Computing first order signatures and making aconstruction of Cochran-Harvey-Leidy explicit.

Christopher William Davis, Rice University

March 10, 2012

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 1 / 1

Outline:

Knot concordance and the solvable filtration.

Robust doubling operators and the construction ofCochran-Harvey-Leidy.

I An outline of the construction and where it fails to be explicit.I The thing that’s missing: a computational tool.

The computational tool: Zeroth order signatures of links are given byintegrals of the Cimasoni-Florens signature function

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 2 / 1

knot concordance and the solvable filtration of the knotconcordance group.

Definition

A knot sitting in S3 = ∂B4 is called slice if bounds an embedded locallyflat disk in B4

Modulo slice knots, knots form an abelian group under connected sum, theKnot Concordance Group, C.

Cochran-Orr-Teichner provide a filtration of C by subgroups:

. . .F1.5 ⊆ F1 ⊆ F0.5 ⊆ F0 ⊆ C

called the solvable filtration of the concordance group. This filtrationcan be thought of as measuring how subtle a knot’s non-sliceness is.

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 3 / 1

knot concordance and the solvable filtration of the knotconcordance group.

Definition

A knot sitting in S3 = ∂B4 is called slice if bounds an embedded locallyflat disk in B4

Modulo slice knots, knots form an abelian group under connected sum, theKnot Concordance Group, C.Cochran-Orr-Teichner provide a filtration of C by subgroups:

. . .F1.5 ⊆ F1 ⊆ F0.5 ⊆ F0 ⊆ C

called the solvable filtration of the concordance group.

This filtrationcan be thought of as measuring how subtle a knot’s non-sliceness is.

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 3 / 1

knot concordance and the solvable filtration of the knotconcordance group.

Definition

A knot sitting in S3 = ∂B4 is called slice if bounds an embedded locallyflat disk in B4

Modulo slice knots, knots form an abelian group under connected sum, theKnot Concordance Group, C.Cochran-Orr-Teichner provide a filtration of C by subgroups:

. . .F1.5 ⊆ F1 ⊆ F0.5 ⊆ F0 ⊆ C

called the solvable filtration of the concordance group. This filtrationcan be thought of as measuring how subtle a knot’s non-sliceness is.

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 3 / 1

The solvable filtration of the knot concordance group.

Definition (Cochran-Orr-Teichner)

A knot is called (n)-solvable, denoted K ∈ Fn, if there exists a smooth4-manifold W , called a (n)-solution, bounded by M(K ) such that:

1 The inclusion induced map H1(M(K ))→ H1(W ) is an isomorphism.

2 H2(W ) has a basis given by smoothly embedded surfaces Li and Di

all disjoint except that Li ∩ Di = one point

3 π1(Li ) sits in the n’th term in the derived series of π1(W ), π1(W )(n).

4 π1(Di ) sits in the n’th term in the derived series of π1(W ).

the derived series of a group G is defined by G (0) = G andG (n+1) =

[G (n),G (n)

].

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 4 / 1

The solvable filtration of the knot concordance group.

Definition (Cochran-Orr-Teichner)

A knot is called (n.5)-solvable, denoted K ∈ Fn.5, if there exists a smooth4-manifold W , called a (n.5)-solution, bounded by M(K ) such that:

1 The inclusion induced map H1(M(K ))→ H1(W ) is an isomorphism.

2 H2(W ) has a basis given by smoothly embedded surfaces Li and Di

all disjoint except that Li ∩ Di = one point

3 π1(Li ) sits in the n + 1’th term in the derived series of π1(W ),π1(W )(n+1).

4 π1(Di ) sits in the n’th term in the derived series of π1(W ).

the derived series of a group G is defined by G (0) = G andG (n+1) =

[G (n),G (n)

].

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 4 / 1

Infection

Take knots R and J and an unknotted curve η in E (R) (the exterior of R).Infection of R by J along η (Rη(J)) can be described as taking the strandsof R which pass through the disk bounded by η and tying them into theknot J.

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 5 / 1

Infection

Take knots R and J and an unknotted curve η in E (R) (the exterior of R).Infection of R by J along η (Rη(J)) can be described as taking the strandsof R which pass through the disk bounded by η and tying them into theknot J.

k twists

η

RJ

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 5 / 1

Infection

Take knots R and J and an unknotted curve η in E (R) (the exterior of R).Infection of R by J along η (Rη(J)) can be described as taking the strandsof R which pass through the disk bounded by η and tying them into theknot J.

J=:

Rη(J)

k twists k twists

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 5 / 1

Infection

When R is slice and η has zero linking with R, the pair (R, η) issometimes called a doubling operator.Infection along doubling operators increases solvability:

Theorem (Cochran-Orr-Teichner)

If (R, η) is a doubling operator, and J is (n)-solvable, then Rη(J) is(n + 1)-solvable.

