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Localized metabelian rho invariants as obstructions totorsion in the knot concordance group.

Chris Davis, Rice University

July 22, 2010

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 1 / 37

Goals

1 Introduce some -invariants corresponding to localizations of theAlexander module of a knot.

2 Prove that they obstruct knots which are torsion in the algebraicconcordance group from being torsion in the concordance group.

3 Prove computable bounds on the value of these invariants.

4 Do a computation to prove the theorem about twist knots which is onthe next slide.

5 State a theorem regarding these invariants and iterated infection.

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 2 / 37

An application: Twist Knots

Theorem (D.)

No nontrivial linear combination of the twist knots,T7,T13,T21 . . .Tx2x1 (x 2), is topologically slice. These knotsare each of algebraic order 2.

This is the mirror image of what others refer to as the n twist knot.

n

Figure: Tn: the n-twist knot.

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 3 / 37

Previous related results

Theorem (A. Tamulis, 2000)

No nontrivial combination of the twist knots Tn with n 3 and 4n + 1prime is topologically slice. Each of these twist knots is algebraically oforder 2.

Theorem (C. Livingston, 2001)

Let pi be an enumeration of the primes congruent to 3 mod 4. Letni = p2i1p2i 1. No nontrivial linear combination of the knots Tni isslice. Each of these twist knots is algebraically of order 4.

Theorem (S. Kim, 2005)

No nontrivial linear combination of the twist knots, except for the 0 twistknot (unknot), the 1 twist knot (figure eight) and the 2 twist knot(stevedore) is ribbon.

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 4 / 37

Twist knots and infection

The computation I do to prove linear independence for these twist knotscan be used to prove the following theorem about infection:

Theorem

For a sufficiently nice slice knot, R, and a sufficiently nice infecting curve,, the set resulting from iterated infection of R along by each ofT7,T21, . . . ,Tx2x1 is linearly independent in the concordance group.

example of niceness: any slice knot with Alexander module of the formQ[t1]p(t)2

with p(t) prime and symmetric and a generator of the Alexander

module.Tristram-Levine signature of the deepest knot in the infection iseverywhere zero.

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 5 / 37

Twist knots and infection

The computation I do to prove linear independence for these twist knotscan be used to prove the following theorem about infection:

Theorem

For a sufficiently nice slice knot, R, and a sufficiently nice infecting curve,, the set resulting from iterated infection of R along by each ofT7,T21, . . . ,Tx2x1 is linearly independent in the concordance group.

example of niceness: any slice knot with Alexander module of the formQ[t1]p(t)2

with p(t) prime and symmetric and a generator of the Alexander

module.

Tristram-Levine signature of the deepest knot in the infection iseverywhere zero.

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 5 / 37

Twist knots and infection

The computation I do to prove linear independence for these twist knotscan be used to prove the following theorem about infection:

Theorem

For a sufficiently nice slice knot, R, and a sufficiently nice infecting curve,, the set resulting from iterated infection of R along by each ofT7,T21, . . . ,Tx2x1 is linearly independent in the concordance group.

example of niceness: any slice knot with Alexander module of the formQ[t1]p(t)2

with p(t) prime and symmetric and a generator of the Alexander

module.Tristram-Levine signature of the deepest knot in the infection iseverywhere zero.

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 5 / 37

Invariants of interest: 0

For L an n component link with zero linking numbers, M(L) denotes zero

surgery along L.

Let : 1(M(L))1(M(L))

1(M(L))(1)= Zn be the

Abelianization map.

Definition

0(L) := (M(L), )

For a knot it is the integral of the Tristram-Levine Signature.(Cochran-Orr-Teichner 04)

For links it is computable: In a nice enough case a computer can betaught how to do it. (More on this later)

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 6 / 37

Invariants of interest: 0

For L an n component link with zero linking numbers, M(L) denotes zero

surgery along L. Let : 1(M(L))1(M(L))

1(M(L))(1)= Zn be the

Abelianization map.

Definition

0(L) := (M(L), )

For a knot it is the integral of the Tristram-Levine Signature.(Cochran-Orr-Teichner 04)

For links it is computable: In a nice enough case a computer can betaught how to do it. (More on this later)

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 6 / 37

Invariants of interest: 0

For L an n component link with zero linking numbers, M(L) denotes zero

surgery along L. Let : 1(M(L))1(M(L))

1(M(L))(1)= Zn be the

Abelianization map.

Definition

0(L) := (M(L), )

For a knot it is the integral of the Tristram-Levine Signature.(Cochran-Orr-Teichner 04)

For links it is computable: In a nice enough case a computer can betaught how to do it. (More on this later)

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 6 / 37

Invariants of interest: 0

For L an n component link with zero linking numbers, M(L) denotes zero

surgery along L. Let : 1(M(L))1(M(L))

1(M(L))(1)= Zn be the

Abelianization map.

Definition

0(L) := (M(L), )

For a knot it is the integral of the Tristram-Levine Signature.(Cochran-Orr-Teichner 04)

Invariants of interest: 0

For L an n component link with zero linking numbers, M(L) denotes zero

surgery along L. Let : 1(M(L))1(M(L))

1(M(L))(1)= Zn be the

Abelianization map.

Definition

0(L) := (M(L), )

For a knot it is the integral of the Tristram-Levine Signature.(Cochran-Orr-Teichner 04)

Invariants of interest: 1

For K a knot, let 1 : 1(M(K ))1(M(K ))

1(M(K ))(2)be the quotient map.

Definition

1(K ) := (M(K ), 1

)Considered by Cochran-Harvey-Leidy 08 together with quotientscorresponding to isotropy of the Blanchfield form as a sliceobstruction.

Hard to compute. There is no known procedure.

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 7 / 37

Invariants of interest: 1

For K a knot, let 1 : 1(M(K ))1(M(K ))

1(M(K ))(2)be the quotient map.

Definition

1(K ) := (M(K ), 1

)

Considered by Cochran-Harvey-Leidy 08 together with quotientscorresponding to isotropy of the Blanchfield form as a sliceobstruction.

Hard to compute. There is no known procedure.

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 7 / 37

Invariants of interest: 1

For K a knot, let 1 : 1(M(K ))1(M(K ))

1(M(K ))(2)be the quotient map.

Definition

1(K ) := (M(K ), 1

)Considered by Cochran-Harvey-Leidy 08 together with quotientscorresponding to isotropy of the Blanchfield form as a sliceobstruction.

Hard to compute. There is no known procedure.

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 7 / 37

Invariants of interest: 1

For K a knot, let 1 : 1(M(K ))1(M(K ))

1(M(K ))(2)be the quotient map.

Definition

1(K ) := (M(K ), 1

)Considered by Cochran-Harvey-Leidy 08 together with quotientscorresponding to isotropy of the Blanchfield form as a sliceobstruction.

Hard to compute. There is no known procedure.

Invariants of interest: 1p

Let p(t) be a polynomial.

Rp :=

{f

g Q(t) : (g , p) = 1

}is the localization of Q[t1] at p.

Let Ap0(K ) = A0(K ) Rp be the localized Alexander module of K .Let 1(M(K ))

(2)p(t) be the kernel of the composition

1(M(K ))(1) 1(M(K ))

(1)

1(M(K ))(2) A0(K )

Id1Ap0(K ).

Let 1p : 1(M(K ))1(M(K ))

1(M(K ))(2)p

be the quotient map.

