Linking Invariant of Knot Theorypantodon.shinshu-u.ac.jp/.../slides/EACAT4slide_Hsieh.pdf ·...

Linking Invariantof Knot Theory

Chun-ChungHsieh

Introduction

Reidemeistermoves

Examples

Higher linkingLn,n−1,...,1,0

Examples

Massey-MilnorlinkingL∗n,n−1,...,1,0

Table of firstnon-vanishinglinking

MATH, Academia Sinica, R.O.C

Linking Invariant of Knot Theory

Chun-Chung Hsieh

Institute of MathematicsAcademia Sinica

Taiwan

EACAT4, 2011Tokyo, Japan


Chun-ChungHsieh

Introduction

Reidemeistermoves

Examples


Examples




Introduction

L = knot = {L0,L1, . . . ,Ln} ≤ R3, each Li is an oriented basedsmooth circle.

�

@@@I

6

?

��

x0 0

n

n − 1 2

1

.. . . . .

.


Chun-ChungHsieh

Introduction

Reidemeistermoves

Examples


Examples




Introduction

Example 1:


Chun-ChungHsieh

Introduction

Reidemeistermoves

Examples


Examples




Introduction

Example 2:


Chun-ChungHsieh

Introduction

Reidemeistermoves

Examples


Examples




Introduction

Problem:

To look for knot invariant(KI, for short), KI:{all knots}→ R such thatKI takes the same value on the equivalence class of L which, bydefinition, is the equivalence class of knots under the ambientmotions of R3.


Chun-ChungHsieh

Introduction

Reidemeistermoves

Examples


Examples




Reidemeister moves

Lemma (Reidemeister) KI:{knots}→ R is a knot invariant iff KI isinvariant under the following 3 moves,


Chun-ChungHsieh

Introduction

Reidemeistermoves

Examples


Examples




Reidemeister moves


Chun-ChungHsieh

Introduction

Reidemeistermoves

Examples


Examples




Examples

Theorem (Gauss)

For a knot L = {L0,L1} of 2 components, the numberL1,0 =

∑L1∧L0

(1,0) is an invariant, where

(1) L1 ∧ L0 refers to the crossing of L1, and L0 when representedby a knot diagram of L, and

(2) (1,0) is the crossing signature/symbol defined as


Chun-ChungHsieh

Introduction

Reidemeistermoves

Examples


Examples




Examples


Chun-ChungHsieh

Introduction

Reidemeistermoves

Examples


Examples




Examples

Theorem (Gauss)

For a knot of 2 components L = {L0,L1}, the above invariantL1,0 =

∫L0

∫L1

< pull-back of the area form on S2 by the mapping

(L0 × L1 ∈ (x0, x1)→ x1 − x0

|x1 − x0|∈ S2) >.


Chun-ChungHsieh

Introduction

Reidemeistermoves

Examples


Examples




Higher linking Ln,n−1,...,1,0

Remark/Definition (1) From now on, we will represent our knot of

(n + 1) components L = {L0,L1, . . . ,Ln} as:

�

@@@I

6

?

��

x0 0

n

n − 1 2

1

.. . . . .

.


Chun-ChungHsieh

Introduction

Reidemeistermoves

Examples


Examples





(2) One of the key aspect in this theory is to show that our knotinvariants are independent of the based points{xi ∈ Li |i = 0,1, . . . ,n}.

(3) Chern-Simons-Witten configuration space integral—CSW, forshort.


Chun-ChungHsieh

Introduction

Reidemeistermoves

Examples


Examples





Given a knot of (n + 1) components L = {L0,L1, . . . ,Ln}represented as a schematic diagram in the introduction and aCSW-graph of degree n, Γ supported on L we define theconfiguration space integral CSW(Γ) as follows.

Definition: a CSW-graph supported on L is a uni-trivalent graphwith univalent vertices supported on L; and degree(Γ) = #{edges of Γ} −#{trivalent vertices of Γ}.


Chun-ChungHsieh

Introduction

Reidemeistermoves

Examples


Examples





(3.1) To each edge AB of Γ, we associate the area form

pulled-back byA− B|A− B|

from the unit sphere in R3.

