M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon Based on the paper by M. Farber and D. Schütz Presented by Yair Carmon

M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon Contents Linkages: Introduction/Reminder Main result: Betti numbers of a planar linkage Remarks on main results (very informal) Outline of proof 2

M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon Acknowledgement I would like to thank my brother Dan for helpful discussions 3

M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon The mechanical linkage “rigid links connected with joints to form a closed chain, or a series of closed chains” Linkages are useful! We limit ourselves to a single 2d chain w/ two fixed joints Definition and animations courtesy of WikipediaWikipedia Klann linkage Peaucellier–Lipkin linkage Chebyshev linkage 4

M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon Configuration space of a planar linkage Informally: the set of angles in an n –edges polygon Formally: Modulo SO(2) is equivalent to two fixed joints 5

M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon Configuration space – properties M l is n –3 dimensional l is called “generic” if M l contains no collinear cfgs For generic l, M l is a closed smooth manifold For non–generic l, M l is a compact manifold with finitely many singular points (the collinear configurations) 6

M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon Configuration space – examples 7

M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon Betti numbers of a planar linkage A subset is called short/median/long if Assume l 1 is maximal, and define, s k ( m k ) – the number of short (median) subsets of size k+1 containing the index 1 Then H k (M l ; Z) is free abelian of rank 8

M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon Back to the examples 9

M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon The non-generic cases (n=4) 10

M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon The non-generic cases (n=4) 11

M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon The non-generic cases (n=4) 12

M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon Remark (1) – order of the edges The order of edges doesn’t matter – not surprising! M l is explicitly independent on edge order: Also, there is a clear isomorphism between permutations of l : 13

M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon Remark (2) – b 0 (M l ) Assume w.l.o For nontrivial M l, s 0 = 1 and m 0 = 0 s n–3 = 1 iff {2,3} is long, and s n–3 = 0 otherwise  b 0 (M l ) = 2 iff {2,3} is long, and = 1 otherwise Geometric meaning: existence of path between a configuration and its reflection 14

M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon Remark (2) – b 0 (M l ), geometric proof w.l.o assume the longest edge is connected to both second longest edges, and their lengths are l 1, l 2, l 3 In a path from a config to its reflection, each angle must pass through 0 or  If {2,3} is long the angle between l 1 and l 2 can’t be  Let’s see if it can be zero: 15

M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon Remark (2’) – topology when b 0 (M l ) If b 0 (M l ) = 2, every set that contains 1 and contains neither 2 nor 3 is short, therefore: The Poincaré polynomial is 2(1+t) n–3 In this case M l was proven to have the topology of two disjoint tori of dimension n–3 : 16

M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon Remark (3) – Equilateral case Nongeneric for even n Betti numbers can be computed from simple combinatorics (boring) Sum of betti numbers is maximal in the equilateral case (proven in the paper) 17

M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon Remark (4) – Euler characteristic In the generic case A striking difference between even and odd n ! For the nongeneric case, add 18

M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon (some of) The ideas behind the proof Warning – hand waving ahead! 19

M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon The short version Define W, the configuration space of an open chain Define W a, by putting a ball in the end of the chain The homology of W is simple to calculate (it’s a torus) The homology of W a is derived (for small enough a ) – using Morse theory and reflection symmetry The homology of M l is derived from their “difference” 20

M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon W – “the robot arm” Not dependant on l This is an n – 1 dimensional cfg space that contains M l W J  W is the subspace for which i.e. locked robot arm W J  M l =  iff J is long 21

M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon f l – the robot arm distance metric a < 0 chosen so that W J  W a iff J is long f l is Morse on W, W a The critical points of f l are collinear configurations p I ( I long) such that, p I  W J iff J  I, and has index n – | I | (for long J ) p J is the global maximum of W J (for long J ) 22

M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon Homology of W Reminder: W is diffeomorphic to T n–1 H i (W ; Z) is free abelian and generated by the homology classes of the sub–tori of dimension i These are exactly the manifolds W J for which 1  J and i = n – |J| Therefore, 23

M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon Homology of W a Everything ( W, W a, W J, f l ) is invariant to reflection! Critical points of f l are fixed points for reflection! Every critical point of f l in W a is contained in a W J in which it’s a global maximum (all the long J ’s) [Apply some fancy Morse theory…] 24

M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon H i (W a ) and H i (W ) revisited Rewrite the homology calsses Where, S i / M i / L i are generated by the set of classes [ W J ] for which J which is short/median/long, |J| = n – i and 1  J is like L i but with 1  J 25

M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon Homomorphism Define a natural homomorphism, By treating every homology class in W a as a class in W Clearly, It can be shown that Therefore, 26

M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon Homology of M l M l is a deformation retract of [somehow] Obtain the short exact sequence: Every point in H i (M l ) can expressed with two “coordinates” from the other groups, and it’s an isomorphism since we’re free abelian 27

M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon Homology of M l – the end Putting it all together, 28

M. Farber and D. Schütz | Homology of Planar Polygon Spaces | Presented by Yair Carmon 29