On the Visibility of Achirality of Alternating Knots

Transcription

1 University of Geneva August 22nd, 2014 Busan, Korea

2 Definition A knot K S 3 is achiral there is an orientation reversing diffeomorphism φ of S 3 s.t. φ(k) = K, φ is a mirror diffeomorphism of K When orienting K, we distinguish 2 cases : K is +achiral if φ preserves the orientation of K. K is achiral if φ reverses the orientation of K.

3 Examples 1 The eight-knot is both + and achiral. 2 The DH-knot described as below is only +achiral

4 Achiral prime alternating knots are periodically achiral since they are hyperbolic (the only non hyperbolic prime alternating knots belong to the class of torus knots which cannot be achiral). Definition The minimum period of periodic mirror diffeomorphisms of a hyperbolic ±achiral knot K is the order of ±achirality of K. Hence if K is prime alternating and : K achiral its order of achirality is 2. K +achiral its order of +achirality is 2 n.

5 Notation Let Π be an oriented projection of K in S 2. Then : ˆΠ denotes the mirror image of Π (obtained by switching all the crossings) by preserving the orientation of Π. ˆΠ is the mirror image by reversing the orientation of Π. Π Π

6 The main tool in our study of achirality is the Tait s Flyping Conjecture proven by Menasco and Thistlethwaite (1991). Recall that a flype is a transformation of projections described as below : A A B (a) B (b)

7 For achirality we will use the following consequence of the Tait s Flyping Conjecture : Theorem (Key Theorem) Let K be a prime oriented alternating knot and Π a minimal projection of K. Then K is ±achiral Π is related to ±ˆΠ by a finite sequence of flypes and of o.p. diffeomorphism of S 2.

8 Question From any minimal projection Π of a prime alternating knot K can we detect the achirality of K? Answer : The answer is rather positive. Applying the Seifert algorithm on any oriented minimal projection Π, one can express K as a diagrammatic Murasugi sum of prime special alternating links L i : K = L 1 L 2... L n (special = all the crossings are of the same sign.) Terminology : L i = blocks (Stoimenow) or Murasugi atoms (Q.-Weber). The Murasugi atoms do not depend on the choice of the minimal projection.

9 Example L 1 K L 2K= L1 L 2 * One has the following criteria for non-achirality : Theorem (Stoimenov, Q.-Weber) If the Murasugi atoms of K cannot be grouped into pairs (L i,ˆl i ) then K is not achiral. (Recall : ˆL i is mirror image of L i ).

10 In fact we have : Theorem Let K be an oriented alternating and achiral knot then K = L ±ˆL Proof Based on the Key Theorem. Theorem (Stoimenov, Q.-Weber) K oriented alternating achiral. Then the leading coefficient of the Conway polynomial is = up to sign to a square of an integer.

11 Question Given a prime achiral alternating knot is there a minimal projection where the achirality is visible? Such a projection is called a minimal achiral projection. Example :

12 Conjecture (Kauffman) A prime alternating achiral knot has a minimal projection Π s.t. the checkboard graph G(Π) and its dual G (Π) are isomorphic. (In fact G (Π) G(ˆΠ)). G(Π)

13 In fact, Theorem (Ermotti-Q.-Weber, 2012) Kauffman s Conjecture is true for prime achiral alternating knots. ( Tait s Conjecture for achiral alternating knots ). However this conjecture is not true in general. The first counterexample is given by Dasbach-Hougardy (1996), the DH-knot, which has 14 crossings.

14 Observations 1 The DH-knot is arborescent The DH-knot is +achiral but not achiral.

15 In 2010 Jablan and Kauffman exhibited infinite families of arborescent achiral alternating knots and put forward the following conjecture : Conjecture (Jablan-Kauffman) K prime achiral alternating knot with no minimal achiral projection. Then K is arborescent. However the Jablan-Kauffman Conjecture is not true.

16 Counterexample : F F^ F

17 Remark A prime alternating +achiral knot which is arborescent has its order of +achirality equal to 4. Replacing in the Jablan-Kauffman Conjecture the conclusion arborescent by order of +achirality 4 one has the following theorem : Theorem (Ermotti-Q.-Weber, 2014) K oriented prime alternating achiral. If K has no minimal achiral projection then K is +achiral and its order of +achirality is 4.

18 Thank you!!!