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Bordered Heegaard Floer Homology and Knot Doubling Operators Adam - - PowerPoint PPT Presentation

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Bordered Heegaard Floer Homology and Knot Doubling Operators Adam Simon Levine Brandeis University Knot Concordance and Homology Cobordism Workshop Wesleyan University July 21, 2010 Adam Simon Levine Bordered HF and Knot Doubling Operators

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Bordered Heegaard Floer Homology and Knot Doubling Operators

Adam Simon Levine

Brandeis University

Knot Concordance and Homology Cobordism Workshop Wesleyan University July 21, 2010

Adam Simon Levine Bordered HF and Knot Doubling Operators

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SLIDE 2

Slice Knots and Links

Definition A knot in S3 is called topologically slice if it is the boundary of a locally flatly embedded disk in B4. smoothly slice if it is the boundary of a smoothly embedded disk in B4. A link is topologically/smoothly slice if it bounds a disjoint union

  • f such disks.

Adam Simon Levine Bordered HF and Knot Doubling Operators

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SLIDE 3

Slice Knots and Links

Definition A knot in S3 is called topologically slice if it is the boundary of a locally flatly embedded disk in B4. smoothly slice if it is the boundary of a smoothly embedded disk in B4. A link is topologically/smoothly slice if it bounds a disjoint union

  • f such disks.

Big question: How do these two notions compare?

Adam Simon Levine Bordered HF and Knot Doubling Operators

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Whitehead and Bing Doubling

Given a knot K, the positive Whitehead double, negative Whitehead double, and Bing double are:

BD(K)

Wh+(K) Wh−(K)

Adam Simon Levine Bordered HF and Knot Doubling Operators

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SLIDE 5

Whitehead and Bing Doubling

Given a knot K, the positive Whitehead double, negative Whitehead double, and Bing double are:

BD(K)

Wh+(K) Wh−(K) We consider only untwisted doubles here.

Adam Simon Levine Bordered HF and Knot Doubling Operators

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When are Whitehead doubles topologically slice?

Theorem (Freedman) The Whitehead double (with either sign) of any knot is topologically slice. More generally, if L is a boundary link, then any Whitehead double of L is topologically slice.

Adam Simon Levine Bordered HF and Knot Doubling Operators

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SLIDE 7

When are Whitehead doubles topologically slice?

Theorem (Freedman) The Whitehead double (with either sign) of any knot is topologically slice. More generally, if L is a boundary link, then any Whitehead double of L is topologically slice. Question (Freedman) Are the Whitehead doubles of a link with trivial linking numbers topologically slice?

Adam Simon Levine Bordered HF and Knot Doubling Operators

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When are Whitehead doubles topologically slice?

Theorem (Freedman) The Whitehead double (with either sign) of any knot is topologically slice. More generally, if L is a boundary link, then any Whitehead double of L is topologically slice. Question (Freedman) Are the Whitehead doubles of a link with trivial linking numbers topologically slice? For two-component links, the answer is yes.

Adam Simon Levine Bordered HF and Knot Doubling Operators

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SLIDE 9

When are Whitehead doubles topologically slice?

Theorem (Freedman) The Whitehead double (with either sign) of any knot is topologically slice. More generally, if L is a boundary link, then any Whitehead double of L is topologically slice. Question (Freedman) Are the Whitehead doubles of a link with trivial linking numbers topologically slice? For two-component links, the answer is yes. It is equivalent to the four-dimensional surgery conjecture.

Adam Simon Levine Bordered HF and Knot Doubling Operators

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SLIDE 10

When are Whitehead doubles topologically slice?

Theorem (Freedman) The Whitehead double (with either sign) of any knot is topologically slice. More generally, if L is a boundary link, then any Whitehead double of L is topologically slice. Question (Freedman) Are the Whitehead doubles of a link with trivial linking numbers topologically slice? For two-component links, the answer is yes. It is equivalent to the four-dimensional surgery conjecture. Most people, including Freedman, think it’s not true.

Adam Simon Levine Bordered HF and Knot Doubling Operators

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When are Whitehead doubles smoothly slice?

Conjecture (Kirby’s problem list) Wh±(K) is smoothly slice if and only if K is (smoothly) slice.

Adam Simon Levine Bordered HF and Knot Doubling Operators

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SLIDE 12

When are Whitehead doubles smoothly slice?

Conjecture (Kirby’s problem list) Wh±(K) is smoothly slice if and only if K is (smoothly) slice. Theorem (Rudolph)

1

If K is a strongly quasipositive knot different from the unknot, then K is not smoothly slice.

Adam Simon Levine Bordered HF and Knot Doubling Operators

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SLIDE 13

When are Whitehead doubles smoothly slice?