Iterating the infection process yields that if J is (n)-solvable, then(Rη)k(J) is (n + k)-solvable.

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 6 / 1

Infection

When R is slice and η has zero linking with R, the pair (R, η) issometimes called a doubling operator.Infection along doubling operators increases solvability:

Theorem (Cochran-Orr-Teichner)

If (R, η) is a doubling operator, and J is (n.5)-solvable, then Rη(J) is(n + 1.5)-solvable.

Iterating the infection process yields that if J is (n.5)-solvable, then(Rη)k(J) is (n + k .5)-solvable.

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 6 / 1

Infection as a path to non-triviality in this filtration: TheCochran-Harvey-Leidy construction

Definition

A doubling operator Rη with R a slice knot is called a robust doublingoperator if

1 The Alexander module of R is given by A0(R) = Q[t±1]/(δ(t)δ(t−1))for a prime polynomial δ and is generated by η

2 For every isotropic submodule P ⊆ A0(R), namely P = (δ(t)),(δ(t−1)) or 0, either P corresponds to a slice disk or the first ordersignature, ρ1

P(R) is nonzero

Theorem (Cochran-Harvey-Leidy)

Let Rη be a robust doubling operator. Let {Kj} be a set of knots in F0

such that the set of zero’th order signatures {ρ0(Kj)}j is rationallylinearly independent of the first order signatures {ρ1

P(R)}P .Then {(Rη)n(Kj)}j is linearly independent in Fn/Fn.5.

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 7 / 1

Infection as a path to non-triviality in this filtration: TheCochran-Harvey-Leidy construction

Definition

A doubling operator Rη with R a slice knot is called a robust doublingoperator if

1 The Alexander module of R is given by A0(R) = Q[t±1]/(δ(t)δ(t−1))for a prime polynomial δ and is generated by η

2 For every isotropic submodule P ⊆ A0(R), namely P = (δ(t)),(δ(t−1)) or 0, either P corresponds to a slice disk or the first ordersignature, ρ1

P(R) is nonzero

Theorem (Cochran-Harvey-Leidy)

Let Rη be a robust doubling operator. Let {Kj} be a set of knots in F0

such that the set of zero’th order signatures {ρ0(Kj)}j is rationallylinearly independent of the first order signatures {ρ1

P(R)}P .Then {(Rη)n(Kj)}j is linearly independent in Fn/Fn.5.

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 7 / 1

Infection as a path to non-triviality in this filtration: TheCochran-Harvey-Leidy construction

Definition

A doubling operator Rη with R a slice knot is called a robust doublingoperator if

1 The Alexander module of R is given by A0(R) = Q[t±1]/(δ(t)δ(t−1))for a prime polynomial δ and is generated by η

2 For every isotropic submodule P ⊆ A0(R), namely P = (δ(t)),(δ(t−1)) or 0, either P corresponds to a slice disk or the first ordersignature, ρ1

P(R) is nonzero

Theorem (Cochran-Harvey-Leidy)

Let Rη be a robust doubling operator. Let {Kj} be a set of knots in F0

such that the set of zero’th order signatures {ρ0(Kj)}j is rationallylinearly independent of the first order signatures {ρ1

P(R)}P .Then {(Rη)n(Kj)}j is linearly independent in Fn/Fn.5.

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 7 / 1

Examples of robust doubling operators.

Theorem (Cochran-Harvey-Leidy)

If J is a trefoil knot, then for each positive integer, k, at least one of thedoubling operators, Rη and R ′η′ is robust.

Theorem (D.)

If k ≥ 3, then Rη is robust.

k twists

η

R

J

k twists

η′

R ′

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 8 / 1

Examples of robust doubling operators.

Theorem (Cochran-Harvey-Leidy)

If J is a trefoil knot, then for each positive integer, k, at least one of thedoubling operators, Rη and R ′η′ is robust.

Theorem (D.)

