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Rectangular Diagrams of Legendrian Graphs Θ−diagrams and bypasses Destabilize the diagram during disc simplification

Rectangular Diagrams and Jones’ Conjecture II: Legendrian Graphs, bypasses, and simplifying discs

Ivan Dynnikov, Maxim Prasolov

Moscow State University

Ivan Dynnikov, Maxim Prasolov Rectangular Diagrams and Jones’ Conjecture II: Legendrian Graphs,

Rectangular Diagrams of Legendrian Graphs Θ−diagrams and bypasses Destabilize the diagram during disc simplification

Rectangular paths

Ivan Dynnikov, Maxim Prasolov Rectangular Diagrams and Jones’ Conjecture II: Legendrian Graphs,

Rectangular Diagrams of Legendrian Graphs Θ−diagrams and bypasses Destabilize the diagram during disc simplification

Bypasses

Rectangular diagram with rectangular path attached = Θ−diagram.

Ivan Dynnikov, Maxim Prasolov Rectangular Diagrams and Jones’ Conjecture II: Legendrian Graphs,

Rectangular Diagrams of Legendrian Graphs Θ−diagrams and bypasses Destabilize the diagram during disc simplification

Standard Contact Structure

The standard contact structure ξstd in R3 is a plane distribution defined by the kernel of a 1-form αstd = dz + xdy.

x y z

Ivan Dynnikov, Maxim Prasolov Rectangular Diagrams and Jones’ Conjecture II: Legendrian Graphs,

Rectangular Diagrams of Legendrian Graphs Θ−diagrams and bypasses Destabilize the diagram during disc simplification

Fronts

Front Projection of a Legendrian graph.

Ivan Dynnikov, Maxim Prasolov Rectangular Diagrams and Jones’ Conjecture II: Legendrian Graphs,

Rectangular Diagrams of Legendrian Graphs Θ−diagrams and bypasses Destabilize the diagram during disc simplification

Front moves

I II III IIG R

• Theorem. (Baader and Ishikawa, 2009) Two generic fronts

represent Legendrian isotopic Legendrian graphs iff they are related by moves which are illustrated on the picture.

Ivan Dynnikov, Maxim Prasolov Rectangular Diagrams and Jones’ Conjecture II: Legendrian Graphs,

Rectangular Diagrams of Legendrian Graphs Θ−diagrams and bypasses Destabilize the diagram during disc simplification

Non-isotopy move: Blow-up and edge contraction

Ivan Dynnikov, Maxim Prasolov Rectangular Diagrams and Jones’ Conjecture II: Legendrian Graphs,

Rectangular Diagrams of Legendrian Graphs Θ−diagrams and bypasses Destabilize the diagram during disc simplification

Generalized Rectangular Diagrams

Ivan Dynnikov, Maxim Prasolov Rectangular Diagrams and Jones’ Conjecture II: Legendrian Graphs,

Rectangular Diagrams of Legendrian Graphs Θ−diagrams and bypasses Destabilize the diagram during disc simplification

Elementary moves: cyclic permutation

Ivan Dynnikov, Maxim Prasolov Rectangular Diagrams and Jones’ Conjecture II: Legendrian Graphs,

Rectangular Diagrams of Legendrian Graphs Θ−diagrams and bypasses Destabilize the diagram during disc simplification

Elementary moves: commutation

Ivan Dynnikov, Maxim Prasolov Rectangular Diagrams and Jones’ Conjecture II: Legendrian Graphs,

Rectangular Diagrams of Legendrian Graphs Θ−diagrams and bypasses Destabilize the diagram during disc simplification

Elementary moves: (de)stabilization

← → ← → ← → ← → type I type II

Ivan Dynnikov, Maxim Prasolov Rectangular Diagrams and Jones’ Conjecture II: Legendrian Graphs,

Rectangular Diagrams of Legendrian Graphs Θ−diagrams and bypasses Destabilize the diagram during disc simplification

Elementary moves: end shift

type I type II

Ivan Dynnikov, Maxim Prasolov Rectangular Diagrams and Jones’ Conjecture II: Legendrian Graphs,

Rectangular Diagrams of Legendrian Graphs Θ−diagrams and bypasses Destabilize the diagram during disc simplification

Elementary moves: inverting an end shift

e.s. I e.s. I com.