Conjecture (Kirby’s problem list) Wh±(K) is smoothly slice if and only if K is (smoothly) slice. Theorem (Rudolph)

1

If K is a strongly quasipositive knot different from the unknot, then K is not smoothly slice.

2

If K is strongly quasipositive, then Wh+(K) is also strongly quasipositive, hence not smoothly slice.

Adam Simon Levine Bordered HF and Knot Doubling Operators

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When are Whitehead doubles smoothly slice?

Conjecture (Kirby’s problem list) Wh±(K) is smoothly slice if and only if K is (smoothly) slice. Theorem (Rudolph)

1

If K is a strongly quasipositive knot different from the unknot, then K is not smoothly slice.

2

If K is strongly quasipositive, then Wh+(K) is also strongly quasipositive, hence not smoothly slice. These were among the first known examples of knots that are topologically but not smoothly slice. (Akbulut, Gompf also found early examples.)

Adam Simon Levine Bordered HF and Knot Doubling Operators

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SLIDE 15

When are Whitehead doubles smoothly slice?

Conjecture (Kirby’s problem list) Wh±(K) is smoothly slice if and only if K is (smoothly) slice. Theorem (Rudolph)

1

If K is a strongly quasipositive knot different from the unknot, then K is not smoothly slice.

2

If K is strongly quasipositive, then Wh+(K) is also strongly quasipositive, hence not smoothly slice. These were among the first known examples of knots that are topologically but not smoothly slice. (Akbulut, Gompf also found early examples.) Bižaca used this to construct explicit examples of exotic smooth structures on R4.

Adam Simon Levine Bordered HF and Knot Doubling Operators

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The Ozsváth–Szabó invariant τ

Knot Floer homology provides a knot invariant τ(K) ∈ Z, which vanishes for any smoothly slice knot.

Adam Simon Levine Bordered HF and Knot Doubling Operators

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The Ozsváth–Szabó invariant τ

Knot Floer homology provides a knot invariant τ(K) ∈ Z, which vanishes for any smoothly slice knot. Theorem (Hedden) τ(Wh+(K)) =

  • 1

τ(K) > 0 τ(K) ≤ 0

Adam Simon Levine Bordered HF and Knot Doubling Operators

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The Ozsváth–Szabó invariant τ

Knot Floer homology provides a knot invariant τ(K) ∈ Z, which vanishes for any smoothly slice knot. Theorem (Hedden) τ(Wh+(K)) =

  • 1

τ(K) > 0 τ(K) ≤ 0 Corollary If K is any knot with τ(K) > 0 (e.g., any strongly quasipositive knot), then any iterated positive Whitehead double of K is not smoothly slice.

Adam Simon Levine Bordered HF and Knot Doubling Operators

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Iterated Bing Doubling

Any binary tree T specifies an iterated Bing double of K, denoted BT(K). K

Adam Simon Levine Bordered HF and Knot Doubling Operators

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Iterated Bing Doubling

Any binary tree T specifies an iterated Bing double of K, denoted BT(K). K

Adam Simon Levine Bordered HF and Knot Doubling Operators

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Iterated Bing Doubling

Any binary tree T specifies an iterated Bing double of K, denoted BT(K). K

Adam Simon Levine Bordered HF and Knot Doubling Operators

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Generalized Borromean Rings

The family of generalized Borromean links consists of all links

  • btained by taking iterated Bing doubles of the components of

the Hopf link.

Adam Simon Levine Bordered HF and Knot Doubling Operators

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Main Theorem

Are Whitehead doubles of iterated Bing doubles smoothly slice?

Adam Simon Levine Bordered HF and Knot Doubling Operators

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Main Theorem

Are Whitehead doubles of iterated Bing doubles smoothly slice? Theorem (L.)

1

Let K be any knot with τ(K) > 0 (e.g., any strongly quasipositive knot), and let T be any binary tree. Then the all-positive Whitehead double of BT(K) is topologically but not smoothly slice.

Adam Simon Levine Bordered HF and Knot Doubling Operators

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Main Theorem

Are Whitehead doubles of iterated Bing doubles smoothly slice? Theorem (L.)

1

Let K be any knot with τ(K) > 0 (e.g., any strongly quasipositive knot), and let T be any binary tree. Then the all-positive Whitehead double of BT(K) is topologically but not smoothly slice.

2

The all-positive Whitehead double of any generalized Borromean link is not smoothly slice.

Adam Simon Levine Bordered HF and Knot Doubling Operators

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Main Theorem

Are Whitehead doubles of iterated Bing doubles smoothly slice? Theorem (L.)

1

Let K be any knot with τ(K) > 0 (e.g., any strongly quasipositive knot), and let T be any binary tree. Then the all-positive Whitehead double of BT(K) is topologically but not smoothly slice.