If k ≥ 3, then Rη is robust.

k twists

η

R

J

k twists

η′

R ′

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 8 / 1

First order signatures and derivatives

The submodules (δ(t)) and (δ(t−1)) are each generated by lifts ofderivatives of R or R ′

A derivative is a g -component link sitting on a genus g Seifert surface forK with zero pairwise linking and zero Seifert framing.

k twists

η

R

J

k twists

η′

R ′

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 9 / 1

First order signatures and derivatives

The submodules (δ(t)) and (δ(t−1)) are each generated by lifts ofderivatives of R or R ′

A derivative is a g -component link sitting on a genus g Seifert surface forK with zero pairwise linking and zero Seifert framing.

k twists

η

R

J

k twists

η′

R ′

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 9 / 1

First order signatures and derivatives

The submodules (δ(t)) and (δ(t−1)) are each generated by lifts ofderivatives of R or R ′

A derivative is a g -component link sitting on a genus g Seifert surface forK with zero pairwise linking and zero Seifert framing.

k twists

η

R

J

k twists

η′

R ′

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 9 / 1

First order signatures and derivatives

The submodules (δ(t)) and (δ(t−1)) are each generated by lifts ofderivatives of R or R ′

A derivative is a g -component link sitting on a genus g Seifert surface forK with zero pairwise linking and zero Seifert framing.

k twists

η

R

J

k twists

η′

R ′

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 9 / 1

First order signatures and derivatives

Theorem (Cochran-Harvey-Leidy)

If P ⊆ A0(R) is generated by the lifts of a g-component derivative, L,then |ρ1

P(R)− ρ0(L)| ≤ g − 1.

Proposition (Cochran-Teichner)

For a knot, K , ρ0(K ) is given by the normalized integral of theTristram-Levine signature

Theorem (D)

For a link, L, ρ0(L) is given by the normalized integral of a signaturefunction defined by Cimasoni-Florens.

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 10 / 1

First order signatures and derivatives

Theorem (Cochran-Harvey-Leidy)

If P ⊆ A0(R) is generated by the lifts of a g-component derivative, L,then |ρ1

P(R)− ρ0(L)| ≤ g − 1.

Proposition (Cochran-Teichner)

For a knot, K , ρ0(K ) is given by the normalized integral of theTristram-Levine signature

Theorem (D)

For a link, L, ρ0(L) is given by the normalized integral of a signaturefunction defined by Cimasoni-Florens.

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 10 / 1

First order signatures and derivatives

Theorem (Cochran-Harvey-Leidy)

If P ⊆ A0(R) is generated by the lifts of a g-component derivative, L,then |ρ1

P(R)− ρ0(L)| ≤ g − 1.

Proposition (Cochran-Teichner)

For a knot, K , ρ0(K ) is given by the normalized integral of theTristram-Levine signature

Theorem (D)

For a link, L, ρ0(L) is given by the normalized integral of a signaturefunction defined by Cimasoni-Florens.

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 10 / 1

Back to the Cochran-Harvey-Leidy construction

k twists

η

R

J

k twists

η′

R ′

They study the doubling operators Rη and R ′η′ (ρ0(J) 6= 0)

For R, (δ(t)) and (δ(t−1)) both correspond to slice disks.For R ′ (δ(t)) corresponds to a slice disk and ρ1

(δ(t−1))(R) = ρ0(J) 6= 0.

Finally, ρ1{0}(R ′) = ρ1

{0}(R) + ρ0(J). At least one of Rη and R ′η is robust.The techniques they use cannot tell which.The trivial submodule is not generated by a derivative.

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 11 / 1

Back to the Cochran-Harvey-Leidy construction

k twists

η

R

J

k twists

η′

R ′

They study the doubling operators Rη and R ′η′ (ρ0(J) 6= 0)

For R, (δ(t)) and (δ(t−1)) both correspond to slice disks.

For R ′ (δ(t)) corresponds to a slice disk and ρ1(δ(t−1))(R) = ρ0(J) 6= 0.

Finally, ρ1{0}(R ′) = ρ1

{0}(R) + ρ0(J). At least one of Rη and R ′η is robust.The techniques they use cannot tell which.The trivial submodule is not generated by a derivative.

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 11 / 1

Back to the Cochran-Harvey-Leidy construction

k twists

η

R

J

k twists

η′

R ′

They study the doubling operators Rη and R ′η′ (ρ0(J) 6= 0)

For R, (δ(t)) and (δ(t−1)) both correspond to slice disks.

For R ′ (δ(t)) corresponds to a slice disk and ρ1(δ(t−1))(R) = ρ0(J) 6= 0.

Finally, ρ1{0}(R ′) = ρ1

{0}(R) + ρ0(J). At least one of Rη and R ′η is robust.The techniques they use cannot tell which.The trivial submodule is not generated by a derivative.

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 11 / 1

Back to the Cochran-Harvey-Leidy construction

k twists

η

R

J

k twists

η′

R ′

They study the doubling operators Rη and R ′η′ (ρ0(J) 6= 0)

For R, (δ(t)) and (δ(t−1)) both correspond to slice disks.

For R ′ (δ(t)) corresponds to a slice disk and ρ1(δ(t−1))(R) = ρ0(J) 6= 0.