(3.2) Do the wedge product over all edges to get a differential formof degree 2e, denoted as WΓ.

(3.3) To each trivalent vertex, we associate a copy of the ambientspace R3.

(3.4) To each univalent vertex on the knot component Lj , weassociate a copy Lj , proper.


Chun-ChungHsieh

Introduction

Reidemeistermoves

Examples


Examples





(3.5) Do the “oriented product” of all spaces over all vertices to geta configuration space of dimension 2e, denoted as CΓ.

(3.6) CSW (Γ) =∫CΓ

WΓ, which by definition is the Feynman

diagram in the Chern-Simons-Witten perturbative quantum fieldtheory.

(4) Chern-Simons-Witten perturbative quantum field theoryChern-Simons-Witten linking Ln,n−1,...,1,0 = sum of all CSW (Γ)over the set of uni-trivalent graphs of degree n.


Chun-ChungHsieh

Introduction

Reidemeistermoves

Examples


Examples




Examples

Exmaples 1.

(Gauss linking) L11.


Chun-ChungHsieh

Introduction

Reidemeistermoves

Examples


Examples




Examples

Exmaples 2. For Borromean ring represented as knot diagram asabove, Gauss Linkings {Li,j} are not good enough to detect itsbeing different from the trivial triple. So, we need higher concept oflinking for knot of 3 components.


Chun-ChungHsieh

Introduction

Reidemeistermoves

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Examples




Examples

Exmaples 3. For L = {L0,L1,L2} for which all lower linkings of Lvanish, we define L2

2:


Chun-ChungHsieh

Introduction

Reidemeistermoves

Examples


Examples




Examples

Exmaples 4. For a knot L = {L0,L1,L2,L3} for which all lowerlinkings vanish, we define L3

3:

0 1

23

0 1

23

0 1

23

0 1

23

0 1

23

0 1

23

01

23

0 1

23

0 1

2

3

0

1

23

x1

x0

x2

x3

x1

x2

x3

x0


Chun-ChungHsieh

Introduction

Reidemeistermoves

Examples


Examples




Examples


Chun-ChungHsieh

Introduction

Reidemeistermoves

Examples


Examples




Examples

0 1

2

3

01

23

0 1

23

0

1

23

x2

x2

x2 x2

x0

x3


Chun-ChungHsieh

Introduction

Reidemeistermoves

Examples


Examples




Examples

0 1

2

3

0

1

2

3

0 1

23

0 1

23

x3 x3

x1

x2

x3 x3

0

1

23

0

1

2

3

0 1

23

0 1

23

x0 x0

x1

x0

x2

x0


Chun-ChungHsieh

Introduction

Reidemeistermoves

Examples


Examples




Examples

Theorem (H-Yang, H-Kauffman-Tsau)

Given a knot of (n + 1) components L = {L0,L1, . . . ,Ln} for whichall linkings of strictly lower degrees vanish then Ln,n−1,...,1,0 is aknot invariant and is independent of the choice of based points.

Note: So in some sense Ln,n−1,...,1,0 is called the firstnon-vanishing linking of L.


Chun-ChungHsieh

Introduction

Reidemeistermoves

Examples


Examples




Massey-Milnor linking L∗n,n−1,...,1,0

Next we introduce another concept of linkings—the so-calledMassey-Milnor linking L∗n,n−1,...,1,0. For this purpose we introducethe following.


Chun-ChungHsieh

Introduction

Reidemeistermoves

Examples


Examples





Definition: Given a knot L = {L0,L1, . . . ,Ln}

(1) For each knot component Lj , we introduce a differential 1-form

as: j(x) =∮

Lj< pull-back of area form from S2 by

(y − x)

|y − x |>,

∀x /∈ Lj .

(2) We define the key stuffs in this aspect—closed differential2-forms (n,n − 1, . . . ,1) and closed differential 1-forms< n,n − 1, . . . ,1 > as following inductively.


Chun-ChungHsieh

Introduction

Reidemeistermoves

Examples


Examples





(2-1)

(2,1) = 2 ∧ 1 + 2(d1)− 1(d2),

< 2,1 >= (2,1) + 21.

