Random planar maps, alternating knots and links Gilles Sc - - PowerPoint PPT Presentation

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Random planar maps, alternating knots and links Gilles Sc - - PowerPoint PPT Presentation

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Random planar maps, alternating knots and links Gilles Sc haeer CNRS Sbastien Kunz-Jacques Join w ork with (LIX Corps des tlcoms) 1 An overview of the talk The enumeration of maps examples of

slide-1
SLIDE 1 Random planar maps, alternating knots and links Gilles Sc haeer
  • CNRS
Join w
  • rk
with Sbastien Kunz-Jacques (LIX
  • Corps
des tlcoms) 1
slide-2
SLIDE 2 An
  • verview
  • f
the talk The enumeration
  • f
maps examples
  • f
algebraic functions Random plana r maps almost sure prop erties Enumerative knot theo ry ? prime alternating links Asymptotic enumeration
  • f
links as an application
  • f
random maps 2
slide-3
SLIDE 3 Ro
  • ted
plana r maps. Denition a plana r map = an em b edding
  • f
a connected graph in the plane. planar map = planar graph + cyclic
  • rder
around v ertices. W e consider r
  • te
d planar maps: a r
  • t
edge is c hosen around the innite face and
  • rien
ted coun terclo c kwise. 3
slide-4
SLIDE 4 Ro
  • ted
plana r maps. Examples The smallest maps: A planar map with
  • nly
  • ne
face is a plane tr e e. A planar map with
  • nly
  • ne
v ertex is a cycle
  • f
lo
  • ps.
4
slide-5
SLIDE 5 Ro
  • ted
plana r maps. On the sphere ? Sometimes I lik e to replace the plane b y a sphere : : : This is equiv alen t but lo
  • ks
more symmetric: all faces are simply connected (=disc). ! nicer pictures but that are more dicult to do : : : 5
slide-6
SLIDE 6 Ro
  • ted
plana r maps. Example
  • f
subfamilies T riangulations 4-regular maps Suc h lo cal restrictions should b e irrelev an t in the large size limit. Compare to simple trees: m-ary trees, plane trees, 1-2 trees ) they usually are all the same. 6
slide-7
SLIDE 7 Enumeration
  • f
maps in combinato rics as
  • pp
  • sed
to physics & enumerative top
  • logy
  • T
utte (1962): a c ensus
  • f
triangulations Originally to attac k the four color theorem via en umeration.
  • Counting
plana r maps (70's): T utte, Bro wn, Mullin, Cori, Liu : : : Results for more than t w en t y subfamilies
  • f
planar maps. : : : Gao-W
  • rmald
(2001) 5-c
  • nne
cte d triangulations
  • Maps
  • n
surfaces (80's), random plana r maps (90's): Bender, Caneld, Arqus, Gao, Ric hmond, W
  • rmald,
: : : F
  • r
instance, Bender-Compton-Ric hmond (1999): 0-1 laws for F O lo gic pr
  • p
erties
  • f
r andom maps
  • n
surfac es. 7
slide-8
SLIDE 8 Enumeration via generating functions just
  • ne
step a w a y from plane trees... 8
slide-9
SLIDE 9 T utte's ro
  • t
deletion metho d. (i) plane trees Usual plane tr e es are exactly maps with
  • ne
face.

T’’ T’ T’ T’’

X T 6 = z jT j = X T ;T 00 z 1+jT j+jT 00 j Th us the equation t(z )
  • 1
= z t(z ) 2 . Plane tr e es (and in gener al simple tr e es) have algebr aic GF. 9
slide-10
SLIDE 10 T utte's ro
  • t
deletion metho d. (ii) lo
  • ps
A map with
  • nly
  • ne
v ertex is a cycle
  • f
lo
  • ps.

5 1 2 3 4 1 2 3

X w 6 = z jw j u d(w ) = X w z 1+jw j (u + u 2 +
  • +
u d(w )+1 ) L(z ; u)
  • 1
= z u 2 u1 L(z ; u)
  • z
u u1 `(z ): A line ar e quation in L(z ; u) with p
  • lynomial
c
  • es
in z , u and `(z ): (u
  • 1
+ z u 2 )L(z ; u) = u
  • 1
  • z
u `(z ): 10
slide-11
SLIDE 11 T utte's ro
  • t
deletion metho d. (iii) all maps The t w
  • previous
cases generalize:

M’ M’’ M’ M’’

