Counting Faces of Polytopes

Transcription

1 Counting Faces of Polytopes Carl Lee University of Kentucky James Madison University March 2015 Carl Lee (UK) Counting Faces of Polytopes James Madison University 1 / 36

2 Convex Polytopes A convex polytope P is the convex hull of a finite set of points in R d. Example: Cube Carl Lee (UK) Counting Faces of Polytopes James Madison University

3 Face-Vectors The face-vector of a d-dimensional polytope is f = (f 0, f 1,..., f d 1 ), where f j is the number of faces of dimension j. Define also f 1 = f d = 1. Example: Cube. f = (8, 12, 6).

4 Face-Vectors The face-vector of a d-dimensional polytope is f = (f 0, f 1,..., f d 1 ), where f j is the number of faces of dimension j. Define also f 1 = f d = 1. Example: Cube. f = (8, 12, 6). 4-Cube. f = (16, 32, 24, 8). Carl Lee (UK) Counting Faces of Polytopes James Madison University 3 / 36

5 Face-Vectors The face-vector of a d-dimensional polytope is f = (f 0, f 1,..., f d 1 ), where f j is the number of faces of dimension j. Define also f 1 = f d = 1. Example: Cube. f = (8, 12, 6). 4-Cube. f = (16, 32, 24, 8). Question: What are the possible face-vectors of polytopes? Carl Lee (UK) Counting Faces of Polytopes James Madison University 3 / 36

6 Three-Dimensional Polytopes Theorem (Euler s Relation) f 0 f 1 + f 2 = 2 for convex 3-polytopes. Example: Cube = 2. Carl Lee (UK) Counting Faces of Polytopes James Madison University 4 / 36

7 Three-Dimensional Polytopes Sketch of proof: Sweep the polytope with a plane in general direction. (Think of immersing in water.) Count vertices, edges, and polygons only when fully swept (under water). Watch how χ = f 0 f 1 + f 2 changes when the plane hits each vertex. Carl Lee (UK) Counting Faces of Polytopes James Madison University 5 / 36

8 Three-Dimensional Polytopes Sketch of proof: Sweep the polytope with a plane in general direction. (Think of immersing in water.) Count vertices, edges, and polygons only when fully swept (under water). Watch how χ = f 0 f 1 + f 2 changes when the plane hits each vertex. Initially χ = 0. Carl Lee (UK) Counting Faces of Polytopes James Madison University 5 / 36

9 Three-Dimensional Polytopes Sketch of proof: Sweep the polytope with a plane in general direction. (Think of immersing in water.) Count vertices, edges, and polygons only when fully swept (under water). Watch how χ = f 0 f 1 + f 2 changes when the plane hits each vertex. Initially χ = 0. Bottom vertex. χ changes by = 1. Carl Lee (UK) Counting Faces of Polytopes James Madison University 5 / 36

10 Three-Dimensional Polytopes Sketch of proof: Sweep the polytope with a plane in general direction. (Think of immersing in water.) Count vertices, edges, and polygons only when fully swept (under water). Watch how χ = f 0 f 1 + f 2 changes when the plane hits each vertex. Initially χ = 0. Bottom vertex. χ changes by = 1. Intermediate vertex with k incident lower edges. χ changes by 1 k + (k 1) = 0. Carl Lee (UK) Counting Faces of Polytopes James Madison University 5 / 36

11 Three-Dimensional Polytopes Sketch of proof: Sweep the polytope with a plane in general direction. (Think of immersing in water.) Count vertices, edges, and polygons only when fully swept (under water). Watch how χ = f 0 f 1 + f 2 changes when the plane hits each vertex. Initially χ = 0. Bottom vertex. χ changes by = 1. Intermediate vertex with k incident lower edges. χ changes by 1 k + (k 1) = 0. Top vertex. If its degree is k, then χ changes by 1 k + k = 1. Carl Lee (UK) Counting Faces of Polytopes James Madison University 5 / 36