Definition

1p(t)(K ) := (M(K ), 1p

)

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 8 / 37

Invariants of interest: 1p

Let p(t) be a polynomial.

Rp :=

{f

g Q(t) : (g , p) = 1

}is the localization of Q[t1] at p.

Let Ap0(K ) = A0(K ) Rp be the localized Alexander module of K .Let 1(M(K ))

(2)p(t) be the kernel of the composition

1(M(K ))(1) 1(M(K ))

(1)

1(M(K ))(2) A0(K )

Id1Ap0(K ).

Let 1p : 1(M(K ))1(M(K ))

1(M(K ))(2)p

be the quotient map.

Definition

1p(t)(K ) := (M(K ), 1p

)

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 8 / 37

Invariants of interest: 1p

Let p(t) be a polynomial.

Rp :=

{f

g Q(t) : (g , p) = 1

}is the localization of Q[t1] at p.

Let Ap0(K ) = A0(K ) Rp be the localized Alexander module of K .Let 1(M(K ))

(2)p(t) be the kernel of the composition

1(M(K ))(1) 1(M(K ))

(1)

1(M(K ))(2) A0(K )

Id1Ap0(K ).

Let 1p : 1(M(K ))1(M(K ))

1(M(K ))(2)p

be the quotient map.

Definition

1p(t)(K ) := (M(K ), 1p

)

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 8 / 37

Invariants of interest: 1p

Let p(t) be a polynomial.

Rp :=

{f

g Q(t) : (g , p) = 1

}is the localization of Q[t1] at p.

Let Ap0(K ) = A0(K ) Rp be the localized Alexander module of K .

Let 1(M(K ))(2)p(t) be the kernel of the composition

1(M(K ))(1) 1(M(K ))

(1)

1(M(K ))(2) A0(K )

Id1Ap0(K ).

Let 1p : 1(M(K ))1(M(K ))

1(M(K ))(2)p

be the quotient map.

Definition

1p(t)(K ) := (M(K ), 1p

)

Invariants of interest: 1p

Let p(t) be a polynomial.

Rp :=

{f

g Q(t) : (g , p) = 1

}is the localization of Q[t1] at p.

Let Ap0(K ) = A0(K ) Rp be the localized Alexander module of K .Let 1(M(K ))

(2)p(t) be the kernel of the composition

1(M(K ))(1) 1(M(K ))

(1)

1(M(K ))(2) A0(K )

Id1Ap0(K ).

Let 1p : 1(M(K ))1(M(K ))

1(M(K ))(2)p

be the quotient map.

Definition

1p(t)(K ) := (M(K ), 1p

)

Invariants of interest: 1p

Let p(t) be a polynomial.

Rp :=

{f

g Q(t) : (g , p) = 1

}is the localization of Q[t1] at p.

Let Ap0(K ) = A0(K ) Rp be the localized Alexander module of K .Let 1(M(K ))

(2)p(t) be the kernel of the composition

1(M(K ))(1) 1(M(K ))

(1)

1(M(K ))(2) A0(K )

Id1Ap0(K ).

Let 1p : 1(M(K ))1(M(K ))

1(M(K ))(2)p

be the quotient map.

Definition

1p(t)(K ) := (M(K ), 1p

)

Invariants of interest: 1p

Let p(t) be a polynomial.

Rp :=

{f

g Q(t) : (g , p) = 1

}is the localization of Q[t1] at p.

Let Ap0(K ) = A0(K ) Rp be the localized Alexander module of K .Let 1(M(K ))

(2)p(t) be the kernel of the composition

1(M(K ))(1) 1(M(K ))

(1)

1(M(K ))(2) A0(K )

Id1Ap0(K ).

Let 1p : 1(M(K ))1(M(K ))

1(M(K ))(2)p

be the quotient map.

Definition

1p(t)(K ) := (M(K ), 1p

)Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 8 / 37

A pair of examples. 1: localize away from the Alexanderpolynomial

Let be the Alexander polynomial of K .

Suppose (p,) = 1. Then1

Rp. annihilates A0, so A0 Rp = Ap0

vanishes.1(M(K ))

(2)p(t) is the kernel of the composition

1(M(K ))(1) 1(M(K ))

(1)

1(M(K ))(2) A0(K )Ap0(K ) = 0.

1(M(K ))(2)p(t) = 1(M(K ))

(1), so 1p(K ) = 0(K )

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 9 / 37

A pair of examples. 2: localize at the Alexanderpolynomial

Suppose p = . Then A0(K ) Id1

A0(K ) Rp is injective.

1(M(K ))(2)p(t) is the kernel of the composition

1(M(K ))(1) 1(M(K ))

(1)

1(M(K ))(2) A0(K )

Id1Ap0(K ).

1(M(K ))(2)p(t) = 1(M(K ))

(2), so 1p(K ) = 1(K )

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 10 / 37

Interactions between 1p and the Alexander polynomial

Proposition

Let be the Alexander polynomial of a knot, K.

For any p which is relatively prime to , 1p(K ) = 0(K ).

1(K ) = 1(K ).

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 11 / 37

1p under infection and connected sums

Proposition (D.)

1p(J#K ) = 1p(J) +

1p(K ).

Proposition (D.)

For an unknot representing an element of the localized Alexander

module of J, 1p(J(K )) =

{1p(J) if = 0 in A

p0(J)

1p(J) + 0(K ) if 6= 0 in Ap0(J)

From the second proposition I get the following discouraging example:

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 12 / 37

1p under infection and connected sums

Proposition (D.)

1p(J#K ) = 1p(J) +

1p(K ).

Proposition (D.)

For an unknot representing an element of the localized Alexander

module of J, 1p(J(K )) =

{1p(J) if = 0 in A

p0(J)

1p(J) + 0(K ) if 6= 0 in Ap0(J)

From the second proposition I get the following discouraging example:

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 12 / 37

A pair of slice knots with differing 1 invariants

Consider the pair of slice knots below. 1(R(A)) = 1(R) + 0(A). In

particular 1(R) and 1(R(A)) cannot both be zero. 1 is not well

defined on the concordance group.

Despite this fact, I will provide in the next few slides a setting in whichthese invariants provide useful concordance information.

A

R R(A)

Figure: A pair of slice knots with differing 1.

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 13 / 37

A pair of slice knots with differing 1 invariants

Consider the pair of slice knots below. 1(R(A)) = 1(R) + 0(A). In

particular 1(R) and 1(R(A)) cannot both be zero. 1 is not well

defined on the concordance group.Despite this fact, I will provide in the next few slides a setting in whichthese invariants provide useful concordance information.

A

R R(A)

Figure: A pair of slice knots with differing 1.

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 13 / 37

Blanchfield forms and isotropy

Q[t1], and thus Rp when p is symmetric, have an involution given byq(t) = q(t1)

With respect to this involution, the localized Alexander module has ahermitian form

Blp : Ap0(K ) Ap0(K )

Q(t)Rp

.

given by extending the Blanchfield form on A0(K )

Definition

P Ap0(K ) is called isotropic if Blp(a, b) = 0 for all a, b P

Definition

K is called p-anisotropic if there is no nontrivial isotropic submodule ofAp0(K ).

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 14 / 37

Blanchfield forms and isotropy

Q[t1], and thus Rp when p is symmetric, have an involution given byq(t) = q(t1)With respect to this involution, the localized Alexander module has ahermitian form

Blp : Ap0(K ) Ap0(K )

Q(t)Rp

.

given by extending the Blanchfield form on A0(K )

Definition

P Ap0(K ) is called isotropic if Blp(a, b) = 0 for all a, b P

Definition

K is called p-anisotropic if there is no nontrivial isotropic submodule ofAp0(K ).