(2-2)

(3,2,1) =(3,2) ∧ 1 + 3 ∧ (2,1) + < 3,2 >(d1)− 31(d2) + < 1,2 >(d3),

< 3,2,1 >= (3,2,1) + (3,2)1 + 3< 2,1 >.


Chun-ChungHsieh

Introduction

Reidemeistermoves

Examples


Examples





(2-3) in general

(n,n − 1, . . . ,2,1) = (n,n − 1, . . . ,2) ∧ 1

+ (n,n − 1, . . . ,3) ∧ (2,1) + . . .

+ (n,n − 1) ∧ (n − 2, . . . ,1) + n ∧ (n − 1, . . . ,1)

+ < n,n − 1, . . . ,2 >(d1)−< n,n − 1, . . . ,3 >1(d2)

+ < n,n − 1, . . . ,4 >< 1,2 >(d3)± . . . . . .

+ (−1)n−1< 1,2, . . . ,n − 1 >(dn);


Chun-ChungHsieh

Introduction

Reidemeistermoves

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Examples





< n,n − 1, . . . ,1 >= (n,n − 1, . . . ,1)

+ (n,n − 1, . . . ,2)1 + (n,n − 1, . . . ,3)< 2,1 >

+ · · ·+ n< n − 1, . . . ,1 >.


Chun-ChungHsieh

Introduction

Reidemeistermoves

Examples


Examples





(2-4)

In the above, over-bar(—) denotes the inverse of differential d−1

whenever well-defined for example: in< n,n − 1, . . . , j + 1 >< 1,2, . . . , j − 1 >(dj), as (dj) is a singularDirac 2-form supported only on Lj , functions< n,n − 1, . . . , j + 1 >(x) and< 1,2, . . . , j − 1 >(x) are well-defined in any tubular neighborhoodof Lj as long as∮

Lj

< n,n − 1, . . . , j + 1 >= 0 =

∮Lj

< 1,2, . . . , j − 1 > .


Chun-ChungHsieh

Introduction

Reidemeistermoves

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(3)

L∗n,n−1,...,1,0 =∮L0

< n,n − 1, . . . ,2,1 >.


Chun-ChungHsieh

Introduction

Reidemeistermoves

Examples


Examples





Theorem (H) L∗n,n−1,...,1,0 is explicitly coded by the set ofChern-Simons-Witten graphs of degree n.

Theorem (H) The above two linkings coincide, namely thatLn,n−1,...,1,0 = L∗n,n−1,...,1,0.

Theorem (H) There are explicit/combinatorial formulae for the firstnon-vanishing linkings Ln,n−1,...,1,0.


Chun-ChungHsieh

Introduction

Reidemeistermoves

Examples


Examples





Example:

L2,1,0 =∑

(2,1)0 +∑1<

02(1,0)(2,0) +

∑2<

10(2,1)(0,1) +

∑0<

21(0,2)(1,2),

where (i , j) denotes the Gauss signature/symbol of crossing asdefined above and (2,1)0 stands for the “residue” defined below.


Chun-ChungHsieh

Introduction

Reidemeistermoves

Examples


Examples






Chun-ChungHsieh

Introduction

Reidemeistermoves

Examples


Examples




Table of first non-vanishing linking

In the following table, n = degree of linking and m + 1 = number ofknot components: Ln

m = linking of degree n supported onL = {L0,L1, . . . ,Lm−1,Lm}.


Chun-ChungHsieh

Introduction

Reidemeistermoves

Examples


Examples





QQ

QQ

QQ

Ln0

L50

L40

L30

L20

L10

Ln1

L51

L41

L31

L21

L11

Ln2

L52

L42

L32

L22

Ln3

L53

L43

L33

Ln4

L54

L44

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.


Chun-ChungHsieh

Introduction

Reidemeistermoves

Examples


Examples





Example L20.


Chun-ChungHsieh

Introduction

Reidemeistermoves

Examples


Examples





Example L30.


Chun-ChungHsieh

Introduction

Reidemeistermoves

Examples


Examples





Example L21.

0 0 0 0

1111

x0 x0 x0 x0

x1x1x1x1

.

Linking Invariant of Knot Theorypantodon.shinshu-u.ac.jp/.../slides/EACAT4slide_Hsieh.pdf ·...

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