2

The all-positive Whitehead double of any generalized Borromean link is not smoothly slice. It is not known whether the links in (2) are topologically slice.

Adam Simon Levine Bordered HF and Knot Doubling Operators

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Doubling operators

Given knots J, K and integers s, t, define the knot DJ,s(K, t) = DK,t(J, s) as the boundary of the plumbing of an s-framed J-annulus and a t-framed K-annulus. J, s K, t

Adam Simon Levine Bordered HF and Knot Doubling Operators

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Doubling operators

Given knots J, K and integers s, t, define the knot DJ,s(K, t) = DK,t(J, s) as the boundary of the plumbing of an s-framed J-annulus and a t-framed K-annulus. J, s K, t So Wh±(K) = DO,∓1(K, 0).

Adam Simon Levine Bordered HF and Knot Doubling Operators

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Doubling operators

Given knots J, K and integers s, t, define the knot DJ,s(K, t) = DK,t(J, s) as the boundary of the plumbing of an s-framed J-annulus and a t-framed K-annulus. J, s K, t So Wh±(K) = DO,∓1(K, 0). When t = 0, we often omit it: DJ,s(K) = DJ,s(K, 0).

Adam Simon Levine Bordered HF and Knot Doubling Operators

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Doubling operators

Proposition (Rudolph, Livingston) If s ≤ TB(J) and t ≤ TB(K), then DJ,s(K, t) is strongly quasipositive, so τ(DJ,s(K, t)) = 1.

Adam Simon Levine Bordered HF and Knot Doubling Operators

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Doubling operators

Proposition (Rudolph, Livingston) If s ≤ TB(J) and t ≤ TB(K), then DJ,s(K, t) is strongly quasipositive, so τ(DJ,s(K, t)) = 1. Theorem (L.) τ(DJ,s(K, t)) =      1 s > 2τ(J), t > 2τ(K) −1 s < 2τ(J), t < 2τ(K)

  • therwise.

Adam Simon Levine Bordered HF and Knot Doubling Operators

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Covering link calculus

Definition A link L in a Z2-homology 3-sphere Y is called Z2-slice if there exists a Z2-homology 4-ball X with ∂X = Y such that L bounds disjoint disks in X.

Adam Simon Levine Bordered HF and Knot Doubling Operators

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Covering link calculus

Definition A link L in a Z2-homology 3-sphere Y is called Z2-slice if there exists a Z2-homology 4-ball X with ∂X = Y such that L bounds disjoint disks in X. Proposition If L′ ⊂ Y ′ is a covering link of L ⊂ Y, and L is Z2-slice, then L′ is Z2-slice.

Adam Simon Levine Bordered HF and Knot Doubling Operators

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Covering link calculus

Definition A link L in a Z2-homology 3-sphere Y is called Z2-slice if there exists a Z2-homology 4-ball X with ∂X = Y such that L bounds disjoint disks in X. Proposition If L′ ⊂ Y ′ is a covering link of L ⊂ Y, and L is Z2-slice, then L′ is Z2-slice. Theorem (Ozsváth-Szabó) If K ⊂ S3 is smoothly Z2-slice, then τ(K) = 0.

Adam Simon Levine Bordered HF and Knot Doubling Operators

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Covering link calculus

Lemma Let L be a link in S3, and suppose there is an unknotted solid torus U ⊂ S3 such that L ∩ U consists of two components K1, K2 embedded as follows: if A1, A2 are the components of the untwisted Bing double of the core C of U, then K1 = DPk,sk ◦ · · · ◦ DP1,s1(A1), K2 = DQl,tl ◦ · · · ◦ DQ1,t1(A2).

Adam Simon Levine Bordered HF and Knot Doubling Operators

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Covering link calculus

Lemma Let L be a link in S3, and suppose there is an unknotted solid torus U ⊂ S3 such that L ∩ U consists of two components K1, K2 embedded as follows: if A1, A2 are the components of the untwisted Bing double of the core C of U, then K1 = DPk,sk ◦ · · · ◦ DP1,s1(A1), K2 = DQl,tl ◦ · · · ◦ DQ1,t1(A2). Let L′ be the link obtained from L by replacing K1 and K2 by C′ = DPk,sk ◦ · · · ◦ DP1,s1 ◦ DR,u(C), where (R, u) =

  • (Q1#Qr

1, 2t1)

l = 1 (DQ1,t1 ◦ · · · ◦ DQl−2,tl−2(DQl−1,tl−1(Ql#Qr

l , 2tl)), 0)

l > 1.