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 14 / 37

Blanchfield forms and isotropy

Q[t1], and thus Rp when p is symmetric, have an involution given byq(t) = q(t1)With respect to this involution, the localized Alexander module has ahermitian form

Blp : Ap0(K ) Ap0(K )

Q(t)Rp

.

given by extending the Blanchfield form on A0(K )

Definition

P Ap0(K ) is called isotropic if Blp(a, b) = 0 for all a, b P

Definition

K is called p-anisotropic if there is no nontrivial isotropic submodule ofAp0(K ).

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 14 / 37

Blanchfield forms and isotropy

Q[t1], and thus Rp when p is symmetric, have an involution given byq(t) = q(t1)With respect to this involution, the localized Alexander module has ahermitian form

Blp : Ap0(K ) Ap0(K )

Q(t)Rp

.

given by extending the Blanchfield form on A0(K )

Definition

P Ap0(K ) is called isotropic if Blp(a, b) = 0 for all a, b P

Definition

K is called p-anisotropic if there is no nontrivial isotropic submodule ofAp0(K ).

Exploration of anisotropy.

If p is a symmetric prime which divides divides with multiplicity 1, then

Ap0(K ) is cyclic: Ap0(K ) =

Rp(p)

. Let generate Ap0(K ).

The Blanchfield form is given by Blp(q, r) =qra

pwith (a, p) = 1. If

Blp(q, r) = 0 then p divides q or r . By symmetry of p, p|q if and onlyif p|qThus Blp(a, b) = 0 if and only in a or b is zero in Ap0(K ). In particularthere are no nontrivial isotropic submodules.This proves the following in the case that p is prime.

Proposition

If the Alexander polynomial of a knot is squarefree, then the knot isp-anisotropic for every symmetric polynomial, p.

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 15 / 37

Exploration of anisotropy.

If p is a symmetric prime which divides divides with multiplicity 1, then

Ap0(K ) is cyclic: Ap0(K ) =

Rp(p)

. Let generate Ap0(K ).

The Blanchfield form is given by Blp(q, r) =qra

pwith (a, p) = 1. If

Blp(q, r) = 0 then p divides q or r . By symmetry of p, p|q if and onlyif p|q

Thus Blp(a, b) = 0 if and only in a or b is zero in Ap0(K ). In particularthere are no nontrivial isotropic submodules.This proves the following in the case that p is prime.

Proposition

If the Alexander polynomial of a knot is squarefree, then the knot isp-anisotropic for every symmetric polynomial, p.

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 15 / 37

Exploration of anisotropy.

If p is a symmetric prime which divides divides with multiplicity 1, then

Ap0(K ) is cyclic: Ap0(K ) =

Rp(p)

. Let generate Ap0(K ).

The Blanchfield form is given by Blp(q, r) =qra

pwith (a, p) = 1. If

Blp(q, r) = 0 then p divides q or r . By symmetry of p, p|q if and onlyif p|qThus Blp(a, b) = 0 if and only in a or b is zero in Ap0(K ). In particularthere are no nontrivial isotropic submodules.

This proves the following in the case that p is prime.

Proposition

If the Alexander polynomial of a knot is squarefree, then the knot isp-anisotropic for every symmetric polynomial, p.

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 15 / 37

Exploration of anisotropy.

If p is a symmetric prime which divides divides with multiplicity 1, then

Ap0(K ) is cyclic: Ap0(K ) =

Rp(p)

. Let generate Ap0(K ).

The Blanchfield form is given by Blp(q, r) =qra

pwith (a, p) = 1. If

Blp(q, r) = 0 then p divides q or r . By symmetry of p, p|q if and onlyif p|qThus Blp(a, b) = 0 if and only in a or b is zero in Ap0(K ). In particularthere are no nontrivial isotropic submodules.This proves the following in the case that p is prime.

Proposition

First big theorem: 1p as an obstruction to linear

dependence in CTheorem (D.)

Given a symmetric polynomial p(t) and (not necessarily distinct)p-anisotropic knots K1, . . .Kn, if K1# . . .#Kn is slice, thenn

i=1

1p(t)(Ki ) = 0

Corollary (D.)

If K is of finite concordance order and is p-anisotropic, then 1p(K ) = 0

Corollary (D.)

If K1, . . . ,Kn are of finite algebraic order and have coprime, squarefreeAlexander polynomials, then if 1(Ki ) is nonzero for each i , {K1, . . . ,Kn}is linearly independent in C.

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 16 / 37

First big theorem: 1p as an obstruction to linear

dependence in CTheorem (D.)

Given a symmetric polynomial p(t) and (not necessarily distinct)p-anisotropic knots K1, . . .Kn, if K1# . . .#Kn is slice, thenn

i=1

1p(t)(Ki ) = 0

Corollary (D.)

If K is of finite concordance order and is p-anisotropic, then 1p(K ) = 0

Corollary (D.)

If K1, . . . ,Kn are of finite algebraic order and have coprime, squarefreeAlexander polynomials, then if 1(Ki ) is nonzero for each i , {K1, . . . ,Kn}is linearly independent in C.

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 16 / 37

First big theorem: 1p as an obstruction to linear

dependence in CTheorem (D.)

Given a symmetric polynomial p(t) and (not necessarily distinct)p-anisotropic knots K1, . . .Kn, if K1# . . .#Kn is slice, thenn

i=1

1p(t)(Ki ) = 0

Corollary (D.)

If K is of finite concordance order and is p-anisotropic, then 1p(K ) = 0

Corollary (D.)

If K1, . . . ,Kn are of finite algebraic order and have coprime, squarefreeAlexander polynomials, then if 1(Ki ) is nonzero for each i , {K1, . . . ,Kn}is linearly independent in C.

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 16 / 37

Proof of second corollary

Let pj the the Alexander polynomial of Kj .

If #aiKi is slice, then

0 =n

i=1

ai1pj

(Ki ) for each j . Since these knots have coprime Alexander

polynomials, 1pj (Ki ) = 0(Ki ) if j 6= i and 1pj (Kj) =

1(Kj) 6= 0. Sinceeach of the knots are of finite algebraic order, 0(Ki ) = 0. Thus,0 = aj

1(Kj). Since 1(Kj) 6= 0, aj = 0.

No nontrivial combinations of the knots Ki is slice and the set is linearlyindependent

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 17 / 37

Proof of second corollary

Let pj the the Alexander polynomial of Kj . If #aiKi is slice, then

0 =n

i=1

ai1pj

(Ki ) for each j .

Since these knots have coprime Alexander

polynomials, 1pj (Ki ) = 0(Ki ) if j 6= i and 1pj (Kj) =

1(Kj) 6= 0. Sinceeach of the knots are of finite algebraic order, 0(Ki ) = 0. Thus,0 = aj

1(Kj). Since 1(Kj) 6= 0, aj = 0.

No nontrivial combinations of the knots Ki is slice and the set is linearlyindependent

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 17 / 37

Proof of second corollary

Let pj the the Alexander polynomial of Kj . If #aiKi is slice, then

0 =n

i=1

ai1pj

(Ki ) for each j . Since these knots have coprime Alexander

polynomials, 1pj (Ki ) = 0(Ki ) if j 6= i and 1pj (Kj) =

1(Kj) 6= 0.