Finally, ρ1{0}(R ′) = ρ1

{0}(R) + ρ0(J). At least one of Rη and R ′η is robust.The techniques they use cannot tell which.The trivial submodule is not generated by a derivative.

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 11 / 1

Back to the Cochran-Harvey-Leidy construction

k twists

η

R

J

k twists

η′

R ′

They study the doubling operators Rη and R ′η′ (ρ0(J) 6= 0)

For R, (δ(t)) and (δ(t−1)) both correspond to slice disks.

For R ′ (δ(t)) corresponds to a slice disk and ρ1(δ(t−1))(R) = ρ0(J) 6= 0.

Finally, ρ1{0}(R ′) = ρ1

{0}(R) + ρ0(J). At least one of Rη and R ′η is robust.The techniques they use cannot tell which.The trivial submodule is not generated by a derivative.

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 11 / 1

Back to the Cochran-Harvey-Leidy construction

k twists

η

R

J

k twists

η′

R ′

They study the doubling operators Rη and R ′η′ (ρ0(J) 6= 0)

For R, (δ(t)) and (δ(t−1)) both correspond to slice disks.For R ′ (δ(t)) corresponds to a slice disk and ρ1

(δ(t−1))(R) = ρ0(J) 6= 0.

Finally, ρ1{0}(R ′) = ρ1

{0}(R) + ρ0(J). At least one of Rη and R ′η is robust.The techniques they use cannot tell which.

The trivial submodule is not generated by a derivative.

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 11 / 1

Back to the Cochran-Harvey-Leidy construction

k twists

η

R

J

k twists

η′

R ′

They study the doubling operators Rη and R ′η′ (ρ0(J) 6= 0)

For R, (δ(t)) and (δ(t−1)) both correspond to slice disks.For R ′ (δ(t)) corresponds to a slice disk and ρ1

(δ(t−1))(R) = ρ0(J) 6= 0.

Finally, ρ1{0}(R ′) = ρ1

{0}(R) + ρ0(J). At least one of Rη and R ′η is robust.The techniques they use cannot tell which.

The trivial submodule is not generated by a derivative.

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 11 / 1

Back to the Cochran-Harvey-Leidy construction

k twists

η

R

J

k twists

η′

R ′

They study the doubling operators Rη and R ′η′ (ρ0(J) 6= 0)

For R, (δ(t)) and (δ(t−1)) both correspond to slice disks.For R ′ (δ(t)) corresponds to a slice disk and ρ1

(δ(t−1))(R) = ρ0(J) 6= 0.

Finally, ρ1{0}(R ′) = ρ1

{0}(R) + ρ0(J). At least one of Rη and R ′η is robust.The techniques they use cannot tell which.

The trivial submodule is not generated by a derivative.

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 11 / 1

Back to the Cochran-Harvey-Leidy construction

k twists

η

R

J

k twists

η′

R ′

They study the doubling operators Rη and R ′η′ (ρ0(J) 6= 0)

For R, (δ(t)) and (δ(t−1)) both correspond to slice disks.For R ′ (δ(t)) corresponds to a slice disk and ρ1

(δ(t−1))(R) = ρ0(J) 6= 0.

Finally, ρ1{0}(R ′) = ρ1

{0}(R) + ρ0(J). At least one of Rη and R ′η is robust.The techniques they use cannot tell which.

The trivial submodule is not generated by a derivative.

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 11 / 1

Back to the Cochran-Harvey-Leidy construction

k twists

η

R

J

k twists

η′

R ′

They study the doubling operators Rη and R ′η′ (ρ0(J) 6= 0)

For R, (δ(t)) and (δ(t−1)) both correspond to slice disks.For R ′ (δ(t)) corresponds to a slice disk and ρ1

(δ(t−1))(R) = ρ0(J) 6= 0.

Finally, ρ1{0}(R ′) = ρ1

{0}(R) + ρ0(J). At least one of Rη and R ′η is robust.The techniques they use cannot tell which.

The trivial submodule is not generated by a derivative.

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 11 / 1

Back to the Cochran-Harvey-Leidy construction

k twists

η

R

J

k twists

η′

R ′

They study the doubling operators Rη and R ′η′ (ρ0(J) 6= 0)

For R, (δ(t)) and (δ(t−1)) both correspond to slice disks.For R ′ (δ(t)) corresponds to a slice disk and ρ1

(δ(t−1))(R) = ρ0(J) 6= 0.

Finally, ρ1{0}(R ′) = ρ1

{0}(R) + ρ0(J). At least one of Rη and R ′η is robust.The techniques they use cannot tell which.The trivial submodule is not generated by a derivative.

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 11 / 1

Additivity of first order signatures

Proposition (D.)