12 Three-Dimensional Polytopes Sketch of proof: Sweep the polytope with a plane in general direction. (Think of immersing in water.) Count vertices, edges, and polygons only when fully swept (under water). Watch how χ = f 0 f 1 + f 2 changes when the plane hits each vertex. Initially χ = 0. Bottom vertex. χ changes by = 1. Intermediate vertex with k incident lower edges. χ changes by 1 k + (k 1) = 0. Top vertex. If its degree is k, then χ changes by 1 k + k = 1. Total change in χ is therefore 2. Carl Lee (UK) Counting Faces of Polytopes James Madison University 5 / 36

13 Three-Dimensional Polytopes Sketch of proof: Sweep the polytope with a plane in general direction. (Think of immersing in water.) Count vertices, edges, and polygons only when fully swept (under water). Watch how χ = f 0 f 1 + f 2 changes when the plane hits each vertex. Initially χ = 0. Bottom vertex. χ changes by = 1. Intermediate vertex with k incident lower edges. χ changes by 1 k + (k 1) = 0. Top vertex. If its degree is k, then χ changes by 1 k + k = 1. Total change in χ is therefore 2. Note: This proof technique generalizes to higher dimensions. Carl Lee (UK) Counting Faces of Polytopes James Madison University 5 / 36

14 Three-Dimensional Polytopes Other necessary conditions: Carl Lee (UK) Counting Faces of Polytopes James Madison University 6 / 36

15 Three-Dimensional Polytopes Other necessary conditions: f 0, f 1, f 2 are positive integers Carl Lee (UK) Counting Faces of Polytopes James Madison University 6 / 36

16 Three-Dimensional Polytopes Other necessary conditions: f 0, f 1, f 2 are positive integers What else? Carl Lee (UK) Counting Faces of Polytopes James Madison University 6 / 36

17 Three-Dimensional Polytopes Other necessary conditions: f 0, f 1, f 2 are positive integers What else? Theorem (Steinitz) A positive integer vector (f 0, f 1, f 2 ) is the face-vector of a 3-polytope if and only if the following conditions hold. f 0 f 1 + f 2 = 2, f 0 2f 2 4, and f 2 2f 0 4. Carl Lee (UK) Counting Faces of Polytopes James Madison University 6 / 36

18 Three-Dimensional Polytopes Carl Lee (UK) Counting Faces of Polytopes James Madison University 7 / 36

19 Four-Dimensional Polytopes What is the characterization of face-vectors of 4-polytopes? Carl Lee (UK) Counting Faces of Polytopes James Madison University 8 / 36

20 Four-Dimensional Polytopes What is the characterization of face-vectors of 4-polytopes? We don t know! Carl Lee (UK) Counting Faces of Polytopes James Madison University 8 / 36

21 Four-Dimensional Polytopes What is the characterization of face-vectors of 4-polytopes? We don t know! But there are some partial results. Carl Lee (UK) Counting Faces of Polytopes James Madison University 8 / 36

22 d-dimensional Polytopes Theorem (Euler-Poincaré Formula) For every d-polytope, d 1 f j = 1 ( 1) d j=0 Carl Lee (UK) Counting Faces of Polytopes James Madison University 9 / 36

23 d-dimensional Polytopes Theorem (Euler-Poincaré Formula) For every d-polytope, d 1 f j = 1 ( 1) d j=0 Early proofs (pre-poincaré) relied upon the implicit or unproven assumption of shellability, not established until 1970 by Bruggesser and Mani. Carl Lee (UK) Counting Faces of Polytopes James Madison University 9 / 36

24 d-dimensional Polytopes Theorem (Euler-Poincaré Formula) For every d-polytope, d 1 f j = 1 ( 1) d j=0 Early proofs (pre-poincaré) relied upon the implicit or unproven assumption of shellability, not established until 1970 by Bruggesser and Mani. Grünbaum developed a sweeping-like proof. Carl Lee (UK) Counting Faces of Polytopes James Madison University 9 / 36

25 d-dimensional Polytopes A facet of a d-polytope is a face of dimension d 1. Carl Lee (UK) Counting Faces of Polytopes James Madison University 10 / 36

