Introduction Topology (triangulations) Geometry (angles) Algebra (TQFT) Analysis (asymptotics) Conclusion The Teichm¨ uller TQFT Volume Conjecture for Twist Knots Fathi Ben Aribi UCLouvain 24th September 2020 (joint work with Fran¸ cois Gu´ eritaud and Eiichi Piguet-Nakazawa) arXiv:1903.09480 1/22 Fathi Ben Aribi The Teichm¨ uller TQFT Volume Conjecture for Twist Knots
Introduction Topology (triangulations) Geometry (angles) Algebra (TQFT) Analysis (asymptotics) Conclusion How you can follow/use this talk : Live : You can download the slides on the K-OS website (helps for following recurring notations). Recurring example (the figure-eight knot): Slides 6, 8, 10, 18. From the future : downloading the slides can also help! (eventual mistakes will have been hopefully corrected at this point). From the future and you are interested in our paper : the pictures and main example in these slides can be a good complement to the technical details in arXiv:1903.09480. 2/22 Fathi Ben Aribi The Teichm¨ uller TQFT Volume Conjecture for Twist Knots
Introduction Topology (triangulations) Geometry (angles) Algebra (TQFT) Analysis (asymptotics) Conclusion Our Goal Proving the Teichm¨ ullerTQFT volume conjecture for twistknots . ··· crossings n 0 Context : quantum topology , volume conjectures . 1 Topology : triangulating the twist knot complements 2 Geometry : the triangulations contain the hyperbolicity 3 Algebra : computing the Teichm¨ uller TQFT 4 Analysis : the hyperbolic volume appears asymptotically (Optional: parts/sketches of proofs, at the audience’s preference) 3/22 Fathi Ben Aribi The Teichm¨ uller TQFT Volume Conjecture for Twist Knots
Introduction Topology (triangulations) Geometry (angles) Algebra (TQFT) Analysis (asymptotics) Conclusion ’84: Jones polynomial , new knot invariant. ’90: Witten retrieves the Jones polynomial via quantum physics . 90s: New topological invariants (TQFTs of Reshitikin-Turaev , Turaev-Viro , . . . ) are discovered via the intuition from physics. Andersen-Kashaev ’11: Teichm¨ uller TQFT of a triangulated 3-manifold M , an ” infinite-dimensional TQFT”. Its partition function { Z b ( M ) ∈ C } b > 0 yields an invariant of M . Volume Conjecture (Andersen-Kashaev ’11) If M is a triangulated hyperbolic knot complement, then its hyperbolic volume Vol ( M ) appears as an exponential decrease rate in Z b ( M ) for the limit b → 0 + . 4/22 Fathi Ben Aribi The Teichm¨ uller TQFT Volume Conjecture for Twist Knots
Introduction Topology (triangulations) Geometry (angles) Algebra (TQFT) Analysis (asymptotics) Conclusion SEMI-CLASSICAL QUANTUM HYPERBOLIC TOPOLOGY LIMIT INVARIANTS GEOMETRY TWIST VOLUME KNOTS K n CONJECTURES vol ( S 3 \ K ) q = e 2 i π/ N , N → ∞ ∼ e N J K ( N , q ) 2 π Diagram Sums Colored Jones polynomials of knot K Hyperbolic Integrals Saddle point Triangulation Teichm¨ uller Volume method Integrals of S 3 \ K TQFT − vol ( S 3 \ K ) 1 Z b ( S 3 \ K ) ∼ e b 2 2 π TH 1 TH 2 b → 0 + TH 4 TH 3 . . . 5/22 Fathi Ben Aribi The Teichm¨ uller TQFT Volume Conjecture for Twist Knots
Introduction Topology (triangulations) Geometry (angles) Algebra (TQFT) Analysis (asymptotics) Conclusion Our tetrahedra have ordered vertices ( ⇒ oriented edges too). ❀ two possible signs ǫ ( T ) ∈ {±} . A triangulation X = ( T 1 , . . . , T N , ∼ ) of a 3-manifold M is the datum of N tetrahedra and a gluing relation ∼ pairing their faces while respecting the vertex order . We consider ideal triangulations of open 3-manifolds, i.e. where the tetrahedra have their vertices removed . 1 1 T + T − B C − 3 2 1 0 C A D B S 3 \ = = 0 0 3 2 2 3 D A 0 1 2 3 + X 3 = { T + , T − } , X 2 = { A , B , C , D } , X 1 = {→ , ։ } , X 0 = {·} face maps x 0 , . . . , x 3 : X 3 → X 2 , for example x 0 ( T + ) = B . 6/22 Fathi Ben Aribi The Teichm¨ uller TQFT Volume Conjecture for Twist Knots
Introduction Topology (triangulations) Geometry (angles) Algebra (TQFT) Analysis (asymptotics) Conclusion Thurston: from a diagram of a knot K , one can construct an ideal triangulation X of the knot complement M = S 3 \ K . − + + + − ··· tetrahedra p crossings n − The n -th twist knot K n and the triangulation X n ( n odd , p = n − 3 2 ) Theorem (TH 1, B.A.-P.N. ’18) For all n � 2 , we construct an ideal triangulation X n of the � n + 4 � complement of the twist knot K n , with tetrahedra. 2 7/22 Fathi Ben Aribi The Teichm¨ uller TQFT Volume Conjecture for Twist Knots