For P ⊆ A0(K #J) ∼= A0(K )⊕ A0(J)

let PK = P ∩ A0(K ) andPJ = P ∩ A0(J). Then ρ1

P(K #J) = ρ1PK

(K ) + ρ1PJ

(J)

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 12 / 1

Additivity of first order signatures

Proposition (D.)

For P ⊆ A0(K #J) ∼= A0(K )⊕ A0(J) let PK = P ∩ A0(K ) andPJ = P ∩ A0(J).

Then ρ1P(K #J) = ρ1

PK(K ) + ρ1

PJ(J)

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 12 / 1

Additivity of first order signatures

Proposition (D.)

For P ⊆ A0(K #J) ∼= A0(K )⊕ A0(J) let PK = P ∩ A0(K ) andPJ = P ∩ A0(J). Then ρ1

P(K #J) = ρ1PK

(K ) + ρ1PJ

(J)

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 12 / 1

Computing first order signatures

Let’s apply this to realize ρ1{0}(R) and a first order signature corresponding

to a submodule generated by a derivative of R#R.

Let P be thesubmodule of A0(R#R) = A0(R)⊕ A0(R) given by⟨(δ(t), δ(t)), (δ(t−1),−δ(t−1))

⟩.

P intersects trivially with both of theA0(R)-summands so 2ρ1

{0}(R) = ρ1P(R#R) which is within 1 of ρ0(L).

If |ρ0(L)| > 1, then Rη is robust. We need a tool to compute zero’th ordersignatures of links.

k twists k twists

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 13 / 1

Computing first order signatures

Let’s apply this to realize ρ1{0}(R) and a first order signature corresponding

to a submodule generated by a derivative of R#R. Let P be thesubmodule of A0(R#R) = A0(R)⊕ A0(R) given by⟨(δ(t), δ(t)), (δ(t−1),−δ(t−1))

⟩.

P intersects trivially with both of theA0(R)-summands so 2ρ1

{0}(R) = ρ1P(R#R) which is within 1 of ρ0(L).

If |ρ0(L)| > 1, then Rη is robust. We need a tool to compute zero’th ordersignatures of links.

k twists k twists

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 13 / 1

Computing first order signatures

Let’s apply this to realize ρ1{0}(R) and a first order signature corresponding

to a submodule generated by a derivative of R#R. Let P be thesubmodule of A0(R#R) = A0(R)⊕ A0(R) given by⟨(δ(t), δ(t)), (δ(t−1),−δ(t−1))

⟩. P intersects trivially with both of the

A0(R)-summands so 2ρ1{0}(R) = ρ1

P(R#R)

which is within 1 of ρ0(L).

If |ρ0(L)| > 1, then Rη is robust. We need a tool to compute zero’th ordersignatures of links.

k twists k twists

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 13 / 1

Computing first order signatures

Let’s apply this to realize ρ1{0}(R) and a first order signature corresponding

to a submodule generated by a derivative of R#R. Let P be thesubmodule of A0(R#R) = A0(R)⊕ A0(R) given by⟨(δ(t), δ(t)), (δ(t−1),−δ(t−1))

⟩. P intersects trivially with both of the

A0(R)-summands so 2ρ1{0}(R) = ρ1

P(R#R) which is within 1 of ρ0(L).

If |ρ0(L)| > 1, then Rη is robust. We need a tool to compute zero’th ordersignatures of links.

k twists k twists

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 13 / 1

Computing first order signatures

Let’s apply this to realize ρ1{0}(R) and a first order signature corresponding

to a submodule generated by a derivative of R#R. Let P be thesubmodule of A0(R#R) = A0(R)⊕ A0(R) given by⟨(δ(t), δ(t)), (δ(t−1),−δ(t−1))

⟩. P intersects trivially with both of the

A0(R)-summands so 2ρ1{0}(R) = ρ1

P(R#R) which is within 1 of ρ0(L).

If |ρ0(L)| > 1, then Rη is robust.

We need a tool to compute zero’th ordersignatures of links.

k twists k twists

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 13 / 1

Computing first order signatures

Let’s apply this to realize ρ1{0}(R) and a first order signature corresponding

to a submodule generated by a derivative of R#R. Let P be thesubmodule of A0(R#R) = A0(R)⊕ A0(R) given by⟨(δ(t), δ(t)), (δ(t−1),−δ(t−1))

⟩. P intersects trivially with both of the

A0(R)-summands so 2ρ1{0}(R) = ρ1

P(R#R) which is within 1 of ρ0(L).

If |ρ0(L)| > 1, then Rη is robust. We need a tool to compute zero’th ordersignatures of links.

k twists k twists

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 13 / 1

Defining the Cimasoni-Florens signature: Links andC-Complexes

Definition

An n color link L = L(1) . . . L(n) is a link whose every component is coloredwith a number 1 through n.