Adam Simon Levine Bordered HF and Knot Doubling Operators

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Covering link calculus

Lemma Let L be a link in S3, and suppose there is an unknotted solid torus U ⊂ S3 such that L ∩ U consists of two components K1, K2 embedded as follows: if A1, A2 are the components of the untwisted Bing double of the core C of U, then K1 = DPk,sk ◦ · · · ◦ DP1,s1(A1), K2 = DQl,tl ◦ · · · ◦ DQ1,t1(A2). Let L′ be the link obtained from L by replacing K1 and K2 by C′ = DPk,sk ◦ · · · ◦ DP1,s1 ◦ DR,u(C), where (R, u) =

  • (Q1#Qr

1, 2t1)

l = 1 (DQ1,t1 ◦ · · · ◦ DQl−2,tl−2(DQl−1,tl−1(Ql#Qr

l , 2tl)), 0)

l > 1. Then L′ is a covering link of L.

Adam Simon Levine Bordered HF and Knot Doubling Operators

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Covering link calculus

P, s Q, t

K1 K2 T

Adam Simon Levine Bordered HF and Knot Doubling Operators

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SLIDE 39

Covering link calculus

P, s Q, t

K1 K2 T

Adam Simon Levine Bordered HF and Knot Doubling Operators

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Covering link calculus

P, s P, s Q, t Q, t

T T

Adam Simon Levine Bordered HF and Knot Doubling Operators

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Covering link calculus

P, s Q#Qr, 2t

T

Adam Simon Levine Bordered HF and Knot Doubling Operators

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Covering link calculus

By iterating this move, we see that there is a knot DPk,sk ◦ · · · ◦ DP1,s1(K) that is a covering link of Wh+(BT(K)).

Adam Simon Levine Bordered HF and Knot Doubling Operators

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Covering link calculus

By iterating this move, we see that there is a knot DPk,sk ◦ · · · ◦ DP1,s1(K) that is a covering link of Wh+(BT(K)). Additionally, si < 2τ(Pi) for all i.

Adam Simon Levine Bordered HF and Knot Doubling Operators

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Covering link calculus

By iterating this move, we see that there is a knot DPk,sk ◦ · · · ◦ DP1,s1(K) that is a covering link of Wh+(BT(K)). Additionally, si < 2τ(Pi) for all i. Thus, τ(DPk ,sk ◦ · · · ◦ DP1,s1(K)) = 1, so DPk,sk ◦ · · · ◦ DP1,s1(K) is not smoothly Z2-slice, so Wh+(BT(K)) is not smoothly slice.

Adam Simon Levine Bordered HF and Knot Doubling Operators

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Covering link calculus

By iterating this move, we see that there is a knot DPk,sk ◦ · · · ◦ DP1,s1(K) that is a covering link of Wh+(BT(K)). Additionally, si < 2τ(Pi) for all i. Thus, τ(DPk ,sk ◦ · · · ◦ DP1,s1(K)) = 1, so DPk,sk ◦ · · · ◦ DP1,s1(K) is not smoothly Z2-slice, so Wh+(BT(K)) is not smoothly slice. If we use a mix of positive and negative Whitehead doubling, this approach fails.

Adam Simon Levine Bordered HF and Knot Doubling Operators

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Heegaard Floer Homology

For a closed 3-manifold Y, we get a chain complex CF(Y), invariant up to chain homotopy. So the homology is an invariant:

  • HF(Y) = H∗(

CF(Y)).

Adam Simon Levine Bordered HF and Knot Doubling Operators

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SLIDE 47

Heegaard Floer Homology

For a closed 3-manifold Y, we get a chain complex CF(Y), invariant up to chain homotopy. So the homology is an invariant:

  • HF(Y) = H∗(

CF(Y)). For a nulhomologous knot K ⊂ Y, we get a filtered chain complex CF(Y, K), invariant up to filtered chain homotopy.

Adam Simon Levine Bordered HF and Knot Doubling Operators

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SLIDE 48

Heegaard Floer Homology

For a closed 3-manifold Y, we get a chain complex CF(Y), invariant up to chain homotopy. So the homology is an invariant:

  • HF(Y) = H∗(

CF(Y)). For a nulhomologous knot K ⊂ Y, we get a filtered chain complex CF(Y, K), invariant up to filtered chain homotopy. The associated graded complex is denoted CFK(Y, K), and its homology is a knot invariant:

  • HFK(Y, K) = H∗(

CFK(Y, K)).

Adam Simon Levine Bordered HF and Knot Doubling Operators

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SLIDE 49

Heegaard Floer Homology

For a closed 3-manifold Y, we get a chain complex CF(Y), invariant up to chain homotopy. So the homology is an invariant:

  • HF(Y) = H∗(

CF(Y)). For a nulhomologous knot K ⊂ Y, we get a filtered chain complex CF(Y, K), invariant up to filtered chain homotopy. The associated graded complex is denoted CFK(Y, K), and its homology is a knot invariant:

  • HFK(Y, K) = H∗(

CFK(Y, K)). There is a spectral sequence with E1 page HFK(Y, K), converging to HF(Y). The whole sequence is an invariant

  • f K.