• dest. I

Ivan Dynnikov, Maxim Prasolov Rectangular Diagrams and Jones’ Conjecture II: Legendrian Graphs,

Rectangular Diagrams of Legendrian Graphs Θ−diagrams and bypasses Destabilize the diagram during disc simplification

From rectangular diagrams to Legendrian graphs

Ivan Dynnikov, Maxim Prasolov Rectangular Diagrams and Jones’ Conjecture II: Legendrian Graphs,

Rectangular Diagrams of Legendrian Graphs Θ−diagrams and bypasses Destabilize the diagram during disc simplification

From rectangular diagrams to Legendrian graphs

• Theorem. The map R → GR induces a bijection between classes of

generalized rectangular diagrams modulo elementary moves of type I and Legendrian graphs modulo Legendrian isotopy and edge contraction / blow-up.

Ivan Dynnikov, Maxim Prasolov Rectangular Diagrams and Jones’ Conjecture II: Legendrian Graphs,

Rectangular Diagrams of Legendrian Graphs Θ−diagrams and bypasses Destabilize the diagram during disc simplification

Bypasses

By a bypass for a rectangular diagram R we call an ordered pair (α, β) of rectangular paths having common ends such that β is a subset of R, and there exists an embedded two-dimensional disc D ⊂ R3 satisfying the following:

◮ the disc boundary ∂D coincides with

α ∪ β;

◮ the intersection D ∩

R coincides with β;

◮ in the link defined by the rectangular diagram

(R \ β) ∪ α ∪ (α ∪ β)ր ∪ (α ∪ β)ւ, the components presented by (R \ β) ∪ α are unlinked with the two others. A bypass (α, β) is called elementary if we have tb(α ∪ β) = 1. The value tb(α ∪ β) will be called the weight of the bypass (α, β).

Ivan Dynnikov, Maxim Prasolov Rectangular Diagrams and Jones’ Conjecture II: Legendrian Graphs,

Rectangular Diagrams of Legendrian Graphs Θ−diagrams and bypasses Destabilize the diagram during disc simplification

Elementary bypass

R1 β1 α1 R′

1

(α1 ∪ β1)ր (α1 ∪ β1)ւ

Ivan Dynnikov, Maxim Prasolov Rectangular Diagrams and Jones’ Conjecture II: Legendrian Graphs,

Rectangular Diagrams of Legendrian Graphs Θ−diagrams and bypasses Destabilize the diagram during disc simplification

Applying an end shift to the Θ−diagram

Ivan Dynnikov, Maxim Prasolov Rectangular Diagrams and Jones’ Conjecture II: Legendrian Graphs,

Rectangular Diagrams of Legendrian Graphs Θ−diagrams and bypasses Destabilize the diagram during disc simplification

An attached path is avoiding the diagram during Legendrian move

ℓ e e′ Y X a) e e′ Y X b) e e′ Y X c)

Ivan Dynnikov, Maxim Prasolov Rectangular Diagrams and Jones’ Conjecture II: Legendrian Graphs,

Rectangular Diagrams of Legendrian Graphs Θ−diagrams and bypasses Destabilize the diagram during disc simplification

An attached path is avoiding the diagram during Legendrian move

e e′ Y X e e′ Y X e e′ Y X

Ivan Dynnikov, Maxim Prasolov Rectangular Diagrams and Jones’ Conjecture II: Legendrian Graphs,

Rectangular Diagrams of Legendrian Graphs Θ−diagrams and bypasses Destabilize the diagram during disc simplification

Bypasses

• Proposition. Let R and R′ be Legendrian equivalent rectangular

diagrams such that α is a bypass of weight b for R. Then there exists a bypass α′ of weight b such that Θ−diagrams R ∪ α and R′ ∪ α′ are Legendrian equivalent.

Ivan Dynnikov, Maxim Prasolov Rectangular Diagrams and Jones’ Conjecture II: Legendrian Graphs,

Rectangular Diagrams of Legendrian Graphs Θ−diagrams and bypasses Destabilize the diagram during disc simplification

Plan

◮ With the rectangular diagram R and the rectangular paths α

and β we associate geometrical objects ˆ R, ˆ α, ˆ β in R3 that are called arc presentations.

