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. Lena Folwaczny (UIC) Finite Type Invariants + + 1 2 ? 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 + + 1 2 ? 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 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 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 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. 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) 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 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 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 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 2 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 2 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.