Adam Simon Levine Bordered HF and Knot Doubling Operators

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SLIDE 50

Heegaard Floer Homology

For a closed 3-manifold Y, we get a chain complex CF(Y), invariant up to chain homotopy. So the homology is an invariant:

  • HF(Y) = H∗(

CF(Y)). For a nulhomologous knot K ⊂ Y, we get a filtered chain complex CF(Y, K), invariant up to filtered chain homotopy. The associated graded complex is denoted CFK(Y, K), and its homology is a knot invariant:

  • HFK(Y, K) = H∗(

CFK(Y, K)). There is a spectral sequence with E1 page HFK(Y, K), converging to HF(Y). The whole sequence is an invariant

  • f K.

If Y = S3, then HF(Y) = F. τ(K) is the least filtration of any element of HFK(Y, K) that survives to the E∞ page.

Adam Simon Levine Bordered HF and Knot Doubling Operators

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Bordered Heegaard Floer homology

Surface F = ⇒

Adam Simon Levine Bordered HF and Knot Doubling Operators

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Bordered Heegaard Floer homology

Surface F = ⇒ DG algebra A(F)

Adam Simon Levine Bordered HF and Knot Doubling Operators

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Bordered Heegaard Floer homology

Surface F = ⇒ DG algebra A(F) Y1, φ1 : F

∼ =

− → ∂Y1 = ⇒

Adam Simon Levine Bordered HF and Knot Doubling Operators

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SLIDE 54

Bordered Heegaard Floer homology

Surface F = ⇒ DG algebra A(F) Y1, φ1 : F

∼ =

− → ∂Y1 = ⇒ Right A∞ module CFA(Y1)A(F)

Adam Simon Levine Bordered HF and Knot Doubling Operators

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SLIDE 55

Bordered Heegaard Floer homology

Surface F = ⇒ DG algebra A(F) Y1, φ1 : F

∼ =

− → ∂Y1 = ⇒ Right A∞ module CFA(Y1)A(F) Y2, φ2 : F

∼ =

− → −∂Y2 = ⇒

Adam Simon Levine Bordered HF and Knot Doubling Operators

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SLIDE 56

Bordered Heegaard Floer homology

Surface F = ⇒ DG algebra A(F) Y1, φ1 : F

∼ =

− → ∂Y1 = ⇒ Right A∞ module CFA(Y1)A(F) Y2, φ2 : F

∼ =

− → −∂Y2 = ⇒ Left DG module A(F) CFD(Y2)

Adam Simon Levine Bordered HF and Knot Doubling Operators

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SLIDE 57

Bordered Heegaard Floer homology

Surface F = ⇒ DG algebra A(F) Y1, φ1 : F

∼ =

− → ∂Y1 = ⇒ Right A∞ module CFA(Y1)A(F) Y2, φ2 : F

∼ =

− → −∂Y2 = ⇒ Left DG module A(F) CFD(Y2) Theorem (Lipshitz–Ozsváth–Thurston) If Y = Y1 ∪φ1◦φ−1

2

Y2, then

  • CFA(Y1) ˜

⊗ CFD(Y2) ≃ CF(Y).

Adam Simon Levine Bordered HF and Knot Doubling Operators

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SLIDE 58

Bordered Heegaard Floer homology

Surface F = ⇒ DG algebra A(F) Y1, φ1 : F

∼ =

− → ∂Y1 = ⇒ Right A∞ module CFA(Y1)A(F) Y2, φ2 : F

∼ =

− → −∂Y2 = ⇒ Left DG module A(F) CFD(Y2) Theorem (Lipshitz–Ozsváth–Thurston) If Y = Y1 ∪φ1◦φ−1

2

Y2, then

  • CFA(Y1) ˜

⊗ CFD(Y2) ≃ CF(Y). Moreover, if K is a nulhomologous knot in either Y1 or Y2, then there is an induced filtration on either CFA(Y1) or CFD(Y2), which induces the filtration on CF(Y, K).

Adam Simon Levine Bordered HF and Knot Doubling Operators

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Bordered Heegaard Floer homology

Can also define bimodules. For example:

Adam Simon Levine Bordered HF and Knot Doubling Operators

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Bordered Heegaard Floer homology

Can also define bimodules. For example: If Y has boundary components parametrized by F1, F2, get a (right, right) bimodule CFAA(Y)A(F1),A(F2).