Since

each of the knots are of finite algebraic order, 0(Ki ) = 0. Thus,0 = aj

1(Kj). Since 1(Kj) 6= 0, aj = 0.

No nontrivial combinations of the knots Ki is slice and the set is linearlyindependent

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 17 / 37

Proof of second corollary

Let pj the the Alexander polynomial of Kj . If #aiKi is slice, then

0 =n

i=1

ai1pj

(Ki ) for each j . Since these knots have coprime Alexander

polynomials, 1pj (Ki ) = 0(Ki ) if j 6= i and 1pj (Kj) =

1(Kj) 6= 0. Sinceeach of the knots are of finite algebraic order, 0(Ki ) = 0.

Thus,0 = aj

1(Kj). Since 1(Kj) 6= 0, aj = 0.

No nontrivial combinations of the knots Ki is slice and the set is linearlyindependent

Proof of second corollary

Let pj the the Alexander polynomial of Kj . If #aiKi is slice, then

0 =n

i=1

ai1pj

(Ki ) for each j . Since these knots have coprime Alexander

polynomials, 1pj (Ki ) = 0(Ki ) if j 6= i and 1pj (Kj) =

1(Kj) 6= 0. Sinceeach of the knots are of finite algebraic order, 0(Ki ) = 0. Thus,0 = aj

1(Kj). Since 1(Kj) 6= 0, aj = 0.

No nontrivial combinations of the knots Ki is slice and the set is linearlyindependent

Proof of second corollary

Let pj the the Alexander polynomial of Kj . If #aiKi is slice, then

0 =n

i=1

ai1pj

(Ki ) for each j . Since these knots have coprime Alexander

polynomials, 1pj (Ki ) = 0(Ki ) if j 6= i and 1pj (Kj) =

1(Kj) 6= 0. Sinceeach of the knots are of finite algebraic order, 0(Ki ) = 0. Thus,0 = aj

1(Kj). Since 1(Kj) 6= 0, aj = 0.

No nontrivial combinations of the knots Ki is slice and the set is linearlyindependent

Comparison of second corollary to polynomial splittingtheorems

The second corollary can be compared to the polynomial splittingtheorems for Casson-Gordon invariants (S. Kim, 2003) and for -invariantscorresponding to isotropy in the Alexander module (S. Kim-T. Kim, 2007).The idea of the comparison: Each theorem says that if some knots withcoprime Alexander polynomials have non-vanishing obstructions then theirconnected sum has non-vanishing obstructions.

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 18 / 37

Example: a linearly independent family of knots in C viainfection.

Consider a family of knots of finitetopological order, J1, J2, . . . withsquare-free, coprime Alexanderpolynomials.

By first corollary, 1(Jn) = 0

Let Kn be given from Jn by infection alonga curve which is nonzero in A0(Jn) using aknot A with 0(A) 6= 0.1(Kn) =

1(Jn) + 0(A) = 0(A) 6= 0

by the second corollary, {K1,K2, . . . } islinearly independent.

n n

A

Jn

Kn

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 19 / 37

Example: a linearly independent family of knots in C viainfection.

Consider a family of knots of finitetopological order, J1, J2, . . . withsquare-free, coprime Alexanderpolynomials.

By first corollary, 1(Jn) = 0

Let Kn be given from Jn by infection alonga curve which is nonzero in A0(Jn) using aknot A with 0(A) 6= 0.1(Kn) =

1(Jn) + 0(A) = 0(A) 6= 0

by the second corollary, {K1,K2, . . . } islinearly independent.

n n

A

Jn

Kn

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 19 / 37

Example: a linearly independent family of knots in C viainfection.

Consider a family of knots of finitetopological order, J1, J2, . . . withsquare-free, coprime Alexanderpolynomials.

By first corollary, 1(Jn) = 0

Let Kn be given from Jn by infection alonga curve which is nonzero in A0(Jn) using aknot A with 0(A) 6= 0.

1(Kn) = 1(Jn) +

0(A) = 0(A) 6= 0by the second corollary, {K1,K2, . . . } islinearly independent.

n n

A

Jn

Kn

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 19 / 37

Example: a linearly independent family of knots in C viainfection.

By first corollary, 1(Jn) = 0

Let Kn be given from Jn by infection alonga curve which is nonzero in A0(Jn) using aknot A with 0(A) 6= 0.1(Kn) =

1(Jn) + 0(A) = 0(A) 6= 0

by the second corollary, {K1,K2, . . . } islinearly independent.

n n

A

Jn

Kn

Example: a linearly independent family of knots in C viainfection.

By first corollary, 1(Jn) = 0

1(Jn) + 0(A) = 0(A) 6= 0

by the second corollary, {K1,K2, . . . } islinearly independent.

n n

A

Jn

Kn

Proof of first big theorem

I start by constructing a 4-manifold whose signature defect is the sum of-invariants in question. Take M(K1) [0, 1] t . . .M(Kn) [0, 1] andconnect it by gluing together neighborhoods of meridians. Glue a slice diskcompliment for K1# . . .#Kn to the M(K1# . . .#Kn)-component of theboundary.It is the anisotropic assumption that lets us cap the cobordism safely.

M(K1) M(K2) M(K3)

M(K1#K2#K3)

W0

W V coral reef upside-down

= jellyfish

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 20 / 37

Proof of first big theorem

I start by constructing a 4-manifold whose signature defect is the sum of-invariants in question.

Take M(K1) [0, 1] t . . .M(Kn) [0, 1] andconnect it by gluing together neighborhoods of meridians. Glue a slice diskcompliment for K1# . . .#Kn to the M(K1# . . .#Kn)-component of theboundary.It is the anisotropic assumption that lets us cap the cobordism safely.

M(K1) M(K2) M(K3)

M(K1#K2#K3)

W0

W V coral reef upside-down

= jellyfish

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 20 / 37

Proof of first big theorem

I start by constructing a 4-manifold whose signature defect is the sum of-invariants in question. Take M(K1) [0, 1] t . . .M(Kn) [0, 1]

andconnect it by gluing together neighborhoods of meridians. Glue a slice diskcompliment for K1# . . .#Kn to the M(K1# . . .#Kn)-component of theboundary.It is the anisotropic assumption that lets us cap the cobordism safely.

M(K1) M(K2) M(K3)

M(K1#K2#K3)

W0

W V coral reef upside-down

= jellyfish

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 20 / 37

Proof of first big theorem

I start by constructing a 4-manifold whose signature defect is the sum of-invariants in question. Take M(K1) [0, 1] t . . .M(Kn) [0, 1] andconnect it by gluing together neighborhoods of meridians.

Glue a slice diskcompliment for K1# . . .#Kn to the M(K1# . . .#Kn)-component of theboundary.It is the anisotropic assumption that lets us cap the cobordism safely.

M(K1) M(K2) M(K3)

M(K1#K2#K3)

W0

W V coral reef upside-down

= jellyfish

Proof of first big theorem

I start by constructing a 4-manifold whose signature defect is the sum of-invariants in question. Take M(K1) [0, 1] t . . .M(Kn) [0, 1] andconnect it by gluing together neighborhoods of meridians. Glue a slice diskcompliment for K1# . . .#Kn to the M(K1# . . .#Kn)-component of theboundary.

It is the anisotropic assumption that lets us cap the cobordism safely.