1 2 3 5 4 6

F (z ; u)
  • 1
= z u 2 F (z ; u) 2 + z u u1 (uF (z ; u)
  • f
(z ))
  • r
equiv alen tly (dep endenc es in z hidden) (u
  • 1)z
u 2 F (u) 2 + (u
  • 1
+ z u 2 )F (u) + (u
  • 1
  • z
uf ) = a quadr atic equation in F (u) with p
  • lynomial
co es in z , u and f . 11
slide-12
SLIDE 12 Linea r equations with a catalytic va riable. The k ernel metho d. The k ernel metho d for K (u)L(u) = q (z ; u; `):
  • L
  • k
for a r
  • t
u
  • f
K (u) such that L(u ) makes sense. Here L(u) 2 C [u][[z ]] and the ro
  • ts
  • f
K (u) are u 1 = 1+ p 14z 2z = 1=z + O (1) and u 2 = 1 p 14z 2z = z + O (z 2 ): L(u 1 ) is not
  • k
but L(u 2 ) con v erges as a formal p
  • w
er series. The substitution u u in the linear equation giv es = u 2
  • 1
+ z u 2 `; so that `(z ) = 1 p 14z 2z (See also Cyril Banderier's talk) 12
slide-13
SLIDE 13 P
  • lynomial
equations with a catalytic va riable. Bousquet-Melou's metho d
  • (extends
k ernel & T utte's quadratic metho ds) F (z ; u) and f (z ) such that ther e is a p
  • lynomial
P (a; b; c) with P (F (u); u; f ) = (dep endenc e in z hidden)
  • Dieren
tiate with resp ect to u: F u (u)P a (F (u); u; f ) + P b (F (u); u; f ) =
  • Supp
  • se
w e nd u = u (z ) suc h that F (u ) is w ell dened and P b (F (u ); u ; f ) = 0: Then P a (F (u ); u ; f ) = and P (F (u ); u ; f ) = 0: A p
  • lynomial
system in F (u ), u , f : algebr aic solutions ! 13
slide-14
SLIDE 14 Ro
  • ted
plana r maps. The solution
  • W
e
  • btain
an algebraic generating function f (z ): f (z ) = X M z jM j = 1
  • 1
  • 18z
  • (1
  • 12z
) 3=2 54z 2 (remark that F (z ; u) is also algebraic ! face degree)
  • T
ransfert theorems (e.g.) yield an asymptotic expansion: #fro
  • ted
maps with n edges g c
  • n
5=2
  • 12
n : The exp
  • nent
5=2 is char acteristic
  • f
planar map emuner ations (c
  • mp
ar e to 3=2 for various simple tr e es). 14
slide-15
SLIDE 15 A rst summa ry .
  • P
  • lynomial
equations with
  • ne
catalytic v ariable should ha v e algebraic solutions (cf. Mireille Bousquet-Mlou): P (F (z ; u); z ; u; f 1 (z ); : : : ; f k (z )) = (if y
  • u
kno w examples, w e are in terested in collecting them !)
  • Ro
  • t
deletion applies to man y families
  • f
maps and yields univ ersal asymptotic b eha vior: #fro
  • ted
F
  • maps
  • f
size ng = c n n 5=2 where c and
  • dep
end
  • n
the family F .
  • In
some cases the explicit form ulas are nice. 15
slide-16
SLIDE 16 Nice fo rmulas, random maps and why plana r maps a re almost Galton-W atson trees 16
slide-17
SLIDE 17 T utte's fo rmulas fo r ro
  • ted
plana r maps. (60's) The ro
  • t
deletion metho d pro vides surprisingly nice form ulas in sev eral cases, among whic h: #ftriangulations with 2n facesg = 2 2n + 2 2 n 2n + 1
  • 3n
n
  • c
1 n 5=2 (27=2) n #f4-regular maps with n v ert.g = 2 n + 2 3 n n + 1
  • 2n
n
  • c
2 n 5=2 12 n All families should b eha v e the same ) concen trate
  • n
those simpler mo dels ! (lik e binary trees in tree en umeration, b ernoulli w alks, : : : ) 17
slide-18
SLIDE 18 T utte's fo rmulae. A bijectiv e pro
  • f
(i). #f 4-regular maps with n v ertices g is 2 n+2
  • 3
n n+1
  • 2n
n
  • .
There are 1 n + 1
  • 2n
n
  • binary
trees with n no des. Suc h trees ha v e n (in ternal) no des and n + 2 lea v es (ro
  • t
included). 18
slide-19
SLIDE 19 T utte's fo rmulae. A bijectiv e pro
  • f
(ii). #f 4-regular maps with n v ertices g is 2 n+2
  • 3
n n+1
  • 2n
n
  • .
On eac h no de, a bud can b e added in three w a ys, giving rise to 3 n n + 1
  • 2n
n
  • blossom
tr e es with n no des. Blossom trees ha v e n buds and n + 2 lea v es around the tree. Up
  • n
matc hing them coun terclo c kwise, t w
  • lea
v es remain unmatche d. 19
