Homology of Planar Polygon Spaces

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

Homology of Planar

Polygon SpacesBased 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 2

Contents

• Linkages: Introduction/Reminder

• Main result: Betti numbers of a planar linkage

• Remarks on main results

• (very informal) Outline of proof


Acknowledgement• I would like to thank my

brother Dan for helpfuldiscussions


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 Wikipedia

Klann linkage Peaucellier–Lipkin linkage

Chebyshev linkage

http://en.wikipedia.org/wiki/Linkage_(mechanical)


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

( ) ( )1 2, ,..., n

nl l l += Îl ¡

( ) 1 11

1,..., | 0

n

n i ii

M u u S S l u=

ì üï ïï ï= Î ´ ´ =í ýï ïï ïî þål L ( )/ 2SO

1l

2l

5l

4l

3l

1u2u

5u4u 3u

1 1u =% 2u%5u%

4u%3u%

( )*1

/ 2i i

SOu uuÛ ¬


Configuration space – properties

• Ml is n–3 dimensional

• l is called “generic” if Ml contains no collinear cfgs

• For generic l, Ml is a closed smooth manifold

• For non–generic l , Ml is a compact manifold with finitely many singular points (the collinear configurations)


Configuration space – examples

@ { } { }0 1È( )3,3,2=l

@4g =( )1,1,1,1,1=l

@ 1S( )5,2,2,2=l


Betti numbers of a planar linkage• A subset is called short/median/long if

• Assume l1 is maximal, and define,sk (mk ) – the number of short (median) subsets of size

k+1 containing the index 1

• Then Hk(Ml ; Z) is free abelian of rank

{ }1,...,J nÍ

jj J

lÎå j

j Jl

Ïå<=>

( ) 3k k k n kb M s m s - -= + +l


Back to the examples

( )3,3,2=l { }0 3 0

0 0

12 , 0

n

k

s s xb b

- -

>

= = =Þ = =

{ }0 1 0

0 0 1 1

1 , , 01

k ks l s mb s s b

f>= = = =Þ = + = =

( )5,2,2,2=l

( )1,1,1,1,1=l 0 1 2

0 0 0 2

1 1 1 1

2 2 1 0

1 , 4, , 0181

k ks s s mb s m sb s m sb s m s

>= = =Þ = + + =Þ = + + =Þ = + + =


The non-generic cases (n=4)( )4,3,2,1=l

@1 2b =



@1 3b =



@1 4b =


Remark (1) – order of the edges• The order of edges doesn’t matter – not surprising!• Ml is explicitly independent on edge order:

• Also, there is a clear isomorphism between permutations of l:

( ) 1 11

1,..., | 0

n

n i ii

M u u S S l u=

ì üï ïï ï= Î ´ ´ =í ýï ïï ïî þål L ( )/ 2SO

1l

2l

5l

4l

3l

1u2u

5u4u 3u

( )1,2,3,4,5

1l

2l

5l

4l3l

1u

2u

5u

4u3u

( )1,5,4,2,3


Remark (2) – b0(Ml)• Assume w.l.o • For nontrivial Ml , s0 = 1 and m0 = 0• sn–3 = 1 iff {2,3} is long, and sn–3 = 0 otherwise

• b0(Ml) = 2 iff {2,3} is long, and = 1 otherwise• Geometric meaning: existence of path between a

configuration and its reflection

1 2 nl l l³ ³ ³L


Remark (2) – b0(Ml), geometric proof• w.l.o assume the longest edge is connected to both

second longest edges, and their lengths are l1,l2,l3• In a path from a config to its reflection, each angle must

pass through 0 or • If {2,3} is long the angle between l1 and l2 can’t be • Let’s see if it can be zero:

1 2l l-2l

3l4l

L nlÛ

{ }

1 2 34

2,3 is not long

n

ii

l l l l=

- + ³åc


Remark (2’) – topology when b0(Ml)• If b0(Ml) = 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 Ml was proven to have the topology of two disjoint tori of dimension n–3:

( )3 3 3 3, 0 23k k k

n n n ns m b Mk k n k k

æ ö æ ö æ ö æ ö- - - -÷ ÷ ÷ ÷ç ç ç ç÷ ÷ ÷ ÷ç ç ç ç= = Þ = + =÷ ÷ ÷ ÷ç ç ç ç÷ ÷ ÷ ÷- -÷ ÷ ÷ ÷ç ç ç çè ø è ø è ø è øl

3 3n nM T T- -@l ò


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)


Remark (4) – Euler characteristic• In the generic case

• A striking difference between even and odd n!

