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 ξ std in R 3 is a plane distribution defined by the kernel of a 1-form α std = dz + xdy . z 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 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 II III I II G 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 dest. I com. 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 �→ G R 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 ⊂ R 3 satisfying the following: α ∪ � ◮ the disc boundary ∂ D coincides with � 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 α 1 ( α 1 ∪ β 1 ) ր β 1 ( α 1 ∪ β 1 ) ւ R ′ R 1 1 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 e ′ ℓ X Y X e e ′ e a) e ′ Y X Y c) e b) 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 ′ X Y X e ′ e e ′ e Y X Y e 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 β in R 3 that are and β we associate geometrical objects ˆ α, ˆ R , ˆ 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 3 3 6 2 6 4 2 4 1 1 5 5 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 1 2 2 3 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 R is isotopic to ˜ ˆ R . Ivan Dynnikov, Maxim Prasolov Rectangular Diagrams and Jones’ Conjecture II: Legendrian Graphs,

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