26 d-dimensional Polytopes A facet of a d-polytope is a face of dimension d 1. Theorem (Upper Bound Theorem, McMullen) The dual to the cyclic d-polytope with n vertices has the largest number of faces of all dimensions of any d-polytope with n facets. Carl Lee (UK) Counting Faces of Polytopes James Madison University 10 / 36

27 Four-Dimensional Polytopes Barnette characterized the sets of various pairs of components of (f 0, f 1, f 2, f 3 ). Carl Lee (UK) Counting Faces of Polytopes James Madison University 11 / 36

28 Simple Polytopes A d-polytope is simple if every vertex is incident to precisely d edges. Carl Lee (UK) Counting Faces of Polytopes James Madison University 12 / 36

29 Simple Polytopes A d-polytope is simple if every vertex is incident to precisely d edges. Example: The cube, (as well as the d-cube for all d). Carl Lee (UK) Counting Faces of Polytopes James Madison University 12 / 36

30 Simple Polytopes Theorem (Lower Bound Theorem, Barnette) The truncation polytope of dimension d with n facets has the smallest number of faces of all dimensions of any simple d-polytope with n facets. Carl Lee (UK) Counting Faces of Polytopes James Madison University 13 / 36

31 h-vectors Sweep a simple d-polytope with a hyperplane in general direction. Orient all edges in the direction of the sweep. Let h i be the number of vertices of indegree i. The h-vector is (h 0, h 1,..., h d ). Note that it is a nonnegative vector of integers. Carl Lee (UK) Counting Faces of Polytopes James Madison University 14 / 36

32 h-vectors Example: Cube Carl Lee (UK) Counting Faces of Polytopes James Madison University 15 / 36

33 h-vectors Example: Cube h = (1, 3, 3, 1) Carl Lee (UK) Counting Faces of Polytopes James Madison University 15 / 36

34 h-vectors Theorem (McMullen) f j = d i=j ( ) i h i, j = 0,..., d j Carl Lee (UK) Counting Faces of Polytopes James Madison University 16 / 36

35 h-vectors Theorem (McMullen) f j = d i=j ( ) i h i, j = 0,..., d j Idea: Every subset of j incoming edges to a vertex corresponds to an i-face. Carl Lee (UK) Counting Faces of Polytopes James Madison University 16 / 36

36 h-vectors These relations are invertible. Theorem (McMullen) h i = d ( ) j ( 1) i+j f j, i = 0,..., d i j=i Carl Lee (UK) Counting Faces of Polytopes James Madison University 17 / 36

37 h-vectors These relations are invertible. Theorem (McMullen) h i = d ( ) j ( 1) i+j f j, i = 0,..., d i j=i This implies that (h 0,..., h d ) is independent of the choice of hyperplane! Carl Lee (UK) Counting Faces of Polytopes James Madison University 17 / 36

38 h-vectors Stanley s trick to convert from the face-factor to the h-vector. Consider the face-vector (36, 108, 141, 102, 43, 10). Carl Lee (UK) Counting Faces of Polytopes James Madison University 18 / 36

39 h-vectors Stanley s trick to convert from the face-factor to the h-vector. Consider the face-vector (36, 108, 141, 102, 43, 10) Carl Lee (UK) Counting Faces of Polytopes James Madison University 18 / 36

40 Dehn-Sommerville Relations Reversing the sweep reverses the directions of all edges, and so swaps indegree with outdegree for each vertex. Carl Lee (UK) Counting Faces of Polytopes James Madison University 19 / 36

41 Dehn-Sommerville Relations Reversing the sweep reverses the directions of all edges, and so swaps indegree with outdegree for each vertex. The invariance of the h-vector then implies Carl Lee (UK) Counting Faces of Polytopes James Madison University 19 / 36

42 Dehn-Sommerville Relations Reversing the sweep reverses the directions of all edges, and so swaps indegree with outdegree for each vertex. The invariance of the h-vector then implies Theorem (Dehn-Sommerville Relations) For every simple d-polytope, h i = h d i for all i. Carl Lee (UK) Counting Faces of Polytopes James Madison University 19 / 36