slide-20
SLIDE 20 T utte's fo rmulae. A bijectiv e pro
  • f
(ii). #f 4-regular maps with n v ertices g is 2 n+2
  • 3
n n+1
  • 2n
n
  • .
On eac h no de, a bud can b e added in three w a ys, giving rise to 3 n n + 1
  • 2n
n
  • blossom
tr e es with n no des. Blossom trees ha v e n buds and n + 2 lea v es around the tree. Up
  • n
matc hing them coun terclo c kwise, t w
  • lea
v es remain unmatche d. 20
slide-21
SLIDE 21 T utte's fo rmulae. A bijectiv e pro
  • f
(ii). #f 4-regular maps with n v ertices g is 2 n+2
  • 3
n n+1
  • 2n
n
  • .
On eac h no de, a bud can b e added in three w a ys, giving rise to 3 n n + 1
  • 2n
n
  • blossom
tr e es with n no des. Blossom trees ha v e n buds and n + 2 lea v es around the tree. Up
  • n
matc hing them coun terclo c kwise, t w
  • lea
v es remain unmatche d. 21
slide-22
SLIDE 22 T utte's fo rmulae. A bijectiv e pro
  • f
(ii). #f 4-regular maps with n v ertices g is 2 n+2
  • 3
n n+1
  • 2n
n
  • .
On eac h no de, a bud can b e added in three w a ys, giving rise to 3 n n + 1
  • 2n
n
  • blossom
tr e es with n no des. Blossom trees ha v e n buds and n + 2 lea v es around the tree. Up
  • n
matc hing them coun terclo c kwise, t w
  • lea
v es remain unmatche d. 22
slide-23
SLIDE 23 T utte's fo rmulae. A bijectiv e pro
  • f
(iii). #f 4-regular maps with n v ertices g is 2 n+2
  • 3
n n+1
  • 2n
n
  • .
The matc hing pro cedure do es not dep end
  • n
whic h leaf is the ro
  • t.
A blossom tree is b alanc e d if its ro
  • t
remains unmatc hed. Eac h c
  • njugacy
class
  • f
tr e es con tains n + 2 blossom trees, 2
  • f
whic h are balanced: the n um b er
  • f
balanced blossom tree is th us 2 n + 2
  • 3
n n + 1
  • 2n
n
  • :
23
slide-24
SLIDE 24 T utte's fo rmulae. A bijectiv e pro
  • f
(iv). #f 4-regular maps with n v ertices g is 2 n+2
  • 3
n n+1
  • 2n
n
  • .
Theo rem (S. 1998): Closure is
  • ne-to-one
b et w een
  • balanced
blossom trees with n no des
  • and
4-regular maps with n v ertices. The con v erse bijection is based
  • n
a bfs tra v ersal
  • f
the dual graph. 24
slide-25
SLIDE 25 Random plana r maps 25
slide-26
SLIDE 26 The random plana r map. Random planar maps are dened b y: the unifo rm distribution
  • n
ro
  • ted
planar maps with n edges. But w e can as w ell use a subfamily:
  • uniform
  • n
4-regular maps with n v ertices
  • uniform
  • n
balanced blossom trees with n no des
  • uniform
  • n
blossom trees with n no des
  • G.W.
trees with 3 t yp es
  • f
  • spring
2, conditioned to ha v e n no des ) map parameters lead to fancy parameters
  • n
trees. 26
slide-27
SLIDE 27 Random plana r maps as random lattices In ph ysics pap ers, they w
  • uld
rather tak e: random 4-regular maps ( 4 lattice mo del).
  • r
random triangulations (dual
  • 3
mo del). 27
slide-28
SLIDE 28 Why random maps in physics ? (a naive p
  • int
  • f
view) Consider a 2d univ erse...
  • Con
v en tional gra vit y: the univ erse is at. ) discretised b y a regular grid.
  • Quan
tum gra vit y: a distribution
  • f
proba
  • n
p
  • ssible
univ erses. ) discretised b y a random map.
  • planar
case is easier ) assume spherical top
  • logy
to start with. This lead some ph ysicists to redisco v er man y form ulas
  • f
T utte using p erturbativ e expansion
  • f
matrix in tegrals. 28
slide-29
SLIDE 29 A gallery
  • f
random maps 29
slide-30
SLIDE 30 What is the t ypical geometry
  • f
a random map ? (o r triangulations
  • r
4-regula r maps, : : : ) 30
slide-31
SLIDE 31 The random plana r map. Prole and diameter (i).
  • X
(k ) n is the n um b er
  • f
v ertices at distance k
  • f
the ro
  • t
  • the
pr
  • le
is then X n = (X (1) n ; X (2) n ; : : : ; X (k ) n ; : : : )