M(K1) M(K2) M(K3)

M(K1#K2#K3)

W0

W V

coral reef upside-down

= jellyfish

Proof of first big theorem

I start by constructing a 4-manifold whose signature defect is the sum of-invariants in question. Take M(K1) [0, 1] t . . .M(Kn) [0, 1] andconnect it by gluing together neighborhoods of meridians. Glue a slice diskcompliment for K1# . . .#Kn to the M(K1# . . .#Kn)-component of theboundary.It is the anisotropic assumption that lets us cap the cobordism safely.

M(K1) M(K2) M(K3)

M(K1#K2#K3)

W0

W V

coral reef upside-down

= jellyfish

Proof of first big theorem

I start by constructing a 4-manifold whose signature defect is the sum of-invariants in question. Take M(K1) [0, 1] t . . .M(Kn) [0, 1] andconnect it by gluing together neighborhoods of meridians. Glue a slice diskcompliment for K1# . . .#Kn to the M(K1# . . .#Kn)-component of theboundary.It is the anisotropic assumption that lets us cap the cobordism safely.

M(K1) M(K2) M(K3)

M(K1#K2#K3)

W0

W V coral reef upside-down

= jellyfish

Proof of first big theorem

M(K1) M(K2) M(K3)

M(K1#K2#K3)

W0

W V coral reef upside-down = jellyfish

Proof of first big theorem

W0W

V

A coefficient system on W must be found.

1(W ) ?

1(M(Ki ))1(M(Ki ))

1(M(Ki ))(2)p

//

OO

//

?

OO

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 21 / 37

Proof of first big theorem

W0W

V

Every inclusion on first homology is anisomorphism. (So Ap0(W ) = H1(W ; Rp) makessense.)

1(V ) H1(V ) = Z

1(M(#Ki )) H1(M(#K1)) = Z

1(W0) H1(W0) = Z

1(M(Ki )) H1(Mi ) = Z

//

OO

//

=

OO =

//

OO

//

OO =

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 21 / 37

Proof of first big theorem

W0W

V

Claim: ker(Ap0(Ki ) Ap0(W )) is isotropic

and thus, zero.

P = ker(Ap0(#Ki ) A

p0(V )

)is isotropic.

Ap0(V )Ap0(Kj)

P

Ap0(#Ki ) Ap0(Kj)

Ap0(W0) Ap0(Kj)

Ap0(Ki )

//=

=

OO

//=

OO

Id

Id

//=

OO ::ttttttt [0,...,0,1,0... ]

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 21 / 37

Proof of first big theorem

W0W

V

Claim: ker(Ap0(Ki ) Ap0(W )) is isotropic

and thus, zero.P = ker

(Ap0(#Ki ) A

p0(V )

)is isotropic.

Ap0(V )Ap0(Kj)

P

Ap0(#Ki ) Ap0(Kj)

Ap0(W0) Ap0(Kj)

Ap0(Ki )

//=

=

OO

//=

OO

Id

Id

//=

OO ::ttttttt [0,...,0,1,0... ]

Proof of first big theorem

W0W

V

Claim: ker(Ap0(Ki ) Ap0(W )) is isotropic

and thus, zero.P = ker

(Ap0(#Ki ) A

p0(V )

)is isotropic.

Ap0(V )Ap0(Kj)

P

Ap0(#Ki ) Ap0(Kj)

Ap0(W0) Ap0(Kj)

Ap0(Ki )

//=

=

OO

//=

OO

Id

Id

//=

OO ::ttttttt [0,...,0,1,0... ]

Proof of first big theorem

W0W

V

1(K )

1(K )(2)p

= H1(M(Ki )) n Ap0(Ki )Z

1(W )1(W )

1(W )(2)p

1(M(Ki ))1(M(Ki ))

1(M(Ki ))(2)p

//

OO

//

?

OO

Proof of first big theorem

W0W

V Thus,1p(Ki ) =

2(W , 1p) (W ) = 0

Twist knotsNeed to compute 1(Tx2x1).

This is hard. The infection trick wont work. Instead try to compute21(Tn) =

1(Tn#Tn).This is algebraically slice, and has a metabolizing link. The followingtheorem will allow us to get a handle on 1

n +1

x red strands x + 1 blue strands

n +1 n +1

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 22 / 37

Twist knotsNeed to compute 1(Tx2x1).This is hard. The infection trick wont work.

Instead try to compute21(Tn) =

1(Tn#Tn).This is algebraically slice, and has a metabolizing link. The followingtheorem will allow us to get a handle on 1

n +1

x red strands x + 1 blue strands

n +1 n +1

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 22 / 37

Twist knotsNeed to compute 1(Tx2x1).This is hard. The infection trick wont work. Instead try to compute21(Tn) =

1(Tn#Tn).

This is algebraically slice, and has a metabolizing link. The followingtheorem will allow us to get a handle on 1

n +1

x red strands x + 1 blue strands

n +1 n +1

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 22 / 37

Twist knotsNeed to compute 1(Tx2x1).This is hard. The infection trick wont work. Instead try to compute21(Tn) =

1(Tn#Tn).This is algebraically slice, and has a metabolizing link. The followingtheorem will allow us to get a handle on 1

n +1

x red strands x + 1 blue strands

n +1 n +1

Second big theorem: computing 1.

Theorem (D.)

Let K1, . . . ,Kn be (not necessarily distinct) p-anisotropic knots . Supposen#i=1

Ki is algebraically slice.

Let F be a genus g seifert surface for #Ki . Let

L be a link of g curves on F representing a metabolizer for the Seifertform, such that the meridians about the bands on which the componentsof L sit form a Z-linearly independent set in Ap0(#Ki )/P, where P is thesubmodule of Ap0(#Ki ) generated by L. Let (L) be the rank of theAlexander module of L.Then (1p(Ki )) 0(L) g 1 (L).1p (a nonabelian -invariant) is approximated by

0 (an abelian-invariant). The latter is more reasonable to expect to compute.

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 23 / 37

Second big theorem: computing 1.

Theorem (D.)

Let K1, . . . ,Kn be (not necessarily distinct) p-anisotropic knots . Supposen#i=1

Ki is algebraically slice. Let F be a genus g seifert surface for #Ki .

Let

L be a link of g curves on F representing a metabolizer for the Seifertform, such that the meridians about the bands on which the componentsof L sit form a Z-linearly independent set in Ap0(#Ki )/P, where P is thesubmodule of Ap0(#Ki ) generated by L. Let (L) be the rank of theAlexander module of L.Then (1p(Ki )) 0(L) g 1 (L).1p (a nonabelian -invariant) is approximated by

0 (an abelian-invariant). The latter is more reasonable to expect to compute.

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 23 / 37

Second big theorem: computing 1.

Theorem (D.)

Let K1, . . . ,Kn be (not necessarily distinct) p-anisotropic knots . Supposen#i=1

Ki is algebraically slice. Let F be a genus g seifert surface for #Ki . Let

L be a link of g curves on F representing a metabolizer for the Seifertform,

such that the meridians about the bands on which the componentsof L sit form a Z-linearly independent set in Ap0(#Ki )/P, where P is thesubmodule of Ap0(#Ki ) generated by L. Let (L) be the rank of theAlexander module of L.Then (1p(Ki )) 0(L) g 1 (L).1p (a nonabelian -invariant) is approximated by

0 (an abelian-invariant). The latter is more reasonable to expect to compute.

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 23 / 37

Second big theorem: computing 1.

Theorem (D.)