43 Dehn-Sommerville Relations Reversing the sweep reverses the directions of all edges, and so swaps indegree with outdegree for each vertex. The invariance of the h-vector then implies Theorem (Dehn-Sommerville Relations) For every simple d-polytope, h i = h d i for all i. Besides being a nonnegative symmetric vector of integers, what other conditions must hold for the h-vector? Carl Lee (UK) Counting Faces of Polytopes James Madison University 19 / 36

44 Canonical Representations For positive integers a and i, a can be written uniquely in the form ( ) ( ) ( ) ai ai 1 aj a = i i 1 j where a i > a i 1 > > a j j 1. This is the i-canonical representation of a. For example, the 4-canonical representation of 26 is 26 = ( ) ( ) 5 3 ( ) Carl Lee (UK) Counting Faces of Polytopes James Madison University 20 / 36

45 Canonical Representations Now define a <i> by adding one to the top and bottom of every binomial coefficient in the i-canonical representation of a. ( ) ( ) ( ) a <i> ai + 1 ai aj + 1 = i + 1 i j + 1 For example, 26 <4> = Define also a <0> = 0. ( ) ( ) ( ) 3 = Carl Lee (UK) Counting Faces of Polytopes James Madison University 21 / 36

46 g-theorem For any symmetric vector (h 0, h 1,..., h d ) define g 0 = h 0 and g i = h i h i 1, i = 1,..., d/2. (That is to say, compute differences up to half way.) Carl Lee (UK) Counting Faces of Polytopes James Madison University 22 / 36

47 g-theorem For any symmetric vector (h 0, h 1,..., h d ) define g 0 = h 0 and g i = h i h i 1, i = 1,..., d/2. (That is to say, compute differences up to half way.) Theorem (g-theorem, Billera-L-Stanley, conjectured by McMullen) A vector (h 0, h 1,..., h d ) of positive integers is the h-vector of a simple d-polytope if and only if the following conditions hold. h i = h d 1, i = 0,..., d, g i 0, i = 0, 1,..., d/2, and g 0 = 1 and g i+1 g <i> i for all i = 1, 2,... d/2 1. Carl Lee (UK) Counting Faces of Polytopes James Madison University 22 / 36

48 g-theorem For example, if we consider the potential face-vector f = (36, 108, 141, 102, 43, 10) we compute h = (1, 4, 8, 10, 8, 4, 1) and g = (1, 3, 4, 2). Now h is symmetric, g is nonnegative, g 0 = 1, 4 3 <1>, and 2 4 <2>, so this is a valid face-vector for a simple polytope. Carl Lee (UK) Counting Faces of Polytopes James Madison University 23 / 36

49 g-theorem The necessity of the conditions comes from considering a certain graded ring associated with the simple polytope (the Stanley-Reisner ring), and its relationship to the cohomology of a certain complex projective toric variety for which the hard Lefschetz Theorem holds. (McMullen later provided a more geometric proof using his polytope algebra.) The sufficiency of the conditions comes from a direct construction. Carl Lee (UK) Counting Faces of Polytopes James Madison University 24 / 36

50 Nonsimple Polytopes It is fruitful to look beyond the face-vectors for nonsimple polytopes, and consider flag-vectors, which count chains of faces of various types and lengths. Carl Lee (UK) Counting Faces of Polytopes James Madison University 25 / 36

51 Flag f -Vector and cd-index Fine; Bayer-Klapper Example: Triangular bipyramid P. Carl Lee (UK) Counting Faces of Polytop

52 Flag f -Vector and cd-index Example: Triangular bipyramid P. f S = numbers of chains of faces of type S. S f S h S w S ccc 4cd 3dc Carl Lee (UK) Counting Faces of Polytopes James Madison University 27 / 36

53 Flag f -Vector and cd-index Example: Triangular bipyramid P. f S = numbers of chains of faces of type S. S f S h S w S ccc 4cd 3dc Carl Lee (UK) Counting Faces of Polytopes James Madison University 28 / 36

