Finite Type Invariants

Lena Folwaczny (UIC)

Joint Meetings, Boston

January 7, 2012

Lena Folwaczny (UIC) Finite Type Invariants 1 3 A knot is a smooth embedding f : S → R . We have various methods of encoding knot informaiton in an eﬀort to help understand them.

Planar Diagrams Braids Tangles

Lena Folwaczny (UIC) Finite Type Invariants Knots as Planar Diagrams

In 1926 Kurt Reidemeister showed that given two knots K1 and K2, then K1 ∼ K2 (K1 is isotopic to K2) if and only if they can be related by a sequence of the three Reidemeister moves and planar isotopy.

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Knots as Gauss Diagrams

To every knot, we can associate a Gauss Code, which we can then draw as a Gauss Diagram. From every Gauss Diagram, we can always ”try” to draw the knot diagram!

Lena Folwaczny (UIC) Finite Type Invariants Knots as Gauss Diagrams

To every knot, we can associate a Gauss Code, which we can then draw as a Gauss Diagram. From every Gauss Diagram, we can always ”try” to draw the knot diagram!

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Lena Folwaczny (UIC) Finite Type Invariants A geometric way to interpret virtual knots is that they are knots drawn on surfaces, and the virtual crossing occurs when project through the surface into the plane, we have an ”appearance” of the crossing a crossing in the planar diagram.

Virtual Knots

We can use a special symbol to denote a crossing that isn’t really there. This knot is now called a virtual knot. Virtual Knots can be though of as the completion of knot diagrams.

Lena Folwaczny (UIC) Finite Type Invariants Virtual Knots

We can use a special symbol to denote a crossing that isn’t really there. This knot is now called a virtual knot. Virtual Knots can be though of as the completion of knot diagrams.

A geometric way to interpret virtual knots is that they are knots drawn on surfaces, and the virtual crossing occurs when project through the surface into the plane, we have an ”appearance” of the crossing a crossing in the planar diagram.

Lena Folwaczny (UIC) Finite Type Invariants Vassiliev Invariants

Deﬁnition A Vassiliev Invariant is an invariant V of singular knots that satisﬁes the following relation:

Notice that invariants of knots are automatically invariants of graphs (singular knots) under this deﬁnition.

Lena Folwaczny (UIC) Finite Type Invariants Let’s calculate the Conway polynomial of the following singular knot.

Lena Folwaczny (UIC) Finite Type Invariants For example, the nth coeﬃcient of the Conway Polynomial is a Vassiliev Invariant of type n. Dror Bar-Natan has related ﬁnite types invariants to derivatives.

Deﬁnition A Vassiliev Invariant V has ﬁnite type ≤ n if it vanishes (i.e. is constantly zero) on all knots that have more than n singular points.

Lena Folwaczny (UIC) Finite Type Invariants Dror Bar-Natan has related ﬁnite types invariants to derivatives.

Deﬁnition A Vassiliev Invariant V has ﬁnite type ≤ n if it vanishes (i.e. is constantly zero) on all knots that have more than n singular points.

For example, the nth coeﬃcient of the Conway Polynomial is a Vassiliev Invariant of type n.

Lena Folwaczny (UIC) Finite Type Invariants Deﬁnition A Vassiliev Invariant V has ﬁnite type ≤ n if it vanishes (i.e. is constantly zero) on all knots that have more than n singular points.

For example, the nth coeﬃcient of the Conway Polynomial is a Vassiliev Invariant of type n. Dror Bar-Natan has related ﬁnite types invariants to derivatives.

Lena Folwaczny (UIC) Finite Type Invariants More Examples of Finite Type Invariants

1 By substituting t = ex into the Jones polynomial,

and considering the formal Taylor series in x of ex , we get a polynomial where the coeﬃcient of xk is a type k Vassiliev invariant. 2 Similar theorems hold for Kauﬀman and HOMFLYPT polynomials, and arbitrary Reshetikhin-Turaev invariants.

Lena Folwaczny (UIC) Finite Type Invariants Conclusion: There are plenty of ﬁnite type invariants! And they are at least as powerful as all the knot polynomial invariants.

Lena Folwaczny (UIC) Finite Type Invariants Let V be a Vassiliev invariant. Considering Rediemeister moves on the diagram, we see that Vassiliev invariants must satisfy the following two relations:

The one term relation is:

Lena Folwaczny (UIC) Finite Type Invariants Similarly, considering the following planar isotopies:

we get the four term relation

Lena Folwaczny (UIC) Finite Type Invariants Proof: Let V be a Vassiliev invariant of order ≤ n. Let K1, K2 be knots with n singular points, where K1 ∼ K2 except for a crossing change somewhere in the diagram. Let’s compare V (K1) and V (K2).

V (K1)- V (K2) = V (Kx ) = 0

Therefore, V (K1) = V (K2)

Theorem The value of a Vassiliev invariant of order ≤ n on a knot with n singular points does not vary when one (or several) crossings are changed to opposite crossings.