Let K1, . . . ,Kn be (not necessarily distinct) p-anisotropic knots . Supposen#i=1

Ki is algebraically slice. Let F be a genus g seifert surface for #Ki . Let

L be a link of g curves on F representing a metabolizer for the Seifertform, such that the meridians about the bands on which the componentsof L sit form a Z-linearly independent set in Ap0(#Ki )/P, where P is thesubmodule of Ap0(#Ki ) generated by L.

Let (L) be the rank of theAlexander module of L.Then (1p(Ki )) 0(L) g 1 (L).1p (a nonabelian -invariant) is approximated by

0 (an abelian-invariant). The latter is more reasonable to expect to compute.

Second big theorem: computing 1.

Theorem (D.)

Let K1, . . . ,Kn be (not necessarily distinct) p-anisotropic knots . Supposen#i=1

Ki is algebraically slice. Let F be a genus g seifert surface for #Ki . Let

L be a link of g curves on F representing a metabolizer for the Seifertform, such that the meridians about the bands on which the componentsof L sit form a Z-linearly independent set in Ap0(#Ki )/P, where P is thesubmodule of Ap0(#Ki ) generated by L. Let (L) be the rank of theAlexander module of L.

Then (1p(Ki )) 0(L) g 1 (L).1p (a nonabelian -invariant) is approximated by

0 (an abelian-invariant). The latter is more reasonable to expect to compute.

Second big theorem: computing 1.

Theorem (D.)

Let K1, . . . ,Kn be (not necessarily distinct) p-anisotropic knots . Supposen#i=1

Ki is algebraically slice. Let F be a genus g seifert surface for #Ki . Let

L be a link of g curves on F representing a metabolizer for the Seifertform, such that the meridians about the bands on which the componentsof L sit form a Z-linearly independent set in Ap0(#Ki )/P, where P is thesubmodule of Ap0(#Ki ) generated by L. Let (L) be the rank of theAlexander module of L.Then (1p(Ki )) 0(L) g 1 (L).

1p (a nonabelian -invariant) is approximated by 0 (an abelian

-invariant). The latter is more reasonable to expect to compute.

Second big theorem: computing 1.

Theorem (D.)

Let K1, . . . ,Kn be (not necessarily distinct) p-anisotropic knots . Supposen#i=1

Ki is algebraically slice. Let F be a genus g seifert surface for #Ki . Let

L be a link of g curves on F representing a metabolizer for the Seifertform, such that the meridians about the bands on which the componentsof L sit form a Z-linearly independent set in Ap0(#Ki )/P, where P is thesubmodule of Ap0(#Ki ) generated by L. Let (L) be the rank of theAlexander module of L.Then (1p(Ki )) 0(L) g 1 (L).1p (a nonabelian -invariant) is approximated by

0 (an abelian-invariant). The latter is more reasonable to expect to compute.

More examples: Infection by a string link.Let L be a 2 component link with zero linking numbers. Jn(L)#Jn isalgebraically slice.

It has L as a metabolizer.1(Jn(L)) + 1(Jn) 0(L) 2 1 (L) 1. Thus, if 0(L) > 1,then 1(Jn(L)) +

1(Jn) 6= 0. As in the previous example, 1(Jn) = 0.Thus, 1(Jn(L)) 6= 0 and {Jn(L) : n 1} is linearly independent.

Jn(L) Jn

n n

L

n n

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 24 / 37

More examples: Infection by a string link.Let L be a 2 component link with zero linking numbers. Jn(L)#Jn isalgebraically slice. It has L as a metabolizer.

1(Jn(L)) + 1(Jn) 0(L) 2 1 (L) 1. Thus, if 0(L) > 1,then 1(Jn(L)) +

1(Jn) 6= 0. As in the previous example, 1(Jn) = 0.Thus, 1(Jn(L)) 6= 0 and {Jn(L) : n 1} is linearly independent.

Jn(L) Jn

n n

L

n n

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 24 / 37

More examples: Infection by a string link.Let L be a 2 component link with zero linking numbers. Jn(L)#Jn isalgebraically slice. It has L as a metabolizer.1(Jn(L)) + 1(Jn) 0(L) 2 1 (L) 1.

Thus, if0(L) > 1,

then 1(Jn(L)) + 1(Jn) 6= 0. As in the previous example, 1(Jn) = 0.

Thus, 1(Jn(L)) 6= 0 and {Jn(L) : n 1} is linearly independent.

Jn(L) Jn

n n

L

n n

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 24 / 37

More examples: Infection by a string link.Let L be a 2 component link with zero linking numbers. Jn(L)#Jn isalgebraically slice. It has L as a metabolizer.1(Jn(L)) + 1(Jn) 0(L) 2 1 (L) 1. Thus, if 0(L) > 1,then 1(Jn(L)) +

1(Jn) 6= 0.

As in the previous example, 1(Jn) = 0.Thus, 1(Jn(L)) 6= 0 and {Jn(L) : n 1} is linearly independent.

Jn(L) Jn

n n

L

n n

More examples: Infection by a string link.Let L be a 2 component link with zero linking numbers. Jn(L)#Jn isalgebraically slice. It has L as a metabolizer.1(Jn(L)) + 1(Jn) 0(L) 2 1 (L) 1. Thus, if 0(L) > 1,then 1(Jn(L)) +

1(Jn) 6= 0. As in the previous example, 1(Jn) = 0.

Thus, 1(Jn(L)) 6= 0 and {Jn(L) : n 1} is linearly independent.

Jn(L) Jn

n n

L

n n

More examples: Infection by a string link.Let L be a 2 component link with zero linking numbers. Jn(L)#Jn isalgebraically slice. It has L as a metabolizer.1(Jn(L)) + 1(Jn) 0(L) 2 1 (L) 1. Thus, if 0(L) > 1,then 1(Jn(L)) +

1(Jn) 6= 0. As in the previous example, 1(Jn) = 0.Thus, 1(Jn(L)) 6= 0 and {Jn(L) : n 1} is linearly independent.

Jn(L) Jn

n n

L

n n

Comparison to Casson-Gordon invariants and Metabelian invariants corresponding to isotropy

(advantage) 1p is not defined in terms of of metabolizers/isotropicsubmodules: only one computation needs to be made, instead of(potentially infinitely) many.

(disadvantage) Only applies to knots whose every prime factor isp-anisotropic. In particular this technique cannot say anything aboutprime algebraically slice knots, for which the other invariants workvery well.

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 25 / 37

The 4-manifold used to prove the second big theorem

Start with W0 as in the previous proof, consider L as a link in +W0. Add2-handles to the zero framing of L. Adding a three handle to this manifoldresults in a manifold with boundary (tM(Ki )) t M(L)

M(K1) M(K2) M(K3)

M(K1#K2#K3)M(L)

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 26 / 37

The 4-manifold used to prove the second big theorem

Start with W0 as in the previous proof,

consider L as a link in +W0. Add2-handles to the zero framing of L. Adding a three handle to this manifoldresults in a manifold with boundary (tM(Ki )) t M(L)

M(K1) M(K2) M(K3)

M(K1#K2#K3)

M(L)

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 26 / 37

The 4-manifold used to prove the second big theorem

Start with W0 as in the previous proof, consider L as a link in +W0.

Add2-handles to the zero framing of L. Adding a three handle to this manifoldresults in a manifold with boundary (tM(Ki )) t M(L)

M(K1) M(K2) M(K3)

M(K1#K2#K3)

M(L)

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 26 / 37

The 4-manifold used to prove the second big theorem

Start with W0 as in the previous proof, consider L as a link in +W0. Add2-handles to the zero framing of L.