◮ We span the trivial knot

α ∪ β by a disc D whose open book foliation obey certain restrictions.

◮ Then we apply induction. For the induction step we modify

ˆ R ∪ D in a certain way so that the disc D gets simpler. As a result, type N destabilizations may occur on the path β, type L destabilizations on α, and commutations as well as cyclic permutations may occur everywhere in the Θ-diagram R ∪ α.

Ivan Dynnikov, Maxim Prasolov Rectangular Diagrams and Jones’ Conjecture II: Legendrian Graphs,

Rectangular Diagrams of Legendrian Graphs Θ−diagrams and bypasses Destabilize the diagram during disc simplification

Arc-presentation

1 2 3 6 5 4 1 2 3 4 5 6

Ivan Dynnikov, Maxim Prasolov Rectangular Diagrams and Jones’ Conjecture II: Legendrian Graphs,

Rectangular Diagrams of Legendrian Graphs Θ−diagrams and bypasses Destabilize the diagram during disc simplification

Arc-presentation

1 2 3 1 2 3

Ivan Dynnikov, Maxim Prasolov Rectangular Diagrams and Jones’ Conjecture II: Legendrian Graphs,

Rectangular Diagrams of Legendrian Graphs Θ−diagrams and bypasses Destabilize the diagram during disc simplification

Exercise

ˆ R is isotopic to ˜ R.

Ivan Dynnikov, Maxim Prasolov Rectangular Diagrams and Jones’ Conjecture II: Legendrian Graphs,

Rectangular Diagrams of Legendrian Graphs Θ−diagrams and bypasses Destabilize the diagram during disc simplification

Restrictions on the disc

If (α, β) is a bypass for R then there exists a disc D such that

◮ the boundary ∂D coincides with

α ∪ β, and the interior of D is disjoint from ˆ R ∪ ˆ α;

◮ D is a smooth image of a polygon whose vertices map exactly

to vertices of α ∪ β;

◮ D intersects the binding of open book transversely finitely

many times;

◮ for any common endpoint of ˆ

α and ˆ β the arc of R \ β coming from the endpoint lies outside the θ−interval occupied by D;

Ivan Dynnikov, Maxim Prasolov Rectangular Diagrams and Jones’ Conjecture II: Legendrian Graphs,

Rectangular Diagrams of Legendrian Graphs Θ−diagrams and bypasses Destabilize the diagram during disc simplification

Suitable disc: the 4th restriction

∂D ∂D

• R

\ β ℓ correct D ∂ D

• R \ β

∂D

ℓ D incorrect

Ivan Dynnikov, Maxim Prasolov Rectangular Diagrams and Jones’ Conjecture II: Legendrian Graphs,

Rectangular Diagrams of Legendrian Graphs Θ−diagrams and bypasses Destabilize the diagram during disc simplification

Suitable disc

◮ the open book foliation defined on D outside the binding has

• nly simple saddle singularities inside D and have no regular

closed fibers;

◮ at intersection points of ∂D with the binding the coorientation

• f D is induced by the vector

∂ ∂z ; ◮ there is exactly one positive (respectively, negative) half-saddle

• f the foliation at every arc of α (respectively, β);

◮ all saddles and half-saddles lie in distinct pages.

Ivan Dynnikov, Maxim Prasolov Rectangular Diagrams and Jones’ Conjecture II: Legendrian Graphs,

Rectangular Diagrams of Legendrian Graphs Θ−diagrams and bypasses Destabilize the diagram during disc simplification

Strips forming the band αրւ ∪ βտց

ℓ ℓ

Ivan Dynnikov, Maxim Prasolov Rectangular Diagrams and Jones’ Conjecture II: Legendrian Graphs,

Rectangular Diagrams of Legendrian Graphs Θ−diagrams and bypasses Destabilize the diagram during disc simplification

Band αրւ ∪ βտց viewing topologically

L

• α
• β

S

Ivan Dynnikov, Maxim Prasolov Rectangular Diagrams and Jones’ Conjecture II: Legendrian Graphs,