Lena Folwaczny (UIC) Finite Type Invariants V (K1)- V (K2) = V (Kx ) = 0

Therefore, V (K1) = V (K2)

Theorem The value of a Vassiliev invariant of order ≤ n on a knot with n singular points does not vary when one (or several) crossings are changed to opposite crossings.

Proof: Let V be a Vassiliev invariant of order ≤ n. Let K1, K2 be knots with n singular points, where K1 ∼ K2 except for a crossing change somewhere in the diagram. Let’s compare V (K1) and V (K2).

Lena Folwaczny (UIC) Finite Type Invariants Theorem The value of a Vassiliev invariant of order ≤ n on a knot with n singular points does not vary when one (or several) crossings are changed to opposite crossings.

V (K1)- V (K2) = V (Kx ) = 0

Therefore, V (K1) = V (K2)

Lena Folwaczny (UIC) Finite Type Invariants The Power of the Theorem

1 The value of an nth order invariant of a knot with n singular points is unaﬀected by changes in crossings. Thus, its value does not depend on the phenomenon of knotting, but only on the order (a combinatorial concept!) in which the singular points appear when following the curve of the knot!

2 Let Vn be the set of all Vassiliev invariants of type ≤ n. Notice that Vn has a natural vector space structure, and that

V0 ⊂ V1 ⊂ V2 ⊂ ...

We would like to better understand the structure of Vn. The theorem suggests that chord diagrams might help us do so.

Lena Folwaczny (UIC) Finite Type Invariants Deﬁnition A chord diagram of order n is an oriented circle with n pairs of distinct points (regarded up to orientation-preserving diﬀeomorphism of the circle).

CDn = set of all chord diagrams of order n.

Lena Folwaczny (UIC) Finite Type Invariants The Algebra of Chord Diagrams

Chord diagrams have a Hopf Algebra Sturcture (a graded bi-algebra). Product: Connected sum of two knots

Co-Product: X 0 δ(D) = DJ ⊗ DJ JC[D]

Lena Folwaczny (UIC) Finite Type Invariants Theorem

The value of v ∈ Vn (Vassiliev invariant of order ≤ n) on a knot K with n singular points depends only on the chord diagram of K.

This theorem tells us we have a well-deﬁned map αn : Vn → CDn. To understand the size and structure of the space Vn, it is useful to have a description of the Kernel and Image of αn. Notice that by deﬁnition of Vassiliev invariant,

Ker(αn) = Vn−1

Singular Knots to Chord Diagrams

Drawing singular knots as chord diagrams is similar to constructing the Gauss diagram. Singular points in the knot diagram get connected by a chord in the chord diagram.

Lena Folwaczny (UIC) Finite Type Invariants This theorem tells us we have a well-deﬁned map αn : Vn → CDn. To understand the size and structure of the space Vn, it is useful to have a description of the Kernel and Image of αn. Notice that by deﬁnition of Vassiliev invariant,

Ker(αn) = Vn−1

Singular Knots to Chord Diagrams

Drawing singular knots as chord diagrams is similar to constructing the Gauss diagram. Singular points in the knot diagram get connected by a chord in the chord diagram.

Theorem

The value of v ∈ Vn (Vassiliev invariant of order ≤ n) on a knot K with n singular points depends only on the chord diagram of K.

Lena Folwaczny (UIC) Finite Type Invariants Singular Knots to Chord Diagrams

Drawing singular knots as chord diagrams is similar to constructing the Gauss diagram. Singular points in the knot diagram get connected by a chord in the chord diagram.

Theorem

The value of v ∈ Vn (Vassiliev invariant of order ≤ n) on a knot K with n singular points depends only on the chord diagram of K.

Ker(αn) = Vn−1

Lena Folwaczny (UIC) Finite Type Invariants Deﬁnition

A weight system of order n is a function f ∈ Fn that satisﬁes the one-term and four-term relations.

Wn is the space of all weight systems of order n.

Notation and Deﬁnitions

Fn = the linear space of all C-valued functions on the set of chord diagrams of order n. Deﬁnition

The symbol of a Vassiliev invariant f ∈ Vn is the function [f ] ∈ Fn deﬁned by [f ](D) = f (K) where K is an arbitrary knot whose chord diagram is D.

Lena Folwaczny (UIC) Finite Type Invariants Notation and Deﬁnitions

Fn = the linear space of all C-valued functions on the set of chord diagrams of order n. Deﬁnition

The symbol of a Vassiliev invariant f ∈ Vn is the function [f ] ∈ Fn deﬁned by [f ](D) = f (K) where K is an arbitrary knot whose chord diagram is D.

Deﬁnition

A weight system of order n is a function f ∈ Fn that satisﬁes the one-term and four-term relations.

Wn is the space of all weight systems of order n.