Adam Simon Levine Bordered HF and Knot Doubling Operators

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Bordered Heegaard Floer homology

Can also define bimodules. For example: If Y has boundary components parametrized by F1, F2, get a (right, right) bimodule CFAA(Y)A(F1),A(F2). If Y has boundary components parametrized by −F1, −F2, get a (left, left) bimodule A(F1),A(F2) CFDD(Y).

Adam Simon Levine Bordered HF and Knot Doubling Operators

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SLIDE 62

Bordered Heegaard Floer homology

Can also define bimodules. For example: If Y has boundary components parametrized by F1, F2, get a (right, right) bimodule CFAA(Y)A(F1),A(F2). If Y has boundary components parametrized by −F1, −F2, get a (left, left) bimodule A(F1),A(F2) CFDD(Y). There are versions of the gluing theorem for bimodules as well.

Adam Simon Levine Bordered HF and Knot Doubling Operators

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SLIDE 63

Bordered Heegaard Floer homology

Let Y s

J , Y t K be the exteriors of J and K, with appropriate

framings.

Adam Simon Levine Bordered HF and Knot Doubling Operators

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SLIDE 64

Bordered Heegaard Floer homology

Let Y s

J , Y t K be the exteriors of J and K, with appropriate

framings. Let B1 ∪ B2 ∪ B3 ⊂ S3 denote the Borromean rings, and let X be the exterior of B1 ∪ B2.

Adam Simon Levine Bordered HF and Knot Doubling Operators

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SLIDE 65

Bordered Heegaard Floer homology

Let Y s

J , Y t K be the exteriors of J and K, with appropriate

framings. Let B1 ∪ B2 ∪ B3 ⊂ S3 denote the Borromean rings, and let X be the exterior of B1 ∪ B2. Then DJ,s(K, t) is the image of B3 in X ∪ Y s

J ∪ Y t K,

Adam Simon Levine Bordered HF and Knot Doubling Operators

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SLIDE 66

Bordered Heegaard Floer homology

Let Y s

J , Y t K be the exteriors of J and K, with appropriate

framings. Let B1 ∪ B2 ∪ B3 ⊂ S3 denote the Borromean rings, and let X be the exterior of B1 ∪ B2. Then DJ,s(K, t) is the image of B3 in X ∪ Y s

J ∪ Y t K, so

  • CF(S3, DJ,s(K, t)) ≃ (

CFAA(X) ˜ ⊗ CFD(Y s

J )) ˜

⊗ CFD(Y t

K ).

Adam Simon Levine Bordered HF and Knot Doubling Operators

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SLIDE 67

Bordered Heegaard Floer homology

Let Y s

J , Y t K be the exteriors of J and K, with appropriate

framings. Let B1 ∪ B2 ∪ B3 ⊂ S3 denote the Borromean rings, and let X be the exterior of B1 ∪ B2. Then DJ,s(K, t) is the image of B3 in X ∪ Y s

J ∪ Y t K, so

  • CF(S3, DJ,s(K, t)) ≃ (

CFAA(X) ˜ ⊗ CFD(Y s

J )) ˜

⊗ CFD(Y t

K ).

We can then follow the spectral sequence from

  • HFK(DJ,s(K, t)) to

HF(S3) carefully to determine τ(DJ,s(K, t)).

Adam Simon Levine Bordered HF and Knot Doubling Operators

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The torus algebra

The algebra A(T 2) is generated over F2 by ι0, ι1, ρ1, ρ2, ρ3, ρ12, ρ23, ρ23

Adam Simon Levine Bordered HF and Knot Doubling Operators

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SLIDE 69

The torus algebra

The algebra A(T 2) is generated over F2 by ι0, ι1, ρ1, ρ2, ρ3, ρ12, ρ23, ρ23 with nonzero multiplications: ι0ι0 = ι0 ι1ι1 = ι1 ρ1ρ2 = ρ12 ρ2ρ3 = ρ23 ρ12ρ3 = ρ123 ρ1ρ23 = ρ123

Adam Simon Levine Bordered HF and Knot Doubling Operators

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SLIDE 70

The torus algebra

The algebra A(T 2) is generated over F2 by ι0, ι1, ρ1, ρ2, ρ3, ρ12, ρ23, ρ23 with nonzero multiplications: ι0ι0 = ι0 ι1ι1 = ι1 ρ1ρ2 = ρ12 ρ2ρ3 = ρ23 ρ12ρ3 = ρ123 ρ1ρ23 = ρ123 ι0ρ1 = ρ1ι1 = ρ1 ι1ρ2 = ρ2ι0 = ρ2 ι0ρ3 = ρ3ι1 = ρ3 ι0ρ12 = ρ12ι0 = ρ12 ι1ρ23 = ρ23ι1 = ρ23 ι0ρ123 = ρ123ι1 = ρ123