54 Flag f -Vector and cd-index Example: Triangular bipyramid P. f S = numbers of chains of faces of type S. S f S h S w S ccc 4cd 3dc h S = T S ( 1) S T f T Carl Lee (UK) Counting Faces of Polytopes James Madison University 29 / 36

55 Flag f -Vector and cd-index Example: Triangular bipyramid P. f S = numbers of chains of faces of type S. S f S h S w S ccc 4cd 3dc 1 1 aaa baa aba aab bba bab abb bbb h S = T S ( 1) S T f T Carl Lee (UK) Counting Faces of Polytopes James Madison University 30 / 36

56 Flag f -Vector and cd-index Example: Triangular bipyramid P. f S = numbers of chains of faces of type S. S f S h S w S ccc 4cd 3dc 1 1 aaa baa aba aab bba bab abb bbb 1 h S = T S ( 1) S T f T c = a + b and d = ab + ba Φ(P) = c 3 + 4cd + 3dc. Carl Lee (UK) Counting Faces of Polytopes James Madison University 31 / 36

57 cd-index Thus the cd-index contains a Fibonacci amount of information. Carl Lee (UK) Counting Faces of Polytopes James Madison University 32 / 36

58 cd-index Thus the cd-index contains a Fibonacci amount of information. Theorem (Bayer-Klapper) The cd-index exists! Carl Lee (UK) Counting Faces of Polytopes James Madison University 32 / 36

59 cd-index Thus the cd-index contains a Fibonacci amount of information. Theorem (Bayer-Klapper) The cd-index exists! Theorem (Stanley) The cd-index is nonnegative. Carl Lee (UK) Counting Faces of Polytopes James Madison University 32 / 36

60 cd-index Thus the cd-index contains a Fibonacci amount of information. Theorem (Bayer-Klapper) The cd-index exists! Theorem (Stanley) The cd-index is nonnegative. The cd-index can be computed by sweeping. Carl Lee (UK) Counting Faces of Polytopes James Madison University 32 / 36

61 cd-index Thus the cd-index contains a Fibonacci amount of information. Theorem (Bayer-Klapper) The cd-index exists! Theorem (Stanley) The cd-index is nonnegative. The cd-index can be computed by sweeping. Ehrenborg has shown how to lift inequalities of cd-indices into new inequalities in higher dimensions. Carl Lee (UK) Counting Faces of Polytopes James Madison University 32 / 36

62 cd-index Thus the cd-index contains a Fibonacci amount of information. Theorem (Bayer-Klapper) The cd-index exists! Theorem (Stanley) The cd-index is nonnegative. The cd-index can be computed by sweeping. Ehrenborg has shown how to lift inequalities of cd-indices into new inequalities in higher dimensions. We still cannot characterize the set of all cd-indices. Carl Lee (UK) Counting Faces of Polytopes James Madison University 32 / 36

63 Four-Dimensional Polytopes What we know about 4-polytopes besides the Euler-Poincaré relation: f 02 3f 2 0 f 02 3f 1 0 f 02 3f 2 + f 1 4f f 1 6f 0 f 02 0 f f 2 f 1 + f 0 0 2(f 02 3f 2 ) + f 1 ( f 0 ) 2 2(f 02 3f 1 ) + f 2 ( f 2 f 1 +f 0 ) 2 f 02 4f 2 + 3f 1 2f 0 ( f 0 ) 2 f 02 + f 2 2f 1 2f 0 ( f 2 f 1 +f 0 2 ) Carl Lee (UK) Counting Faces of Polytopes James Madison University 33 / 36

64 Four-Dimensional Polytopes But there are still some huge gaps in what we know to be true about flag-vectors and the 4-polytopes we know how to construct. Carl Lee (UK) Counting Faces of Polytopes James Madison University 34 / 36

65 What I am Skipping About Counting Faces Shellings Winding numbers in Gale transforms Dimensions of stress and motion spaces Formulas for volumes of polytopes The toric h-vector Hopf algebras Much more... Carl Lee (UK) Counting Faces of Polytopes James Madison University 35 / 36

66 Thank you! Carl Lee (UK) Counting Faces of Polytopes James Madison University 36 / 36