Introduction Topology (triangulations) Geometry (angles) Algebra (TQFT) Analysis (asymptotics) Conclusion Sketch of proof of TH1: First draw a tetrahedron around each crossing of K , whose diagram lives in the equatorial plane of S 3 . ≻ (observer) B 3 ( · + F A B B A E E F . . . B 3 − Then collapse the tetrahedra into segments ( K ❀ · ). Hence the collapsed S 3 decomposes into two polyhedra . Finally, triangulate the two polyhedra (several possible ways). 8/22 Fathi Ben Aribi The Teichm¨ uller TQFT Volume Conjecture for Twist Knots
Introduction Topology (triangulations) Geometry (angles) Algebra (TQFT) Analysis (asymptotics) Conclusion (2 , 3)- Pachner moves are moves between ideal triangulations . Matveev-Piergallini: X and X ′ triangulate the same M if and only if they are related by a finite sequence of Pachner moves. ⇒ Useful for constructing topological invariants for M . source of the picture: Wikipedia 9/22 Fathi Ben Aribi The Teichm¨ uller TQFT Volume Conjecture for Twist Knots
Introduction Topology (triangulations) Geometry (angles) Algebra (TQFT) Analysis (asymptotics) Conclusion A X is the space of angle structures on X = ( T 1 , . . . , T N , ∼ ), i.e. of 3 N -tuples α ∈ (0 , π ) 3 N of dihedral angles on edges, such that the angle sum is π at each vertex and 2 π around each edge. 1 1 T + T − B C S 3 \ = α + α + α + α − α − α − 2 1 3 3 1 2 C A D B α + α + α − α − 0 0 3 2 2 3 3 D 2 2 A 3 α + α − 1 1 α + � α + 1 + α + 2 + α + 3 = π 1 � α − 1 + α − 2 + α − α + � 3 = π π 2 � 3 α + � . ∈ (0 , π ) 6 � 3 . A X = α = ∋ α − 3 + 2 α − 2 + α − . � ( → ) 2 α + 1 + α + 3 = 2 π � 1 α − � π 3 � 2 α − � 3 + 2 α − 1 + α − ( ։ ) 2 α + 2 + α + 3 = 2 π � 3 α fixed ❀ angle maps α 1 , α 2 , α 3 : X 3 → R , for example α 2 ( T + ) = α + 2 . 10/22 Fathi Ben Aribi The Teichm¨ uller TQFT Volume Conjecture for Twist Knots
Introduction Topology (triangulations) Geometry (angles) Algebra (TQFT) Analysis (asymptotics) Conclusion The 3-dimensional hyperbolic space is H 3 = R 2 × R > 0 with ( ds ) 2 = ( dx ) 2 + ( dy ) 2 + ( dz ) 2 , z 2 a metric which has constant curvature − 1. A knot is hyperbolic if its complement M can be endowed with a complete hyperbolic metric of finite volume Vol ( M ). ❀ a specific α ∈ A X on X = ( T 1 , . . . , T N , ∼ ) triangulation of M . ∞ α 1 α 3 α 2 α 2 α 3 α 1 � α 1 + α 2 + α 3 = π edge α j = 2 π (+ others) → H 3 T ֒ gluing gives a manifold 11/22 Fathi Ben Aribi The Teichm¨ uller TQFT Volume Conjecture for Twist Knots
Introduction Topology (triangulations) Geometry (angles) Algebra (TQFT) Analysis (asymptotics) Conclusion For all n � 2, the twist knot K n is hyperbolic . Theorem (TH2, B.A.-G.-P.N. ’20) For all n � 2 , the triangulation X n of S 3 \ K n is geometric , i.e. it admits an angle structure α 0 ∈ A X n corresponding to the complete hyperbolic structure on the complement of K n . X geometric ⇔ ∃ solution to the nonlinear gluing equations of X (difficult!) Casson-Rivin, Futer-Gu´ eritaud: approach via A X , the solutions to the linear part: maximising the volume fonctional . 12/22 Fathi Ben Aribi The Teichm¨ uller TQFT Volume Conjecture for Twist Knots
Introduction Topology (triangulations) Geometry (angles) Algebra (TQFT) Analysis (asymptotics) Conclusion � z 0 log(1 − u ) du Dilogarithm function: Li 2 ( z ) = − for z ∈ C \ [1 , ∞ ). u Volume functional Vol : A X → R � 0 ( strictly concave ) is: � Vol ( α ) := ℑ Li 2 ( z ( T )) + arg(1 − z ( T )) log | z ( T ) | , T ∈ X 3 ǫ ( T ) � sin α 3 ( T ) � e i α 1 ( T ) ∈ R + i R > 0 encodes the angles of T . where z ( T ) = sin α 2 ( T ) Theorem (TH2, B.A.-G.-P.N. ’20) For all n � 2 , the triangulation X n of S 3 \ K n is geometric , i.e. it admits an angle structure α 0 ∈ A X n corresponding to the complete hyperbolic structure on the complement of K n . Sketch of proof of TH2: Check that the open polyhedron A X is non-empty . General fact: the complete structure α 0 exists ⇔ max Vol is reached in A X . A X Prove that max Vol cannot be on ∂ A X (case-by-case). A X 13/22 Fathi Ben Aribi The Teichm¨ uller TQFT Volume Conjecture for Twist Knots
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