Adam Simon Levine Bordered HF and Knot Doubling Operators

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SLIDE 71
  • CFD of knot complements

For K ⊂ S3, CFD(X t

K ) is determined by the following data

coming from CFK−(S3, K):

Adam Simon Levine Bordered HF and Knot Doubling Operators

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SLIDE 72
  • CFD of knot complements

For K ⊂ S3, CFD(X t

K ) is determined by the following data

coming from CFK−(S3, K): Two bases {˜ η0, . . . , ˜ η2n} and {˜ ξ0, . . . , ˜ ξ2n} for CFK−(S3, K);

Adam Simon Levine Bordered HF and Knot Doubling Operators

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SLIDE 73
  • CFD of knot complements

For K ⊂ S3, CFD(X t

K ) is determined by the following data

coming from CFK−(S3, K): Two bases {˜ η0, . . . , ˜ η2n} and {˜ ξ0, . . . , ˜ ξ2n} for CFK−(S3, K); Vertical arrows ˜ ξ2j−1 → ˜ ξ2j of length kj ∈ N;

Adam Simon Levine Bordered HF and Knot Doubling Operators

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SLIDE 74
  • CFD of knot complements

For K ⊂ S3, CFD(X t

K ) is determined by the following data

coming from CFK−(S3, K): Two bases {˜ η0, . . . , ˜ η2n} and {˜ ξ0, . . . , ˜ ξ2n} for CFK−(S3, K); Vertical arrows ˜ ξ2j−1 → ˜ ξ2j of length kj ∈ N; Horizontal arrows ˜ ξ2j−1 → ˜ ξ2j of length lj ∈ N.

Adam Simon Levine Bordered HF and Knot Doubling Operators

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SLIDE 75
  • CFD of knot complements

Lipshitz, Ozsváth, and Thurston proved: ι0 CFD(X t

K ) is generated by {ξ0, . . . , ξ2n} or by

{η0, . . . , η2n}.

Adam Simon Levine Bordered HF and Knot Doubling Operators

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SLIDE 76
  • CFD of knot complements

Lipshitz, Ozsváth, and Thurston proved: ι0 CFD(X t

K ) is generated by {ξ0, . . . , ξ2n} or by

{η0, . . . , η2n}. ι1 CFD(X t

K ) is generated by

{γ1, . . . , γr} ∪

n

  • j=1

{κj

1, . . . , κj kj} ∪ n

  • j=1

{λj

1, . . . , λj lj}.

where r = |2τ(K) − t|.

Adam Simon Levine Bordered HF and Knot Doubling Operators

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SLIDE 77
  • CFD of knot complements

Vertical stable chains: ξ2j

ρ123

− − → κj

1 ρ23

− − → · · ·

ρ23

− − → κj

kj ρ1

← − ξ2j−1.

Adam Simon Levine Bordered HF and Knot Doubling Operators

slide-78
SLIDE 78
  • CFD of knot complements

Vertical stable chains: ξ2j

ρ123

− − → κj

1 ρ23

− − → · · ·

ρ23

− − → κj

kj ρ1

← − ξ2j−1. Horizonal stable chains: η2j−1

ρ3

− → λj

1 ρ23

− − → · · ·

ρ23

− − → λj

lj ρ2

− → η2j,

Adam Simon Levine Bordered HF and Knot Doubling Operators

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SLIDE 79
  • CFD of knot complements

Vertical stable chains: ξ2j

ρ123

− − → κj

1 ρ23

− − → · · ·

ρ23

− − → κj

kj ρ1

← − ξ2j−1. Horizonal stable chains: η2j−1

ρ3

− → λj

1 ρ23

− − → · · ·

ρ23

− − → λj

lj ρ2

− → η2j, Unstable chain:      η0

ρ3

− → γ1

ρ23

− − → · · ·

ρ23

− − → γr

ρ1

← − ξ0 t < 2τ(K) ξ0

ρ12

− − → η0 t = 2τ(K) ξ0

ρ123

− − → γ1

ρ23

− − → · · ·

ρ23

− − → γr

ρ2

− → η0 t > 2τ(K).