Lena Folwaczny (UIC) Finite Type Invariants Two Assertions

1 (V. Vassiliev) The symbol of every Vassiliev invariant is a weight system. 2 (M. Kontsevich) Every weight system is a symbol of a certain Vassiliev invariant.

Kontsevich’s Theorem

Theorem ∼ Vn/Vn−1 = Wn

Lena Folwaczny (UIC) Finite Type Invariants Kontsevich’s Theorem

Theorem ∼ Vn/Vn−1 = Wn

Two Assertions

1 (V. Vassiliev) The symbol of every Vassiliev invariant is a weight system. 2 (M. Kontsevich) Every weight system is a symbol of a certain Vassiliev invariant.

Lena Folwaczny (UIC) Finite Type Invariants Theorem ∼ Vn/Vn−1 = An

With the help of a supercomputer at Harvard in 1995, Bar-Natan succeeded in ﬁnding the dimenion of An for n = 0,1, ... , 9. The values are 1,0,1,1,1,3,4,9,14,27 and 44, respectively.

Another approach to weight systems

Deﬁnition

Let An denote the space of all chord diagrams of order n modulo the subspace spanned by all 4-term and 1-term relations.

Lena Folwaczny (UIC) Finite Type Invariants Another approach to weight systems

Deﬁnition

Let An denote the space of all chord diagrams of order n modulo the subspace spanned by all 4-term and 1-term relations.

Theorem ∼ Vn/Vn−1 = An

With the help of a supercomputer at Harvard in 1995, Bar-Natan succeeded in ﬁnding the dimenion of An for n = 0,1, ... , 9. The values are 1,0,1,1,1,3,4,9,14,27 and 44, respectively.

Lena Folwaczny (UIC) Finite Type Invariants 2 Every ﬁnite type invariant can be represented as a linear combination of such coeﬃcients (this is why the Kontsevich integral is called the Universal Vassiliev Invariant). 3 The Kontsevich Integral of a knot is a far-reaching generalization of the Gauss Integral Formula.

Some facts about the Kontsevich Integral

1 Every coeﬃcient of the Kontsevich integral is a ﬁnite type invariant.

Lena Folwaczny (UIC) Finite Type Invariants 3 The Kontsevich Integral of a knot is a far-reaching generalization of the Gauss Integral Formula.

Some facts about the Kontsevich Integral

1 Every coefficient of the Kontsevich integral is a finite type invariant. 2 Every finite type invariant can be represented as a linear combination of such coefficients (this is why the Kontsevich integral is called the Universal Vassiliev Invariant).

Lena Folwaczny (UIC) Finite Type Invariants Some facts about the Kontsevich Integral

Lena Folwaczny (UIC) Finite Type Invariants The Kontsevich Integral

Linear Combination of Morse knot Chord Diagrams

Lena Folwaczny (UIC) Finite Type Invariants The Kontsevich Integral

Lena Folwaczny (UIC) Finite Type Invariants For ”braids”, yes (Kohno, ’95). For ”string links up to homotopy”, yes (X. S. Lin, ’92). Open: Do Vassiliev invariants detect knot orientation? Open: Can Vassiliev invariants detect mutant knots? (There is no ﬁnite type invariant of degree < 11 which can distinguish mutant knots.) Open: Can the signature of a knot be expressed as a Vassiliev Invariant?

Conjectures and Open Problems

Conjecture: Finite type Vassiliev invariants can distinguish all knots.

Lena Folwaczny (UIC) Finite Type Invariants Open: Do Vassiliev invariants detect knot orientation? Open: Can Vassiliev invariants detect mutant knots? (There is no ﬁnite type invariant of degree < 11 which can distinguish mutant knots.) Open: Can the signature of a knot be expressed as a Vassiliev Invariant?

Conjectures and Open Problems

Conjecture: Finite type Vassiliev invariants can distinguish all knots. For ”braids”, yes (Kohno, ’95). For ”string links up to homotopy”, yes (X. S. Lin, ’92).

Lena Folwaczny (UIC) Finite Type Invariants Open: Can Vassiliev invariants detect mutant knots? (There is no ﬁnite type invariant of degree < 11 which can distinguish mutant knots.) Open: Can the signature of a knot be expressed as a Vassiliev Invariant?

Conjectures and Open Problems

Lena Folwaczny (UIC) Finite Type Invariants Open: Can the signature of a knot be expressed as a Vassiliev Invariant?

Conjectures and Open Problems

Conjecture: Finite type Vassiliev invariants can distinguish all knots. For ”braids”, yes (Kohno, ’95). For ”string links up to homotopy”, yes (X. S. Lin, ’92). Open: Do Vassiliev invariants detect knot orientation? Open: Can Vassiliev invariants detect mutant knots? (There is no ﬁnite type invariant of degree < 11 which can distinguish mutant knots.)

Lena Folwaczny (UIC) Finite Type Invariants Conjectures and Open Problems

Lena Folwaczny (UIC) Finite Type Invariants Thank you!

Lena Folwaczny (UIC) Finite Type Invariants