Knots in S 3 and minimal surfaces in B 4 joint work with Marc Soret Marina Ville Universit´ e de Tours, France Institut Henri Poincar´ e, June 22th, 2018
Paul Laurain, Image des maths Institut Henri Poincar´ e, June 22th, 2018 Knots in S 3 and minimal surfaces in B 4 Marina Ville (Universit´ e de Tours, France) / 32
Minimal surfaces in R 4 critical point for the area in any deformation with compact support � � � � d ( area ( S t )) | t =0 = 0 dt Institut Henri Poincar´ e, June 22th, 2018 Knots in S 3 and minimal surfaces in B 4 Marina Ville (Universit´ e de Tours, France) / 32
Minimal surfaces in R 4 critical point for the area in any deformation with compact support Harmonic map → C 2 = R 4 D − � � z �→ ( e ( z ) + ¯ f ( z ) , g ( z ) + ¯ h ( z )) � � e, f, g, h holomorphic d ( area ( S t )) | t =0 = 0 dt Institut Henri Poincar´ e, June 22th, 2018 Knots in S 3 and minimal surfaces in B 4 Marina Ville (Universit´ e de Tours, France) / 32
Minimal surfaces in R 4 critical point for the area in any deformation with compact support Harmonic map → C 2 = R 4 D − � � z �→ ( e ( z ) + ¯ f ( z ) , g ( z ) + ¯ h ( z )) � � e, f, g, h holomorphic Conformality condition d ( area ( S t )) e ′ f ′ + g ′ h ′ = 0 | t =0 = 0 dt Institut Henri Poincar´ e, June 22th, 2018 Knots in S 3 and minimal surfaces in B 4 Marina Ville (Universit´ e de Tours, France) / 32
Minimal surfaces in R 4 critical point for the area in any deformation with compact support Harmonic map → C 2 = R 4 D − � � z �→ ( e ( z ) + ¯ f ( z ) , g ( z ) + ¯ h ( z )) � � e, f, g, h holomorphic Conformality condition d ( area ( S t )) e ′ f ′ + g ′ h ′ = 0 | t =0 = 0 dt EXEMPLE. Complex curves in C 2 = R 4 . Institut Henri Poincar´ e, June 22th, 2018 Knots in S 3 and minimal surfaces in B 4 Marina Ville (Universit´ e de Tours, France) / 32
Ribbon knots K in R 3 (or S 3 ) is ribbon if K bounds a disk with ribbon singularities Institut Henri Poincar´ e, June 22th, 2018 Knots in S 3 and minimal surfaces in B 4 Marina Ville (Universit´ e de Tours, France) / 32
Ribbon knots TH (Hass, 1983): a knot in S 3 is K in R 3 (or S 3 ) is ribbon if K ribbon iff it bounds an bounds a disk with embedded minimal disk ∆ in B 4 REMARK. Harmonic parametrization == > the restriction of d (0 , . ) to ∆ has no ribbon singularities local maxima. Institut Henri Poincar´ e, June 22th, 2018 Knots in S 3 and minimal surfaces in B 4 Marina Ville (Universit´ e de Tours, France) / 32
Torus knots K (3 , 7) torus knot In R 3 , the parameter goes N times around a circle C in a vertical plane while C rotates p times around Oz . Institut Henri Poincar´ e, June 22th, 2018 Knots in S 3 and minimal surfaces in B 4 Marina Ville (Universit´ e de Tours, France) / 32
Torus knots K (3 , 7) torus knot In R 3 , the parameter goes N times around a circle C in a vertical plane while C rotates p times around Oz . In S 3 , K ( N, p ) : S 1 − → S 3 e iθ �→ ( 1 2 e Niθ , 1 2 e piθ ) √ √ inside the Clifford torus Institut Henri Poincar´ e, June 22th, 2018 Knots in S 3 and minimal surfaces in B 4 Marina Ville (Universit´ e de Tours, France) / 32
Torus knots Algebraic curves C N,p = { ( z 1 , z 2 ) z p 1 = z N 2 } K (3 , 7) torus knot In R 3 , the parameter goes N times around a circle C in a vertical plane while C rotates p times around Oz . In S 3 , K ( N, p ) : S 1 − → S 3 e iθ �→ ( 1 2 e Niθ , 1 2 e piθ ) √ √ inside the Clifford torus Institut Henri Poincar´ e, June 22th, 2018 Knots in S 3 and minimal surfaces in B 4 Marina Ville (Universit´ e de Tours, France) / 32
Torus knots Algebraic curves C N,p = { ( z 1 , z 2 ) z p 1 = z N 2 } parametrized near (0 , 0) by K (3 , 7) torus knot z �→ ( z N , z p ) In R 3 , the parameter goes N times around a circle C in a vertical plane while C rotates p times around Oz . In S 3 , K ( N, p ) : S 1 − → S 3 e iθ �→ ( 1 2 e Niθ , 1 2 e piθ ) √ √ inside the Clifford torus Institut Henri Poincar´ e, June 22th, 2018 Knots in S 3 and minimal surfaces in B 4 Marina Ville (Universit´ e de Tours, France) / 32
Torus knots Algebraic curves C N,p = { ( z 1 , z 2 ) z p 1 = z N 2 } parametrized near (0 , 0) by K (3 , 7) torus knot z �→ ( z N , z p ) In R 3 , the parameter goes N times around a circle C in a vertical plane while C ex: cusp z 3 1 = z 2 2 (drawn in R 2 !) rotates p times around Oz . In S 3 , � K ( N, p ) : S 1 − → S 3 K ( N, p ) = C N,p ∩ S 3 ����� ǫ e iθ �→ ( 1 2 e Niθ , 1 2 e piθ ) NB. (0 , 0) is a branch point; C N,p is not √ √ a smooth near (0 , 0) but it has a inside the Clifford torus tangent plane Institut Henri Poincar´ e, June 22th, 2018 Knots in S 3 and minimal surfaces in B 4 Marina Ville (Universit´ e de Tours, France) / 32