Adding a three handle to this manifoldresults in a manifold with boundary (tM(Ki )) t M(L)

M(K1) M(K2) M(K3)

M(K1#K2#K3)M(L)

The 4-manifold used to prove the second big theorem

Start with W0 as in the previous proof, consider L as a link in +W0. Add2-handles to the zero framing of L. Adding a three handle to this manifoldresults in a manifold with boundary (tM(Ki )) t M(L)

M(K1) M(K2) M(K3)

M(K1#K2#K3)

M(L)

The 4-manifold used to prove the second big theorem

1(M(L) H1(M(L))

1(W )1(W )

1(W )(2)p

1(M(Ki ))1(M(Ki ))

1(M(Ki ))(2)p

//

_

//

OO

//

?

OO

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 27 / 37

The 4-manifold used to prove the second big theorem

Thus,1p(Ki ) 0(L) = 2(W , 1p) (W )

(W ) = 02(W , 1p) rank( H2 (W ; )H2 (W ; ))

= g 1 (L)

where = Q

[1(W )

1(W )(2)p

]

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 27 / 37

The 4-manifold used to prove the second big theorem

Thus,1p(Ki ) 0(L) = 2(W , 1p) (W )

(W ) = 02(W , 1p) rank( H2 (W ; )H2 (W ; ))

= g 1 (L)

where = Q

[1(W )

1(W )(2)p

]

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 27 / 37

Proof of main application.

For n = x2 x 1, Tn#Tn is algebraically slice.

It has a metabolizerwhich I call Lx . Thus we are in the setting of the second big theorem.

x red strands x + 1 blue strands

n +1 n +1

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 28 / 37

Proof of main application.

For n = x2 x 1, Tn#Tn is algebraically slice. It has a metabolizerwhich I call Lx .

Thus we are in the setting of the second big theorem.

x red strands x + 1 blue strands

n +1 n +1

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 28 / 37

Proof of main application.

For n = x2 x 1, Tn#Tn is algebraically slice. It has a metabolizerwhich I call Lx . Thus we are in the setting of the second big theorem.

x red strands x + 1 blue strands

n +1 n +1

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 28 / 37

Computation.

If0(Lx) > 1 then 1(Tn) 6= 0, since 21(Tn) 0(Lx) < 2 1 0

and the proof of linear independence will be complete, since they havedistinct, prime Alexander polynomials.

The computation of 0(Lx) can be reduced to planar considerations,algorithmized, and implemented on a computer. The javaimplementation may be found here.

I will do the computations by hand for an easier example, and thenstate the results for Lx . In the end,

0(Lx) < 1 for each x , and theproof will be complete

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 29 / 37

http://math.rice.edu/~cwd1/research/computeRho0

Computation.

If0(Lx) > 1 then 1(Tn) 6= 0, since 21(Tn) 0(Lx) < 2 1 0

and the proof of linear independence will be complete, since they havedistinct, prime Alexander polynomials.

The computation of 0(Lx) can be reduced to planar considerations,algorithmized, and implemented on a computer. The javaimplementation may be found here.

I will do the computations by hand for an easier example, and thenstate the results for Lx . In the end,

0(Lx) < 1 for each x , and theproof will be complete

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 29 / 37

http://math.rice.edu/~cwd1/research/computeRho0

Computation.

If0(Lx) > 1 then 1(Tn) 6= 0, since 21(Tn) 0(Lx) < 2 1 0

and the proof of linear independence will be complete, since they havedistinct, prime Alexander polynomials.

The computation of 0(Lx) can be reduced to planar considerations,algorithmized, and implemented on a computer. The javaimplementation may be found here.

I will do the computations by hand for an easier example, and thenstate the results for Lx . In the end,

0(Lx) < 1 for each x , and theproof will be complete

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 29 / 37

http://math.rice.edu/~cwd1/research/computeRho0

Simplified exampleLet K be the knot with the surgery description below. It happens tobe the left handed trefoil.

This surgery description gives M(K ) as the boundary of the4-manifold, W . The associated inclusion is an isomorphism on firsthomology.Thus, 0(K ) = 2(W , ) (W ).By inspection, (W ) = 1.

1

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 30 / 37

Simplified exampleLet K be the knot with the surgery description below. It happens tobe the left handed trefoil.This surgery description gives M(K ) as the boundary of the4-manifold, W . The associated inclusion is an isomorphism on firsthomology.

Thus, 0(K ) = 2(W , ) (W ).By inspection, (W ) = 1.

1

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 30 / 37

Simplified exampleLet K be the knot with the surgery description below. It happens tobe the left handed trefoil.This surgery description gives M(K ) as the boundary of the4-manifold, W . The associated inclusion is an isomorphism on firsthomology.Thus, 0(K ) = 2(W , ) (W ).

By inspection, (W ) = 1.

1

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 30 / 37

Simplified exampleLet K be the knot with the surgery description below. It happens tobe the left handed trefoil.This surgery description gives M(K ) as the boundary of the4-manifold, W . The associated inclusion is an isomorphism on firsthomology.Thus, 0(K ) = 2(W , ) (W ).By inspection, (W ) = 1.

1

Computing 2(W ):

the curve, , lifts to the universal abelian cover of M(unknot) whereit bounds an embedded disk, D. This disk together with the core ofthe 2-handle glued to form the 2-sphere S which generatesH2(W ; Q[Z]).The twisted intersection form of W is given by the self intersection ofS .The self intersection of S in H2(W ; Q[Z]) is the same as theintersection (with coefficients) between D and the 1 push-off of inthe cover of M(unknot).counting these, S ,S = t t1 + 1, so that [t t1 + 1] is thetwisted intersection matrix of W

1

t1

t

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 31 / 37

Computing 2(W ):the curve, , lifts to the universal abelian cover of M(unknot)

whereit bounds an embedded disk, D. This disk together with the core ofthe 2-handle glued to form the 2-sphere S which generatesH2(W ; Q[Z]).The twisted intersection form of W is given by the self intersection ofS .The self intersection of S in H2(W ; Q[Z]) is the same as theintersection (with coefficients) between D and the 1 push-off of inthe cover of M(unknot).counting these, S ,S = t t1 + 1, so that [t t1 + 1] is thetwisted intersection matrix of W

1

t1

t

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 31 / 37

Computing 2(W ):the curve, , lifts to the universal abelian cover of M(unknot) whereit bounds an embedded disk, D. This disk together with the core ofthe 2-handle glued to form the 2-sphere S which generatesH2(W ; Q[Z]).

The twisted intersection form of W is given by the self intersection ofS .The self intersection of S in H2(W ; Q[Z]) is the same as theintersection (with coefficients) between D and the 1 push-off of inthe cover of M(unknot).counting these, S ,S = t t1 + 1, so that [t t1 + 1] is thetwisted intersection matrix of W

1

t1

t

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 31 / 37

Computing 2(W ):the curve, , lifts to the universal abelian cover of M(unknot) whereit bounds an embedded disk, D. This disk together with the core ofthe 2-handle glued to form the 2-sphere S which generatesH2(W ; Q[Z]).The twisted intersection form of W is given by the self intersection ofS .