Adam Simon Levine Bordered HF and Knot Doubling Operators

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SLIDE 80
  • CFA for the Whitehead double

Let Wh ⊂ S1 × D2 be the pattern for the positive Whitehead

  • double. Then

CFA(S1 × D2, Wh) has the following form:

−1 1 c c′

  • b

ρ3,ρ2,ρ1

  • ρ1
  • b′
  • ρ3,ρ2,ρ1
  • ρ1
  • ρ12
  • ρ123
  • a

a′

1+ρ23

  • ρ2
  • d

ρ3

  • Adam Simon Levine

Bordered HF and Knot Doubling Operators

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SLIDE 81
  • CFA for the Whitehead double

Let Wh ⊂ S1 × D2 be the pattern for the positive Whitehead

  • double. Then

CFA(S1 × D2, Wh) has the following form:

−1 1 c c′

  • b

ρ3,ρ2,ρ1

  • ρ1
  • b′
  • ρ3,ρ2,ρ1
  • ρ1
  • ρ12
  • ρ123
  • a

a′

1+ρ23

  • ρ2
  • d

ρ3

  • In other words, for instance:

m1(b′) = b m2(b, ρ1) = a m4(b, ρ3, ρ2, ρ1) = c

Adam Simon Levine Bordered HF and Knot Doubling Operators

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SLIDE 82

Proving Hedden’s formula for τ(Wh+(K, t))

We split CFA(S1 × D2, Wh) ⊠ CFD(X t

K ) into direct summands

according to the horizontal and vertical chains: Cj

vert =

  • b, b′

⊠

  • ξ2j−1, ξ2j
  • +
  • a, a′, c, c′

⊠

  • κj

i | 1 ≤ i ≤ kj

  • Cj

hor = d ⊠

  • η2j−1, η2j
  • +
  • a, a′, c, c′

⊠

  • λj

i | 1 ≤ i ≤ lj

  • Cunst =
  • b ⊠ ξ0, b′ ⊠ ξ0, d ⊠ η0
  • +
  • a, a′, c, c′

⊠ λi | 1 ≤ i ≤ r .

Adam Simon Levine Bordered HF and Knot Doubling Operators

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SLIDE 83

Proving Hedden’s formula for τ(Wh+(K, t))

We split CFA(S1 × D2, Wh) ⊠ CFD(X t

K ) into direct summands

according to the horizontal and vertical chains: Cj

vert =

  • b, b′

⊠

  • ξ2j−1, ξ2j
  • +
  • a, a′, c, c′

⊠

  • κj

i | 1 ≤ i ≤ kj

  • Cj

hor = d ⊠

  • η2j−1, η2j
  • +
  • a, a′, c, c′

⊠

  • λj

i | 1 ≤ i ≤ lj

  • Cunst =
  • b ⊠ ξ0, b′ ⊠ ξ0, d ⊠ η0
  • +
  • a, a′, c, c′

⊠ λi | 1 ≤ i ≤ r . What’s special here is that we actually get a direct sum

  • decomposition. Almost. The single F that survives in homology

always comes from Cunst.

Adam Simon Levine Bordered HF and Knot Doubling Operators

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SLIDE 84

Proving Hedden’s formula for τ(Wh+(K, t))

In the case where s < 2τ(K):

d ⊠ η0

  • c ⊠ γ1

c′ ⊠ γ1

  • a ⊠ γ1

a′ ⊠ γ1

  • c ⊠ γr

c′ ⊠ γr

  • a ⊠ γr

a′ ⊠ γr

  • b ⊠ ξ0
  • b′ ⊠ ξ0
  • Adam Simon Levine

Bordered HF and Knot Doubling Operators

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SLIDE 85

Poetic Conclusion

Our goal is one whose application’s nice For smooth four-manifold topology: To tell if certain knots and links are slice With bordered Heegaard Floer homology.

Adam Simon Levine Bordered HF and Knot Doubling Operators

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SLIDE 86

Poetic Conclusion

Our goal is one whose application’s nice For smooth four-manifold topology: To tell if certain knots and links are slice With bordered Heegaard Floer homology. We seek concordance data that detect Some links obtained by Whitehead doublings, As well as knots we get when we infect Along two of the three Borromean rings.

Adam Simon Levine Bordered HF and Knot Doubling Operators

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SLIDE 87

Poetic Conclusion

Our goal is one whose application’s nice For smooth four-manifold topology: To tell if certain knots and links are slice With bordered Heegaard Floer homology. We seek concordance data that detect Some links obtained by Whitehead doublings, As well as knots we get when we infect Along two of the three Borromean rings. Some lengthy work with bordered Floer then proves How τ for satellites like these is found. We see, by this result and cov’ring moves, That smooth slice disks our links can never bound.

Adam Simon Levine Bordered HF and Knot Doubling Operators

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SLIDE 88

Poetic Conclusion

Our goal is one whose application’s nice For smooth four-manifold topology: To tell if certain knots and links are slice With bordered Heegaard Floer homology. We seek concordance data that detect Some links obtained by Whitehead doublings, As well as knots we get when we infect Along two of the three Borromean rings. Some lengthy work with bordered Floer then proves How τ for satellites like these is found. We see, by this result and cov’ring moves, That smooth slice disks our links can never bound. The theorem’s proved, the dissertation’s done, But all the work ahead has just begun.

Adam Simon Levine Bordered HF and Knot Doubling Operators