• For the nongeneric case, add

( ) ( ) ( ) ( )33

00

2 1 , odd10 , even

n kn k k

kkk

s nM b Mn

c-

-

==

ìïï -ïï@ - = íïïïïî

åål l

( )3

01

n k

kk

m-

=-å


(some of) The ideas behind the proof• Warning – hand waving ahead!


The short version• Define W, the configuration space of an open chain• Define Wa, 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 Wa is derived (for small enough a)

– using Morse theory and reflection symmetry

• The homology of Ml is derived from their “difference”

1l2l 4l

3l

aW aW


W – “the robot arm”

• Not dependant on l• This is an n – 1 dimensional cfg space that contains Ml • WJ W is the subspace for which

i.e. locked robot arm• WJ Ml = iff J is long

( ){ } ( )( ){ }

1 11

1 1 11 1

,..., / 2,..., | 1

nn

n

W u u S S SOu u S S u T -

= Î ´ ´= Î ´ ´ = @

LL

1l2l

4l

3l

,n mu u n m J= " Î


fl – the robot arm distance metric

•

• a < 0 chosen so that WJ Wa iff J is long

• fl is Morse on W, Wa

• The critical points of fl are collinear configurations pI (I long) such that,

• pI WJ iff J I , and has index n – | I | (for long J )• pJ is the global maximum of WJ (for long J )

( )2

11

: , ,...,n

n i ii

f W f u u l u=

® = - å¡

(1 ,aW f a- ù= - ¥ úûl

1 , , 1 , i iu i I u i I¢ ¢= ± " Î = " Ïm


Homology ofW• Reminder: W is diffeomorphic to Tn–1

• Hi(W ; Z) is free abelian and generated by the homology classes of the sub–tori of dimension i

• These are exactly the manifolds WJ for which 1 J and i = n – |J|

• Therefore,

( ) s.t.

1

;i JJi n J

J

H W W= -

Î

é ù= Å ê úë û¢


Homology of Wa

• Everything (W , Wa ,WJ , fl ) is invariant to reflection!

• Critical points of fl are fixed points for reflection!

• Every critical point of fl in Wa is contained in a WJ in which it’s a global maximum (all the long J’s)

• [Apply some fancy Morse theory…]

( ) s.t.

is long

;ai JJ

i n JJ

H W W= -

é ù= Å ê úë û¢


Hi(Wa ) and Hi(W ) revisited• Rewrite the homology calsses

• Where, Si / Mi / Li are generated by the set of classes [WJ

] for which J which is short/median/long , |J| = n – i and 1 J

• is like Li but with 1 J

( )( )

;;

ai i i

i i i i

H W L LH W L S M

= Å= Å Å

%¢¢

iL%


Homomorphism• Define a natural homomorphism,

By treating every homology class inWa as a class inW

• Clearly,

• It can be shown that

• Therefore,

( ) ( ): ; ;ai i iH W H Wf ®¢ ¢

( )i i iL Lf Í%( )i i iL Lf =

( ) ( )( )

11

1 1

ker rk kercoker rk coker

i i i in n i

i i i i n i n i

L s sS M s m

ffff

-- - -

- - - -

Þ = =Å Þ = +

%;;


Homology of Ml• Ml is a deformation retract of • [somehow] Obtain the short exact sequence:

( )1\ ,0aW W f a- é ù= ê úë ûl

( )1 20 coker ker 0in i n iH Mff - - - -® ® ® ®l

Im Kerg h=

- 1:1g

- untoh

Every point in Hi(Ml) can expressed with two “coordinates”from the other groups, and it’s an isomorphism since we’re free abelian

( ) 1 2coker kerin i n iH M ff - - - -Þ Ål ;

( )iH M l 2ker n if - -1coker n if - -


Homology of Ml – the end• Putting it all together,

( ) ( ) ( ) ( )( ) ( )

( )1 2

11

1 1

rk rk rk coker rk kerrk ker

rk coker

ii n i n i

i in n i

i n i n i

H M H Ms ss m

ffff

- - - -

-- - -

- - - -

= = += == +ß

l l

( ) ( ) 3rki i i i n ib M H M s m s - -= = + +l l


?

Homology of Planar Polygon Spaces

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