Direct computation of knot Floer homology and the Upsilon invariant - - PowerPoint PPT Presentation

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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 the G0 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 from Conc(S3) to Z,
• 2. Vk sequence, determines the d-invariants of S3

p/q(K), p/q > 0,

• 3. Υ invariant, a homomorphism from Conc(S3) to PL([0, 2], R).

Main Result

We have determined Υ for 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 homology HFK is a knot homology theory, originated from the Heegaard Floer homology. Although HFK involves heavy analytic machineries, a purely combinatorial description was lately

• found. It uses the grid diagram for the construction, hence also

called the grid homology.

Figure 1: Grid diagram and the corresponding knot

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Construction of the chain complex (sketch)

Let G be a grid diagram of a knot, and N be its grid number. The complex C −(G) is a finitely generated free module over F2[U1, · · · , UN], where: ◮ the generators {x} are given by permutations of length N. (Each x can be drawn as N-tuple of points on the lattice), ◮ the homological degree of x is given by the Maslov function MG(x), with each factor Ui contributing to degree −2, and ◮ the differential ∂ : C −

k (G) → C − k−1(G) is given by

∂x =

• y
• r∈Recto(x,y)

Uε1

1 · · · UεN N y

where r runs over the empty rectangles connecting x to 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 =

• y
• r∈Recto(x,y)

Uε1

1 · · · UεN N y

= x = y

r

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Computing H−

∗ (G) algorithmically

Since the ground ring F2[U1, · · · , UN] is not a field (nor a PID), we cannot use computational methods to calculate the homology. However, since the generators are finite and deg Ui = −2, we may regard each k-th chain module C −

k (G) as a finitely generated free

✿✿✿✿✿✿✿✿✿✿

F2-module with generators of the form: Ua1

1 · · · UaN N x

where deg x − 2

iai = k.

With these inflated generators, it becomes possible to compute the homology group H−

k (G) for each k ∈ Z.

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Example (G = 31, N = 5)

k

• 6
• 5
• 4
• 3
• 2
• 1

#{x} 2 10 27 40 30 10 1 rank C −

k

622 360 192 90 35 10 1 rank H−

k

1 1 1 1

Example (G = 61, N = 8)

k

• 11

...

• 3
• 2
• 1

1 #{x} 1 ... 8,379 4,949 1,873 402 36 rank C −

k

5,321,071 ... 24,659 8,165 2,161 402 36 rank H−

k

... 1 1 # of inflated generators explodes as k 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 a bifiltration. Namely, every inflated generator

Ua1

1 · · · UaN N x ∈ C − ∗ (G),

is assigned a bidegree (i, j) ∈ Z2 as ◮ i = −a1, the exponent of U1 in its coefficient, and ◮ j = AG(x) −

ℓ aℓ, where AG is the Alexander function.

The first degree i is called the algebraic degree, and the second j is called the Alexander degree. It can be proved that ∂ is

✿✿✿✿✿✿✿✿✿✿✿✿✿

non-increasing for both i and j. Important knot concordance invariants such as τ, Vk and Υ can be

• btained from this bifiltration. In year 2019, K. Sato introduced a

new invariant G0(K) that unifies these invariants.

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Closed region and Z2-filtration

We call a subset R ⊂ Z2 an closed region iff: If (i, j) ∈ R and (i′ ≤ i, j′ ≤ j) then (i′, j′) ∈ R. For any closed region R, there is a corresponding subcomplex FRC − := SpanF2{ z ∈ C − | (i(z), j(z)) ∈ R } where z is a monomial of the form z = Ua1

1 · · · UaN N x. For another

closed region R′ ⊂ R, we have FR′C − ⊂ FRC −. The differential ∂ is closed in FRC −. Thus C −

∗ (G) admits a

✿✿✿✿✿✿✿✿✿✿✿

Z2-filtration with respect to the partial order ≤.

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C −

0 (G)

i j R R′

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The invariant G0(K)

Recall that H−

0 (G) ∼

= F2. The set G(0)

0 (K) is defined as the set of

minimal closed regions, each containing a homological generator of H−

0 (G), namely,

R ∈ G(0)

0 (K)

⇔

• ∃z ∈ FRC −

0 (G) s.t. 0 = [z] ∈ H− 0 (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: Determining τ, Vk and Υ from G0(K).

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Main result

Theorem (S.-Sato)

There is an algorithm for computing G0(K).

Corollary

There is an algorithm for computing τ, Vk and Υ. 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 G0(K)

First we compute one homological generator z ∈ C −

0 (G), i.e. a

cycle whose homology class generates H−

0 (G) ∼

= F2. For any closed region R, it contains a homological iff: ∃c ∈ C1 s.t. z − ∂c ∈ FRC0. Let QRC∗ = C∗/FRC∗, then the above condition is equivalent to: ∃c ∈ QRC1 s.t. ∂c = z ∈ QRC0 Each QRCi is a finite dimensional F2-vector space. We represent ∂ by a matrix A, then the c above corresponds to the solution x of the linear system: Ax = vec(z).

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We minimize the region containing z by “sweeping” its components into a smaller region. The invariant G0 is the set of all such minimal regions.

• z
• z′
• c

∂

sweep

R R′

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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 any R ∈ 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 candidate R. Take the basis of QRC0 = C0/FRC0 by modding out the generators of C0 that lie in R. Step 4. Compute the matrix A representing 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 realizable R’s that are minimal w.r.t. the inclusion ⊂, and we obtain G(0)

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. The ν+-equivalence classes of genus one knots. 2019.

arXiv: 1907.09116 [math.GT].