L(k) refers to the sub-link of L of componentscolored with a k. They are considered up to color preserving isotopy.

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 14 / 1

Defining the Cimasoni-Florens signature: Links andC-Complexes

Definition

An n color link L = L(1) . . . L(n) is a link whose every component is coloredwith a number 1 through n. L(k) refers to the sub-link of L of componentscolored with a k.

They are considered up to color preserving isotopy.

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 14 / 1

Defining the Cimasoni-Florens signature: Links andC-Complexes

Definition

An n color link L = L(1) . . . L(n) is a link whose every component is coloredwith a number 1 through n. L(k) refers to the sub-link of L of componentscolored with a k. They are considered up to color preserving isotopy.

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 14 / 1

Defining the Cimasoni-Florens signature: Links andC-Complexes

Definition (Cimasoni, 2004)

For an n colored Link L a C-Complex for L is a union of surfacesS = S1 . . . Sn in S3 with

1 ∂Si = L(i)

2 Si and Sj intersect in arcs with one endpoint on ∂Si and the other in∂Sj (a “clasp”)

3 Si ∩ Sj ∩ Sk = ∅

A Clasp:

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 15 / 1

Defining the Cimasoni-Florens signature: Links andC-Complexes

Definition (Cimasoni, 2004)

For an n colored Link L a C-Complex for L is a union of surfacesS = S1 . . . Sn in S3 with

1 ∂Si = L(i)

2 Si and Sj intersect in arcs with one endpoint on ∂Si and the other in∂Sj (a “clasp”)

3 Si ∩ Sj ∩ Sk = ∅

A Clasp:

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 15 / 1

Defining the Cimasoni-Florens signature: Links andC-Complexes

Definition (Cimasoni, 2004)

For an n colored Link L a C-Complex for L is a union of surfacesS = S1 . . . Sn in S3 with

1 ∂Si = L(i)

2 Si and Sj intersect in arcs with one endpoint on ∂Si and the other in∂Sj (a “clasp”)

3 Si ∩ Sj ∩ Sk = ∅

A Clasp:

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 15 / 1

Defining the Cimasoni-Florens signature: Links andC-Complexes

Definition (Cimasoni, 2004)

For an n colored Link L a C-Complex for L is a union of surfacesS = S1 . . . Sn in S3 with

1 ∂Si = L(i)

2 Si and Sj intersect in arcs with one endpoint on ∂Si and the other in∂Sj (a “clasp”)

3 Si ∩ Sj ∩ Sk = ∅

A Clasp:

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 15 / 1

Defining the Cimasoni-Florens signature: The linking form

Definition (Cimasoni-Florens)

A simple closed curve γ on a C-complex S is called a loop if whenever itintersects a clasp, it looks like:

For a loop γ in S1 . . . Sn and ε = 〈ε1, . . . , εn〉 ∈ {±1}n, γε is the curve inS3 − S gotten by pushing the part of γ in Si in the εi normal direction:

γ++: γ+−: γ−+: γ−−:

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 16 / 1

Defining the Cimasoni-Florens signature: The linking form

Definition (Cimasoni-Florens)

A simple closed curve γ on a C-complex S is called a loop if whenever itintersects a clasp, it looks like:

For a loop γ in S1 . . . Sn and ε = 〈ε1, . . . , εn〉 ∈ {±1}n, γε is the curve inS3 − S gotten by pushing the part of γ in Si in the εi normal direction:

γ++: γ+−: γ−+: γ−−:

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 16 / 1

Defining the Cimasoni-Florens signature: The linking form

Definition (Cimasoni-Florens)

A simple closed curve γ on a C-complex S is called a loop if whenever itintersects a clasp, it looks like:

For a loop γ in S1 . . . Sn and ε = 〈ε1, . . . , εn〉 ∈ {±1}n, γε is the curve inS3 − S gotten by pushing the part of γ in Si in the εi normal direction:

γ++: γ+−: γ−+: γ−−:

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 16 / 1

Defining the Cimasoni-Florens signature: The linking form

Definition (Cimasoni-Florens)

A simple closed curve γ on a C-complex S is called a loop if whenever itintersects a clasp, it looks like:

For a loop γ in S1 . . . Sn and ε = 〈ε1, . . . , εn〉 ∈ {±1}n, γε is the curve inS3 − S gotten by pushing the part of γ in Si in the εi normal direction:

γ++:

γ+−: γ−+: γ−−:

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 16 / 1

Defining the Cimasoni-Florens signature: The linking form

Definition (Cimasoni-Florens)

