0(0)(K). Continue the same process for higher shift numbers.
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DEMO.
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Thank you!
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References I
[1] M. Khovanov.“A categorification of the Jones polynomial”.In:
Duke Math. J. 101.3 (2000), pp. 359–426.
[2] E. S. Lee.“An endomorphism of the Khovanov invariant”.In:
Adv. Math. 197.2 (2005), pp. 554–586.
[3] J. Rasmussen.“Khovanov homology and the slice genus”. In:
Invent. Math. 182.2 (2010), pp. 419–447.
[4] C. Manolescu, P. Ozsv´ath, and S. Sarkar.“A combinatorial description of knot Floer homology”.In: Ann. of Math. (2) 169.2 (2009), pp. 633–660.
[5] C. Manolescu et al.“On combinatorial link Floer homology”.In:
Geom. Topol. 11 (2007), pp. 2339–2412.
[6] K. Sato.Theν+-equivalence classes of genus one knots.2019.
arXiv:1907.09116 [math.GT].