Rectangular Diagrams of Legendrian Graphs Θ−diagrams and bypasses Destabilize the diagram during disc simplification

A disc D near the boundary

ℓ

• α

∪ β D D ∂D

Ivan Dynnikov, Maxim Prasolov Rectangular Diagrams and Jones’ Conjecture II: Legendrian Graphs,

Rectangular Diagrams of Legendrian Graphs Θ−diagrams and bypasses Destabilize the diagram during disc simplification

Simplifying the charactaristic foliation: rearranging saddles

• α
• α
• β
• β
• α
• β
• α
• β

Ivan Dynnikov, Maxim Prasolov Rectangular Diagrams and Jones’ Conjecture II: Legendrian Graphs,

Rectangular Diagrams of Legendrian Graphs Θ−diagrams and bypasses Destabilize the diagram during disc simplification

Simplifying the charactaristic foliation: removing a 2-valent vertex

Ivan Dynnikov, Maxim Prasolov Rectangular Diagrams and Jones’ Conjecture II: Legendrian Graphs,

Rectangular Diagrams of Legendrian Graphs Θ−diagrams and bypasses Destabilize the diagram during disc simplification

Simplifying the charactaristic foliation: removing a 2-valent vertex

• α
• β
• α
• β

Ivan Dynnikov, Maxim Prasolov Rectangular Diagrams and Jones’ Conjecture II: Legendrian Graphs,

Rectangular Diagrams of Legendrian Graphs Θ−diagrams and bypasses Destabilize the diagram during disc simplification

Rearranging of saddles

S1 S2 ℓ Pθ1 Pθ2 ∆ S1 S2 ℓ ∆ Pθ1 Pθ2 K1 K2

Ivan Dynnikov, Maxim Prasolov Rectangular Diagrams and Jones’ Conjecture II: Legendrian Graphs,

Rectangular Diagrams of Legendrian Graphs Θ−diagrams and bypasses Destabilize the diagram during disc simplification

Rearranging of saddles at boundary

• α

S1 S2 ℓ Pθ1 Pθ2 ∆

• α′

ℓ

Ivan Dynnikov, Maxim Prasolov Rectangular Diagrams and Jones’ Conjecture II: Legendrian Graphs,

Rectangular Diagrams of Legendrian Graphs Θ−diagrams and bypasses Destabilize the diagram during disc simplification

Wrinkle

∆2 ∆1 ℓ

Ivan Dynnikov, Maxim Prasolov Rectangular Diagrams and Jones’ Conjecture II: Legendrian Graphs,

Rectangular Diagrams of Legendrian Graphs Θ−diagrams and bypasses Destabilize the diagram during disc simplification

Removing a wrinkle

P1 P0 P2

Ivan Dynnikov, Maxim Prasolov Rectangular Diagrams and Jones’ Conjecture II: Legendrian Graphs,

Rectangular Diagrams of Legendrian Graphs Θ−diagrams and bypasses Destabilize the diagram during disc simplification

Wrinkle near the boundary

• α
• α ∪ β
• α ∪ β

Ivan Dynnikov, Maxim Prasolov Rectangular Diagrams and Jones’ Conjecture II: Legendrian Graphs,

Rectangular Diagrams of Legendrian Graphs Θ−diagrams and bypasses Destabilize the diagram during disc simplification

Removing a wrinkle near the boundary yields a destabilization

θ z z1 z2 π

Ivan Dynnikov, Maxim Prasolov Rectangular Diagrams and Jones’ Conjecture II: Legendrian Graphs,

Rectangular Diagrams of Legendrian Graphs Θ−diagrams and bypasses Destabilize the diagram during disc simplification

The disc consisting of a single wrinkle

• α
• R \ β
• R \ β
• β

Ivan Dynnikov, Maxim Prasolov Rectangular Diagrams and Jones’ Conjecture II: Legendrian Graphs,

Rectangular Diagrams of Legendrian Graphs Θ−diagrams and bypasses Destabilize the diagram during disc simplification

The disc consisting of a single wrinkle

π z1 z2 z θ β α R \ β R \ β

Ivan Dynnikov, Maxim Prasolov Rectangular Diagrams and Jones’ Conjecture II: Legendrian Graphs,