Minimal knots → R 4 minimal F : D − Institut Henri Poincar´ e, June 22th, 2018 Knots in S 3 and minimal surfaces in B 4 Marina Ville (Universit´ e de Tours, France) / 32
Minimal knots → R 4 minimal F : D − If 0 is a critical point of F , it is a branch point (lowest order term is conformal): in a neighbourhood of F , F ( z ) = ( z N + o ( z N ) , o ( z N )) Institut Henri Poincar´ e, June 22th, 2018 Knots in S 3 and minimal surfaces in B 4 Marina Ville (Universit´ e de Tours, France) / 32
Minimal knots → R 4 minimal F : D − If 0 is a critical point of F , it is a branch point (lowest order term is conformal): in a neighbourhood of F , F ( z ) = ( z N + o ( z N ) , o ( z N )) Assume that F is injective in a neighbourhood of 0 (i.e. F ( D ) has no codimension 1 singularities). For a small ǫ > 0, set K ǫ = F ( D ) ∩ S 3 ǫ Institut Henri Poincar´ e, June 22th, 2018 Knots in S 3 and minimal surfaces in B 4 Marina Ville (Universit´ e de Tours, France) / 32
Minimal knots → R 4 minimal F : D − If 0 is a critical point of F , it is a branch point (lowest order term is conformal): in a neighbourhood of F , F ( z ) = ( z N + o ( z N ) , o ( z N )) Assume that F is injective in a neighbourhood of 0 (i.e. F ( D ) has no codimension 1 singularities). For a small ǫ > 0, set K ǫ = F ( D ) ∩ S 3 ǫ For ǫ small enough, the type of the knot does not depend on ǫ . There is a homeomorphism ǫ , K ǫ ) ∼ Cone ( S 3 = ( B 4 , F ( D )) WHO ARE THE KNOTS OF BRANCH POINTS OF MINIMAL DISKS??? CAN THEY BE ALL THE KNOTS?????? Institut Henri Poincar´ e, June 22th, 2018 Knots in S 3 and minimal surfaces in B 4 Marina Ville (Universit´ e de Tours, France) / 32
Constructing the knot RECALL −− > Coordinate functions of a minimal surfaces are harmonic. So Each of the 4 components of the minimal disk is a series in z = re iθ and ¯ z = re − iθ . We truncate each component by larger and larger powers of r : as soon as we get something injective, we can stop and we have the knot type. Institut Henri Poincar´ e, June 22th, 2018 Knots in S 3 and minimal surfaces in B 4 Marina Ville (Universit´ e de Tours, France) / 32
Constructing the knot RECALL −− > Coordinate functions of a minimal surfaces are harmonic. So Each of the 4 components of the minimal disk is a series in z = re iθ and ¯ z = re − iθ . We truncate each component by larger and larger powers of r : as soon as we get something injective, we can stop and we have the knot type. SIMPLEST CASE. We can stop at the lowest order term of each of the 4 components. ( r N cos( Nθ ) , r N sin( Nθ ) , r p cos( pθ + φ ) , r q sin( qθ )) Institut Henri Poincar´ e, June 22th, 2018 Knots in S 3 and minimal surfaces in B 4 Marina Ville (Universit´ e de Tours, France) / 32
Constructing the knot RECALL −− > Coordinate functions of a minimal surfaces are harmonic. So Each of the 4 components of the minimal disk is a series in z = re iθ and ¯ z = re − iθ . We truncate each component by larger and larger powers of r : as soon as we get something injective, we can stop and we have the knot type. SIMPLEST CASE. We can stop at the lowest order term of each of the 4 components. ( r N cos( Nθ ) , r N sin( Nθ ) , r p cos( pθ + φ ) , r q sin( qθ )) p = q , ( N, q ) torus knot. Institut Henri Poincar´ e, June 22th, 2018 Knots in S 3 and minimal surfaces in B 4 Marina Ville (Universit´ e de Tours, France) / 32
Constructing the knot RECALL −− > Coordinate functions of a minimal surfaces are harmonic. So Each of the 4 components of the minimal disk is a series in z = re iθ and ¯ z = re − iθ . We truncate each component by larger and larger powers of r : as soon as we get something injective, we can stop and we have the knot type. SIMPLEST CASE. We can stop at the lowest order term of each of the 4 components. ( r N cos( Nθ ) , r N sin( Nθ ) , r p cos( pθ + φ ) , r q sin( qθ )) p = q , ( N, q ) torus knot. p � = q Lissajous toric knot Institut Henri Poincar´ e, June 22th, 2018 Knots in S 3 and minimal surfaces in B 4 Marina Ville (Universit´ e de Tours, France) / 32
Lissajous toric knots Lissajous curve C q,p,φ in a vertical plane t �→ (sin qt, cos( pθ + φ )) Type I: (sin(2 t ) ; cos(3 t )) ; 0 � t � 2 � http://mathserver.neu.edu / bridger/U170/Lissajous/Lissajous.pdf Institut Henri Poincar´ e, June 22th, 2018 Knots in S 3 and minimal surfaces in B 4 Marina Ville (Universit´ e de Tours, France) / 32
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