The self intersection of S in H2(W ; Q[Z]) is the same as theintersection (with coefficients) between D and the 1 push-off of inthe cover of M(unknot).counting these, S ,S = t t1 + 1, so that [t t1 + 1] is thetwisted intersection matrix of W

1

t1

t

Computing 2(W ):the curve, , lifts to the universal abelian cover of M(unknot) whereit bounds an embedded disk, D. This disk together with the core ofthe 2-handle glued to form the 2-sphere S which generatesH2(W ; Q[Z]).The twisted intersection form of W is given by the self intersection ofS .The self intersection of S in H2(W ; Q[Z]) is the same as theintersection (with coefficients) between D and the 1 push-off of inthe cover of M(unknot).

counting these, S ,S = t t1 + 1, so that [t t1 + 1] is thetwisted intersection matrix of W

1

t1

t

Computing 2(W ):the curve, , lifts to the universal abelian cover of M(unknot) whereit bounds an embedded disk, D. This disk together with the core ofthe 2-handle glued to form the 2-sphere S which generatesH2(W ; Q[Z]).The twisted intersection form of W is given by the self intersection ofS .The self intersection of S in H2(W ; Q[Z]) is the same as theintersection (with coefficients) between D and the 1 push-off of inthe cover of M(unknot).counting these, S , S = t t1 + 1, so that [t t1 + 1] is thetwisted intersection matrix of W

1

t1

t

Computing 2(W ):

The transform from l2(Z) to L2(S1) (S1 C) sending tn to zn is anisomorphism: [1 t t1] has the same signature as[1 z z1] = [1 2 Re(z)] which is the integral of sign(1 2 Re(z))over S1, which is

1

3.

thus, 0(K ) =1

3 (1) = 4

3

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 32 / 37

Computation for the link, L2In general this can be duplicated for any link arising as 1 surgery alongcommutator curves on the unlink.

With this in mind, L2 can be realizedas such.

+1 +1

+1+1+1+1

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 33 / 37

Computation for the link, L2In general this can be duplicated for any link arising as 1 surgery alongcommutator curves on the unlink. With this in mind, L2 can be realizedas such.

+1 +1

+1+1+1+1

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 33 / 37

Computation for the link, L2In general this can be duplicated for any link arising as 1 surgery alongcommutator curves on the unlink. With this in mind, L2 can be realizedas such.

+1 +1+1+1+1+1

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 33 / 37

Computation for the link, L2In general this can be duplicated for any link arising as 1 surgery alongcommutator curves on the unlink. With this in mind, L2 can be realizedas such.

+1 +1+1+1+1+1

Computation for the link, L2In general this can be duplicated for any link arising as 1 surgery alongcommutator curves on the unlink. With this in mind, L2 can be realizedas such. The resulting 4-manifold has regular intersection form (byinspection)

1 0 0 00 1 0 00 0 1 00 0 0 1

,

+1 +1+1+1+1+1

Computation for the link, L2In general this can be duplicated for any link arising as 1 surgery alongcommutator curves on the unlink. With this in mind, L2 can be realizedas such. The resulting 4-manifold has regular intersection form (byinspection)

1 0 0 00 1 0 00 0 1 00 0 0 1

,and twisted intersection form (computed by taking the algorithm outlinedfor the trefoil knot and implementing it on a computer) 1 + xy 2 + y + x1y2 + y1 xy y2 xy + y2y 2 x1y1 y + y1 y + x1 y1

y 2 + x1y1 y + x y1 y + y1

+1 +1+1+1+1+1

Computation for the link, L2In general this can be duplicated for any link arising as 1 surgery alongcommutator curves on the unlink. With this in mind, L2 can be realizedas such. The resulting 4-manifold has regular intersection form (byinspection)

1 0 0 00 1 0 00 0 1 00 0 0 1

,and twisted intersection form (computed by taking the algorithm outlinedfor the trefoil knot and implementing it on a computer) 1 + xy 2 + y + x1y2 + y1 xy y2 xy + y2y 2 x1y1 y + y1 y + x1 y1

y 2 + x1y1 y + x y1 y + y1

which have signatures 4 and .38 (numerically integrating via computer).

+1 +1+1+1+1+1

1 0 0 00 1 0 00 0 1 00 0 0 1

,and twisted intersection form (computed by taking the algorithm outlinedfor the trefoil knot and implementing it on a computer) 1 + xy 2 + y + x1y2 + y1 xy y2 xy + y2y 2 x1y1 y + y1 y + x1 y1

y 2 + x1y1 y + x y1 y + y1

which have signatures 4 and .38 (numerically integrating via computer).Thus, 0(L2) 3.62 and 21(K7) . 2.62 (7 = 22 2 1), and inparticular is not zero.

+1 +1+1+1+1+1

Computation for the link, Lx

Lx+1 can be realized from Lx by +1 surgery along 2x 1 new commutatorcurves.

Proposition

If a link L is realized as +1 surgery along commutator curves on anotherlink L, then 0(L) 0(L)

Thus 0(Lx) . 3.62 for all x 2 and 1(Tx2x1) 6= 0. The knotshave distinct, prime Alexander polynomials and so are linearly independent.

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 34 / 37

Computation for the link, Lx

Lx+1 can be realized from Lx by +1 surgery along 2x 1 new commutatorcurves.

Proposition

If a link L is realized as +1 surgery along commutator curves on anotherlink L, then 0(L) 0(L)

Thus 0(Lx) . 3.62 for all x 2 and 1(Tx2x1) 6= 0. The knotshave distinct, prime Alexander polynomials and so are linearly independent.

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 34 / 37

Computation for the link, Lx

Lx+1 can be realized from Lx by +1 surgery along 2x 1 new commutatorcurves.

Proposition

If a link L is realized as +1 surgery along commutator curves on anotherlink L, then 0(L) 0(L)

Thus 0(Lx) . 3.62 for all x 2 and 1(Tx2x1) 6= 0.

The knotshave distinct, prime Alexander polynomials and so are linearly independent.

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 34 / 37

Computation for the link, Lx

Lx+1 can be realized from Lx by +1 surgery along 2x 1 new commutatorcurves.

Proposition

If a link L is realized as +1 surgery along commutator curves on anotherlink L, then 0(L) 0(L)

Thus 0(Lx) . 3.62 for all x 2 and 1(Tx2x1) 6= 0. The knotshave distinct, prime Alexander polynomials and so are linearly independent.

Anisotropic knots and infection.

Definition

1p is some other localized -invariant for knots.

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 35 / 37

Anisotropic knots and infection.

Theorem

Given slice knots R1, . . .Rn, infecting curves 1, . . . n representingelements of A0(Rn) which sit in the sum of no pair of isotropic submodulesand p-anisotropic knots K1, . . . ,Km, ifn#i=1

R1(1, (R2(2, . . . (Rn(n(Ki )) . . . ))) is slice

thenn

i=1

1p(Ki ) = 0

The following is an immediate application.

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 36 / 37

Anisotropic knots and infection.

Corollary

Let R be a slice knot whose Alexander module has a unique nontrivialisotropic submodule. Let be an arc representing an element of A0(R)which is not isotropic. Let Kn be a sequence of anisotropic knots withsquare-free, strongly coprime Alexander polynomials and nonzero 1

invariants (take Tn(x), for example).Then {R(,R(, . . .R((Kn)) . . . ))} is linearly independent, whereinfection is understood to take place some fixed number of times.

Chris Davis, Rice University () Localized metabelian rho invariants as obstructions to torsion in the knot concordance group.July 22, 2010 37 / 37