Algebraic unknotting number and 4-manifolds joint with S. Friedl Maciej Borodzik

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joint with S. Friedl

Maciej Borodzik

Institute of Mathematics, University of Warsaw

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Algebraic unknotting

Unknotting number: how many crossing changes makeK the unknot.

Maciej Borodzik Algebraic unknotting number and 4-manifolds

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Algebraic unknotting

Algebraic unknotting numberua:how many crossing changes makeK a knotLwith∆(L) ≡1.

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Algebraic unknotting

Defined by Murakami and Fogel in 1993.

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Algebraic unknotting

Murakami and Saeki considered anan algebraic unknotting operation on Seifert matrices.

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Algebraic unknotting

Murakami and Saeki considered anan algebraic unknotting operation on Seifert matrices.

uadepends only on the Seifert matrix. For example, if

∆(K) ≡1, thenu_a=0.

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Surgery presentation

A unknotting move can be regarded as a±1surgery on a suitable link.

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Surgery presentation

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Surgery presentation

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Surgery presentation

Asurgery presentationis a collection of such circles and numbers±1, such that a simultaneous surgery transforms the knot into the unknot.

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A manifold with boundary S

₀³

(K )

Consider a surgery presentationc₁, . . . ,c_r,n₁, . . . ,n_r.

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A manifold with boundary S

₀³

(K )

Consider a surgery presentationc₁, . . . ,c_r,n₁, . . . ,n_r. The cyclesc₁, . . . ,c_r may be choosen to lie on the boundary of the tubular neighbourhood ofK.

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A manifold with boundary S

₀³

(K )

Consider them as a cycles onS₀³(K). Surgery on them yieldsS³₀(unknot) =S²×S¹.

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A manifold with boundary S

₀³

(K )

These surgeries induce a cobordism ofS₀³(K)withS²×S¹ with only 2-handles. We glueD³×S¹at the end.

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A manifold with boundary S

₀³

(K )

These surgeries induce a cobordism ofS₀³(K)withS²×S¹ with only 2-handles. We glueD³×S¹at the end.

We obtainW with∂W =S₀³(K).

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Properties of W

π1(W) = Z;

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Properties of W

π1(W) = Z;

H₁(M) →H₁(W)is an isomorphism;

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Properties of W

π1(W) = Z;

b₂(W) =r;

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Properties of W

π1(W) = Z;

b₂(W) =r;

The intersection pairing onW is diagonalizable;

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Properties of W

π1(W) = Z;

b₂(W) =r;

Remark

If the surgery onc₁, . . . ,c_r yields a knot with Alexander polynomial1, then suchW still exists, but it is a topological manifold in general.

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Properties of W

π1(W) = Z;

b₂(W) =r;

Remark

If the surgery onc₁, . . . ,c_r yields a knot with Alexander polynomial1, then suchW still exists, but it is a topological manifold in general.

Definition

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Off-topic

This formula appears in almost every talk here, so I will write it.

. . . F_(n

.5)⊂ F_(n)⊂ . . . ⊂ F₍₀₎⊂ C

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Setup

For a knotK we considerX =X(K)its complement andXe its infinite cyclic cover.

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Setup

DenoteΛ = Z[t^±¹]

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Setup

H₁(X; Λ)as homologies ofXe regarded as aΛ-module.

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Setup

Lemma (Blanchfield, 1959)

There exists a pairingH₁(X; Λ) ×H₁(X; Λ) → Q(t)/Λ.

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Setup

Lemma (Blanchfield, 1959)

There exists a pairingH₁(X; Λ) ×H₁(X; Λ) → Q(t)/Λ.

The construction resembles the standard construction of a

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Presentation matrix

Definition

We say that a squarek×k matrixAoverΛrepresents the Blanchfield pairing if

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Presentation matrix

Definition

H₁(X; Λ) ∼= Λ^k/AΛ^k;

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Presentation matrix

Definition

H₁(X; Λ) ∼= Λ^k/AΛ^k;

the pairing is(a,b) →a·A⁻¹bunder the above identification.

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Presentation matrix

Definition

H₁(X; Λ) ∼= Λ^k/AΛ^k;

the pairing is(a,b) →a·A⁻¹bunder the above identification.

Lemma (Kearton 1975)

A Seifert matrix gives rise to a presentation matrix of the same

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Presentation matrix and W

Lemma (—,Friedl 2012)

IfW strictly coboundsM(K)andB is a matrix of the intersection form onH₂(W; Λ), thenBrepresents also the Blanchfield pairing forK.

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Presentation matrix and W

Corollary

Letn(K)be the minimal size of a matrixArepresenting the Blanchfield pairing (such thatA(1)is diagonal). Then n(K) ≤u_a(K).

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Presentation matrix and W

Corollary

Theorem (–,Friedl 2013)

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Presentation matrix and W

Corollary

Theorem (–,Friedl 2013)

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Consequences

Given the knotK, the following four numbers are equal.

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Consequences

The minimal number of crossings needed to changeK into an Alexander polynomial 1 knot;

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Consequences

The minimal number of algebraic crossing changes on the Seifert matrix, which make the Seifert matrix trivial;

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Consequences

The minimal size of a matrixArepresenting the Blanchfield pairing;

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Consequences

The minimal size of a matrixArepresenting the Blanchfield pairing;

The minimalb₂(W)for a manifoldW strictly cobounding M(K);

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Computing n(K )

Lower bounds.

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Computing n(K )

Lower bounds.

n(K)is not smaller than theNakanishi index;

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Computing n(K )

Lower bounds.

n(K) ≥ |σ_K(z)|, in fact we can take thespan of T-L signature;

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Computing n(K )

Lower bounds.

n(K)contains theLickorish and Jabuka obstructionto u(K) =1;

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Computing n(K )

Lower bounds.

n(K)contains theStoimenowu(K) =2obstruction;

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Computing n(K )

Lower bounds.

a new obstruction from careful reading of Owens’ paper;

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Computing n(K )

Lower bounds.

Upper bounds.

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Computing n(K )

Lower bounds.

Upper bounds.

the unknotting number;

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Computing n(K )

Lower bounds.

Upper bounds.

the unknotting number;

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Open questions

Isn(K)mutation invariant?

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Open questions

Is the conditionA(1)is diagonal important?

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Open questions

What if we requireW to be smooth?

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Open questions

What if we requireW to be smooth?

Does this generalize to higher dimensions? We have a notion of a zero-surgery onSⁿ⁻²⊂Sⁿ.