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Lattice structures of multidimensional continued fractions Oleg Karpenkov, University of Liverpool 8 October 2014 Oleg Karpenkov, University of Liverpool Lattice structure of MCF Contents I. Introduction. II. Klein continued fractions. III.

  1. Next step: empty marked pyramids A marked pyramid is empty if all lattice points distinct to the vertex are in the base. O O B B C C A A Lattice distance equals 1 – any base. Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  2. Next step: empty marked pyramids A marked pyramid is empty if all lattice points distinct to the vertex are in the base. O O B B C C A A Lattice distance equals 1 – any base. Lattice distance is greater than 1 – ??? Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  3. Awful slide Theorem (Karpenkov, 2008) A complete list of 3D empty marked multistory pyramids. — the quadrangular marked pyramids M a , b , with b ≥ a ≥ 1 ; — triangular T ξ a , r , where a ≥ 1 , and gcd( ξ, r ) = 1 , r ≥ 2 , and 0 < ξ ≤ r / 2 ; — the triangular marked pyramids U b , where b ≥ 1 ; — two triangular marked pyramids V and W . Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  4. Awful slide Theorem (Karpenkov, 2008) A complete list of 3D empty marked multistory pyramids. — the quadrangular marked pyramids M a , b , with b ≥ a ≥ 1 ; — triangular T ξ a , r , where a ≥ 1 , and gcd( ξ, r ) = 1 , r ≥ 2 , and 0 < ξ ≤ r / 2 ; — the triangular marked pyramids U b , where b ≥ 1 ; — two triangular marked pyramids V and W . Vertex at the origin. Bases M a , b : (2 , − 1 , 0), (2 , − a − 1 , 1), (2 , − 1 , 2), (2 , b − 1 , 1) T ξ a , r : ( ξ, r − 1 , − r ), ( a + ξ, r − 1 , − r ), ( ξ, r , − r ) U b : (2 , 1 , b − 1), (2 , 2 , − 1), (2 , 0 , − 1) V : (2 , − 2 , 1), (2 , − 1 , − 1), (2 , 1 , 2) W : (3 , 0 , 2), (3 , 1 , 1), (3 , 2 , 3) Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  5. Bases empty marked pyramids (0 , 1) (0 , 1) ( − a, 0) ( b, 0) (0 , − 1) (0 , 0) ( a, 0) T ξ M a,b a,r (2 , 1) (0 , 1) (0 , 1) ( − 1 , 0) ( b, 0) (1 , 0) (0 , − 1) (0 , − 2) ( − 1 , − 1) U b V W Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  6. Bases empty marked pyramids (0 , 1) (0 , 1) ( − a, 0) ( b, 0) (0 , − 1) (0 , 0) ( a, 0) T ξ M a,b a,r (2 , 1) (0 , 1) (0 , 1) ( − 1 , 0) ( b, 0) (1 , 0) (0 , − 1) (0 , − 2) ( − 1 , − 1) U b V W Corollary Any face of MCF at distance > 1 from O is from the list above . This corollary is used in for the algorithm to construct MCF. Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  7. Empty 4D simplices Problem (unsolved, 1964) What happens in 4D with empty simplices? Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  8. Empty 4D simplices Problem (unsolved, 1964) What happens in 4D with empty simplices? Useful filtrations: volume and widths of pyramids or of their faces, distances to the base. Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  9. Empty 4D simplices Problem (unsolved, 1964) What happens in 4D with empty simplices? Useful filtrations: volume and widths of pyramids or of their faces, distances to the base. Problem What faces on distance 1 three dimensional MCF can have? Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  10. Empty 4D simplices Problem (unsolved, 1964) What happens in 4D with empty simplices? Useful filtrations: volume and widths of pyramids or of their faces, distances to the base. Problem What faces on distance 1 three dimensional MCF can have? Problem What about 3D faces? Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  11. Part III III. Minkovskii-Voronoi continued fractions. Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  12. Coaxial sets in general position. Definition A subset S ⊂ R n ≥ 0 is axial if S contains points on each of the coordinate axes. Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  13. Coaxial sets in general position. Definition A subset S ⊂ R n ≥ 0 is axial if S contains points on each of the coordinate axes. Definition An axial subset is in general position if: ◮ Each coordinate plane contains exactly n − 1 points of S none of which are at the origin; these points are on different coordinate axes. ◮ No two points on other plane parallel to a coordinate plane. Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  14. Coaxial sets in general position. Definition A subset S ⊂ R n ≥ 0 is axial if S contains points on each of the coordinate axes. Definition An axial subset is in general position if: ◮ Each coordinate plane contains exactly n − 1 points of S none of which are at the origin; these points are on different coordinate axes. ◮ No two points on other plane parallel to a coordinate plane. Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  15. Minkovskii-Voronoi minima and minimal sets Set max( A , i ) = max { x i | ( x 1 , . . . , x n ) ∈ A } and define the parallelepiped Π( A ) = { ( x 1 , . . . , x n ) | 0 ≤ x i ≤ max( A , i ) , i = 1 , . . . , n } . Π(Red dots) . Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  16. Minkovskii-Voronoi minima and minimal sets Set max( A , i ) = max { x i | ( x 1 , . . . , x n ) ∈ A } and define the parallelepiped Π( A ) = { ( x 1 , . . . , x n ) | 0 ≤ x i ≤ max( A , i ) , i = 1 , . . . , n } . Π(Red dots) . Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  17. Minkovskii-Voronoi minima and minimal sets Definition Let S be an arbitrary subset of R n ≥ 0 (csgp). An element γ ∈ S is called a Voronoi relative minimum if the parallelepiped Π( { γ } ) contains no points of S \ { γ } . Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  18. Minkovskii-Voronoi minima and minimal sets Definition Let S be an arbitrary subset of R n ≥ 0 (csgp). An element γ ∈ S is called a Voronoi relative minimum if the parallelepiped Π( { γ } ) contains no points of S \ { γ } . Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  19. Minkovskii-Voronoi minima and minimal sets Definition Let S be an arbitrary subset of R n ≥ 0 (csgp). An element γ ∈ S is called a Voronoi relative minimum if the parallelepiped Π( { γ } ) contains no points of S \ { γ } . Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  20. Minkovskii-Voronoi minima and minimal sets Definition Let S be an arbitrary subset of R n ≥ 0 (csgp). An element γ ∈ S is called a Voronoi relative minimum if the parallelepiped Π( { γ } ) contains no points of S \ { γ } . Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  21. Minkovskii-Voronoi minima and minimal sets Definition Let S be an arbitrary subset of R n ≥ 0 (csgp). An element γ ∈ S is called a Voronoi relative minimum if the parallelepiped Π( { γ } ) contains no points of S \ { γ } . Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  22. Minkovskii-Voronoi minima and minimal sets Definition Let S be an arbitrary subset of R n ≥ 0 (csgp). An element γ ∈ S is called a Voronoi relative minimum if the parallelepiped Π( { γ } ) contains no points of S \ { γ } . Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  23. Minkovskii-Voronoi minima and minimal sets Definition Let S be an arbitrary subset of R n ≥ 0 (csgp). An element γ ∈ S is called a Voronoi relative minimum if the parallelepiped Π( { γ } ) contains no points of S \ { γ } . Definition A finite subset F ⊂ Vrm( S ) is called minimal if the parallelepiped Π( F ) contains no Voronoi relative minima of Vrm( S ) \ F . Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  24. Minkovskii-Voronoi minima and minimal sets Definition Let S be an arbitrary subset of R n ≥ 0 (csgp). An element γ ∈ S is called a Voronoi relative minimum if the parallelepiped Π( { γ } ) contains no points of S \ { γ } . Definition A finite subset F ⊂ Vrm( S ) is called minimal if the parallelepiped Π( F ) contains no Voronoi relative minima of Vrm( S ) \ F . Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  25. Minkovskii-Voronoi minima and minimal sets Definition Let S be an arbitrary subset of R n ≥ 0 (csgp). An element γ ∈ S is called a Voronoi relative minimum if the parallelepiped Π( { γ } ) contains no points of S \ { γ } . Definition A finite subset F ⊂ Vrm( S ) is called minimal if the parallelepiped Π( F ) contains no Voronoi relative minima of Vrm( S ) \ F . Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  26. Minkovskii-Voronoi minima and minimal sets Definition Let S be an arbitrary subset of R n ≥ 0 (csgp). An element γ ∈ S is called a Voronoi relative minimum if the parallelepiped Π( { γ } ) contains no points of S \ { γ } . Definition A finite subset F ⊂ Vrm( S ) is called minimal if the parallelepiped Π( F ) contains no Voronoi relative minima of Vrm( S ) \ F . Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  27. Minkovskii-Voronoi complex Definition MV-complex is an ( n − 1)-dimensional complex such that ◮ the k -dimensional faces are enumerated by the minimal ( n − k )-element subsets ◮ a face with minimal subset F 1 is adjacent to a face with a minimal subset F 2 � = F 1 if and only if F 1 ⊂ F 2 . Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  28. Minkovskii-Voronoi complex Definition MV-complex is an ( n − 1)-dimensional complex such that ◮ the k -dimensional faces are enumerated by the minimal ( n − k )-element subsets ◮ a face with minimal subset F 1 is adjacent to a face with a minimal subset F 2 � = F 1 if and only if F 1 ⊂ F 2 . Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  29. Minkovskii-Voronoi complex Definition MV-complex is an ( n − 1)-dimensional complex such that ◮ the k -dimensional faces are enumerated by the minimal ( n − k )-element subsets ◮ a face with minimal subset F 1 is adjacent to a face with a minimal subset F 2 � = F 1 if and only if F 1 ⊂ F 2 . Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  30. Example of the MV-complex Consider � � S 0 = γ 1 , γ 2 , γ 3 , γ 4 , γ 5 , γ 6 , where γ 1 = (3 , 0 , 0) , γ 2 = (0 , 3 , 0) , γ 3 = (0 , 0 , 3) , γ 4 = (2 , 1 , 2) , γ 5 = (1 , 2 , 1) , γ 6 = (2 , 3 , 4) . Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  31. Example of the MV-complex Consider � � S 0 = γ 1 , γ 2 , γ 3 , γ 4 , γ 5 , γ 6 , where γ 1 = (3 , 0 , 0) , γ 2 = (0 , 3 , 0) , γ 3 = (0 , 0 , 3) , γ 4 = (2 , 1 , 2) , γ 5 = (1 , 2 , 1) , γ 6 = (2 , 3 , 4) . Relative minima: γ 1 , . . . , γ 5 . Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  32. Example of the MV-complex MV-complex contains 5 vertices, 6 edges, and 5 faces. Vertices: v 1 = { γ 1 , γ 3 , γ 4 } , v 2 = { γ 3 , γ 4 , γ 5 } , v 3 = { γ 1 , γ 4 , γ 5 } , v 4 = { γ 2 , γ 3 , γ 5 } , v 5 = { γ 1 , γ 2 , γ 5 } . Edges: e 1 = { γ 1 , γ 3 } , e 2 = { γ 3 , γ 2 } , e 3 = { γ 1 , γ 2 } , e 4 = { γ 3 , γ 4 } , e 5 = { γ 1 , γ 4 } , e 6 = { γ 4 , γ 5 } , e 7 = { γ 3 , γ 5 } , e 8 = { γ 1 , γ 5 } , e 9 = { γ 2 , γ 5 } . Faces: f 1 = { γ 1 } , f 2 = { γ 2 } , f 3 = { γ 3 } , f 4 = { γ 4 } , f 5 = { γ 5 } . Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  33. Example of the MV-complex f 3 e 1 v 1 v 2 v 4 e 4 e 7 e 2 f 4 f 5 e 5 e 6 e 9 v 3 e 8 f 1 f 2 v 5 e 3 MV ( S ) as a tessellation of an open two-dimensional disk. Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  34. Tessellations of the plane Question: How to describe MV-complexes in 3D? Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  35. Tessellations of the plane Question: How to describe MV-complexes in 3D? Useful tools: Minkowski polyhedron for an arbitrary S ; Tessellations of the plane. Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  36. Tessellations of the plane z f 3 v 1 v 2 v 4 f 4 v 3 f 5 f 1 f 2 v 5 y x Minkowski polyhedron for a set S (some sort of convex hull): S ⊕ R 3 ≥ 0 = { s + r | s ∈ S , r ∈ R 3 ≥ 0 } . Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  37. Tessellations of the plane x + y + z = 0 z f 3 f 3 v 1 v 2 v 4 v 1 v 2 v 4 f 4 f 5 v 3 f 4 f 1 f 2 v 3 f 5 f 1 f 2 v 5 v 5 y x ◮ The Minkowski polyhedron (left) ◮ Minkowski–Voronoi tessellation (right). Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  38. Tessellations of the plane x + y + z = 0 z f 3 f 3 v 1 v 2 v 4 v 1 v 2 v 4 f 4 f 5 v 3 f 4 f 1 f 2 v 3 f 5 f 1 f 2 v 5 v 5 y x Definition Step 1. Project the Minkowski polyhedron to x + y + z = 0. Step 2. Remove relative minima (i.e., minima of x + y + z ). Remove also all edges adjacent to them. Step 3. Rays to vertices of valence 1. Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  39. Linearisation of faces γ c γ c γ 0 γ 0 γ a γ b γ b γ a Linearisation laws for edges Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  40. Linearisation of faces γ c Linearisation laws γ c γ 0 γ 0 γ a γ b γ b γ a Theorem Every linearized finite face is as follows ( up to size rescaling ) : n 3 n 2 n 1 where n 1 , n 2 , n 3 ≥ 0 . Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  41. Linearisation of faces γ c Linearisation laws γ c γ 0 γ 0 γ a γ b γ b γ a Theorem Every linearized finite face is as follows ( up to size rescaling ) : n 3 n 2 n 1 where n 1 , n 2 , n 3 ≥ 0 . In our example: n 1 = 0, n 2 = 4, and n 3 = 2. Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  42. Diagrams of the tessellation Definition A diagram of a tessellation is canonical if all its faces are linearized. Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  43. Diagrams of the tessellation Definition A diagram of a tessellation is canonical if all its faces are linearized. Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  44. Diagrams of the tessellation Definition A diagram of a tessellation is canonical if all its faces are linearized. Proposition Every vertex of the MV-complex that is one of one of . Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  45. Diagrams of the tessellation Definition A diagram of a tessellation is canonical if all its faces are linearized. Proposition Every vertex of the MV-complex that is one of one of . Proposition Every finite tessellation of the plane admits a canonical diagram. Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  46. MV for lattices Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  47. MV for lattices Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  48. MV for lattices Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  49. MV for lattices Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  50. MV for lattices Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  51. MV for lattices Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  52. MV for lattices Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  53. MV for lattices The lattice is generated by (8 , 0) and (5 , 1). Here 8 5 = [1 : 1; 1; 1; 1] . Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  54. MV for lattices The lattice is generated by (8 , 0) and (5 , 1). Here 8 5 = [1 : 1; 1; 1; 1] . Remark. Here the continued fraction has 5 elements. Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  55. MV for lattices Theorem on combinatorics of continued fractions. The number of relative minima for a general lattice generated by ( N , 0) and ( a , 1) coincides with the number of elements for the longest continued fractions of a N . Oleg Karpenkov, University of Liverpool Lattice structure of MCF

  56. Lattice examples in 3D Oleg Karpenkov, University of Liverpool Lattice structure of MCF

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