A simple closed curve γ on a C-complex S is called a loop if whenever itintersects a clasp, it looks like:

For a loop γ in S1 . . . Sn and ε = 〈ε1, . . . , εn〉 ∈ {±1}n, γε is the curve inS3 − S gotten by pushing the part of γ in Si in the εi normal direction:

γ++: γ+−:

γ−+: γ−−:

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 16 / 1

Defining the Cimasoni-Florens signature: The linking form

Definition (Cimasoni-Florens)

A simple closed curve γ on a C-complex S is called a loop if whenever itintersects a clasp, it looks like:

For a loop γ in S1 . . . Sn and ε = 〈ε1, . . . , εn〉 ∈ {±1}n, γε is the curve inS3 − S gotten by pushing the part of γ in Si in the εi normal direction:

γ++: γ+−: γ−+: γ−−:

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 16 / 1

Defining the Cimasoni-Florens signature: The linking formPick a basis for H1(S) given by loops γ1 . . . γk .

For each ε ∈ {±1}n, define

Aε :=(

lnk(γi , γεj ))

. For ω = (ω1, . . . , ωn) ∈ Tn∗ define

H(ω) :=∑ε

∏i

(1− ω−εii )Aε.

Tn∗ := {ω = (ω1, . . . , ωn) : for all i , |ωi | = 1 but , ωi 6= 1}

If ωi = 1 for some i , then H(ω) = 0

Definition (The Cimasoni-Florens signature)

For ω ∈ Tn∗ define The Cimasoni-Florens signature of L at ω by

σL(ω) := σ(H(ω)) =number of positive eigenvalues− number of negative eigenvalues

Theorem (Cimasoni-Florens)

σL(ω) depends only on ω and L.

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 17 / 1

Defining the Cimasoni-Florens signature: The linking formPick a basis for H1(S) given by loops γ1 . . . γk . For each ε ∈ {±1}n, define

Aε :=(

lnk(γi , γεj ))

.

For ω = (ω1, . . . , ωn) ∈ Tn∗ define

H(ω) :=∑ε

∏i

(1− ω−εii )Aε.

Tn∗ := {ω = (ω1, . . . , ωn) : for all i , |ωi | = 1 but , ωi 6= 1}

If ωi = 1 for some i , then H(ω) = 0

Definition (The Cimasoni-Florens signature)

For ω ∈ Tn∗ define The Cimasoni-Florens signature of L at ω by

σL(ω) := σ(H(ω)) =number of positive eigenvalues− number of negative eigenvalues

Theorem (Cimasoni-Florens)

σL(ω) depends only on ω and L.

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 17 / 1

Defining the Cimasoni-Florens signature: The linking formPick a basis for H1(S) given by loops γ1 . . . γk . For each ε ∈ {±1}n, define

Aε :=(

lnk(γi , γεj ))

. For ω = (ω1, . . . , ωn) ∈ Tn∗ define

H(ω) :=∑ε

∏i

(1− ω−εii )Aε.

Tn∗ := {ω = (ω1, . . . , ωn) : for all i , |ωi | = 1 but , ωi 6= 1}

If ωi = 1 for some i , then H(ω) = 0

Definition (The Cimasoni-Florens signature)

For ω ∈ Tn∗ define The Cimasoni-Florens signature of L at ω by

σL(ω) := σ(H(ω)) =number of positive eigenvalues− number of negative eigenvalues

Theorem (Cimasoni-Florens)

σL(ω) depends only on ω and L.

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 17 / 1

Defining the Cimasoni-Florens signature: The linking formPick a basis for H1(S) given by loops γ1 . . . γk . For each ε ∈ {±1}n, define

Aε :=(

lnk(γi , γεj ))

. For ω = (ω1, . . . , ωn) ∈ Tn∗ define

H(ω) :=∑ε

∏i

(1− ω−εii )Aε.

Tn∗ := {ω = (ω1, . . . , ωn) : for all i , |ωi | = 1 but , ωi 6= 1}

If ωi = 1 for some i , then H(ω) = 0

Definition (The Cimasoni-Florens signature)

For ω ∈ Tn∗ define The Cimasoni-Florens signature of L at ω by

σL(ω) := σ(H(ω)) =number of positive eigenvalues− number of negative eigenvalues

Theorem (Cimasoni-Florens)

σL(ω) depends only on ω and L.

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 17 / 1

Defining the Cimasoni-Florens signature: The linking formPick a basis for H1(S) given by loops γ1 . . . γk . For each ε ∈ {±1}n, define

Aε :=(

lnk(γi , γεj ))

. For ω = (ω1, . . . , ωn) ∈ Tn∗ define

H(ω) :=∑ε

∏i

(1− ω−εii )Aε.

Tn∗ := {ω = (ω1, . . . , ωn) : for all i , |ωi | = 1 but , ωi 6= 1}

If ωi = 1 for some i , then H(ω) = 0

Definition (The Cimasoni-Florens signature)

For ω ∈ Tn∗ define The Cimasoni-Florens signature of L at ω by

σL(ω) := σ(H(ω)) =number of positive eigenvalues− number of negative eigenvalues

Theorem (Cimasoni-Florens)

σL(ω) depends only on ω and L.

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 17 / 1

ρ0 via the Cimasoni-Florens signature function.

Theorem (D.)

For an n-component link, L = L1, . . . , Ln with zero pairwise linkingnumbers, ρ0(L) = 1

(2π)n

∫Tn∗

σL(ω)dω.

In order to make sense of the right

hand side, regard L as a colored link by taking L(k) = Lk .

Something better is true. All abelian ρ-invariants can be computed byintegrating the Cimasoni-Florens signature over a subtorus of Tn

∗.

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 18 / 1

ρ0 via the Cimasoni-Florens signature function.

Theorem (D.)

For an n-component link, L = L1, . . . , Ln with zero pairwise linkingnumbers, ρ0(L) = 1

(2π)n

∫Tn∗

σL(ω)dω. In order to make sense of the right

hand side, regard L as a colored link by taking L(k) = Lk .

Something better is true. All abelian ρ-invariants can be computed byintegrating the Cimasoni-Florens signature over a subtorus of Tn

∗.

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 18 / 1

ρ0 via the Cimasoni-Florens signature function.

Theorem (D.)

For an n-component link, L = L1, . . . , Ln with zero pairwise linkingnumbers, ρ0(L) = 1

(2π)n

∫Tn∗

σL(ω)dω. In order to make sense of the right

hand side, regard L as a colored link by taking L(k) = Lk .

Something better is true. All abelian ρ-invariants can be computed byintegrating the Cimasoni-Florens signature over a subtorus of Tn

∗.

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 18 / 1

Back to the candidate robust doubling operator:computing ρ0(L)

The link L:

Isotope

This is the link whose ρ0-invariant we compute.

k twists k twists

−k −k

−k

−k

−k

−k−k

−k

−k

−k

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 19 / 1

Back to the candidate robust doubling operator:computing ρ0(L)

The link L:

Isotope

This is the link whose ρ0-invariant we compute.

k twists k twists

−k −k

−k

−k

−k

−k−k

−k

−k

−k

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 19 / 1

Back to the candidate robust doubling operator:computing ρ0(L)

The link L:

Isotope

This is the link whose ρ0-invariant we compute.

k twists k twists−k −k

−k

−k

−k

−k−k

−k

−k

−k

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 19 / 1

Back to the candidate robust doubling operator:computing ρ0(L)

The link L:

Isotope

This is the link whose ρ0-invariant we compute.

k twists k twists−k −k

−k

−k

−k

−k

−k

−k

−k

−k

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 19 / 1

Back to the candidate robust doubling operator:computing ρ0(L)

The link L:

Isotope

This is the link whose ρ0-invariant we compute.

k twists k twists−k −k

−k

−k

−k

−k

−k

−k

−k

−k

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 19 / 1

Back to the candidate robust doubling operator:computing ρ0(L)

The link L:

Isotope

This is the link whose ρ0-invariant we compute.

k twists k twists−k −k

−k

−k

−k

−k−k

−k

−k

−k

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 19 / 1

Back to the candidate robust doubling operator:computing ρ0(L)

The link L:

Isotope

This is the link whose ρ0-invariant we compute.

k twists k twists−k −k

−k

−k

−k

−k−k

−k

−k

−k

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 19 / 1

Computing ρ0(L)

ρ0(L) =∫T2∗σL(ω)dω.

We use the following C-complex and loops toperform these computations.

α1 α2 . . .αk−1

β1 β2 . . .βk−1γS1

S2

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 20 / 1

Computing ρ0(L)

ρ0(L) =∫T2∗σL(ω)dω. We use the following C-complex and loops to

perform these computations.

α1 α2 . . .αk−1

β1 β2 . . .βk−1γS1

S2

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 20 / 1

doing the computation

Integrating the signature of the resulting linking matrix shows that∫T2∗σL(ω)dω < −1 when k ≥ 3. This means ρ1

{0}(R) < 0 and we concludethat

Theorem

For k ≥ 3, the doubling operator Rη is robust.

k twists

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 21 / 1

Thank you.

Christopher William Davis, Rice University Computing first order signatures and making a construction of Cochran-Harvey-Leidy explicit.March 10, 2012 22 / 1