6 5 4 D=8 4 1 2 3 4 dist X

(k)

= h=4

  • h
n is the heigh t (maximal distance from the ro
  • t)
  • D
n is the diameter
  • f
a random n-triangulation In particular h n
  • D
n
  • 2h
n . 31
slide-32
SLIDE 32 The random plana r map. Distances and diameter (ii). Exp erimen tation using random sampling algorithms: Six random proles:

500 1000 1500 2000 2500 50 100 150 200 250

All for maps
  • f
size n = 100; 000. A v eraged proles:

500 1000 1500 2000 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380

F
  • r
v arious n (100 to 100; 000). )Conjecture (S. 1998) The correct scaling is k = tn 1=4 . F
  • r
this scaling I exp ect normalised X (k ) n to con v erge to a random pro cess X (t) supp
  • rted
  • n
R + . 32
slide-33
SLIDE 33 The random plana r map. Distances and diameter (iii). In particular this should imply
  • T
w
  • b
eaut yful heuristic calculations b y ph ysicists W atabiki, Ambj rn et al. (1994:) The Hausdor dimension is 1=4 : meaning for k
  • n
1=4 ; E ( R k X (i) n )
  • k
4 ; for k
  • n
1=4 ; E (X (k ) n ) is exp. decreasing
  • Conjecture
(S. 2001): E (h n )
  • n
1=4 e (e=n) 1=4 where
  • =
q 2 + 13 6 p 3 . (constan t
  • giv
en here for lo
  • pless
cubic maps). 33
slide-34
SLIDE 34 The random plana r map. A ten tativ e picture
  • f
distances.

n

1/4

n

1/4

2 n

3/4

k

3

exp

>> >>

34
slide-35
SLIDE 35 The random plana r map. A ten tativ e picture
  • f
cuts.

n

2/3

n

2/3

O( )

in n/2 + x baby universes of size cte size cuts (linear number) (Conjecture: Gao, S. 01) logarithmic −separator ? mother universe of size scaling with stable law 3/2 (BaFlScSo00) α each but summing to (n) Θ

35
slide-36
SLIDE 36 A second summa ry
  • The
random planar maps mo del has man y v arian ts (triangulations, bipartite maps, con v ex p
  • lyhedra,
: : : )
  • P
arameters
  • f
in terests ha v e similar a v
  • r
as for simple trees (prole, heigh t, maximal degree, 0-1 la ws, : : : )
  • All
kno ws results satisfy the exp ected univ ersalit y: critical exp
  • nen
ts agree for dieren t families. 36
slide-37
SLIDE 37 An application to knot theo ry: the asymptotic numb er
  • f
p rime alternating links join w
  • rk
with Sbastien Kunz-Jacques. 37
slide-38
SLIDE 38 Knots and links.
  • The
unknot is the simplest knot : : : A knot is made
  • f
  • ne
lace, a link ma y ha v e more.
  • A
plana r diagram
  • f
a link: a generic pro jection.
  • The
size
  • f
a link is its minimal n um b er
  • f
crossings in a planar diagram.
  • The
3 Reidemeister moves connect all its diagrams. 38
slide-39
SLIDE 39 Knots and links. Prime facto r decomp
  • sition
The p ro duct
  • f
t w
  • knots.

+ =

Knots and links ha v e a unique decomp
  • sition
in p rime factors. Prime links cannot b e decomp
  • sed:
no 2-cut in their minimal diagrams Example
  • f
defect
  • f
primalit y
  • !
39
slide-40
SLIDE 40 Knots and links. Enumerative knot theo ry ? Coun t prime knots and links w.r.t. n um b er
  • f
crossings !
  • r
e quivalently Coun t equiv alence classes
  • f
diagrams under Reidemeister mo v es. This seems to b e a v ery dicult problem... W e shall restrict
  • ur
atten tion to easier sub classes. 40
slide-41
SLIDE 41 Knots and links. Alternating links Alternating diagrams: eac h edge is undercrossing at
  • ne
end, and
  • v
ercrossing at the
  • ther.

I II

Find which
  • ne
is alternating ! An alternating link is
  • ne
that admits an alternating diagram. Not al l knots ar e alternating, but these have nic er pr
  • p
erties than gener al knots... 41
slide-42
SLIDE 42 Knots and links. Flyp e and T ait's conjecture A yp e transforms
  • ne
diagram in to another: Theo rem (Menasco and Thistleth w aite, 1993) A ny two prime alternating diagr ams
  • f
a prime alternating link ar e c
  • nne
cte d by a se quenc e
  • f
yp es. Co rolla ry . A l l prime alternating diagr ams
  • f
a prime alternating link have the same size (= numb er
  • f
cr
  • ssings).
42
slide-43
SLIDE 43 The numb er
  • f
p rime alternating links A simpler p roblem ? Coun t prime alternating links
  • r
e quivalently Coun t equiv alence classes
  • f
diagrams under the action
  • f
yp es. Theo rem (Sundb erg and Thistleth w aite, 1998) The numb er A n
  • f
prime alternating links
  • f
size n satises c 1
  • n
n 7=2
  • A
n
  • c
2
  • n
n 5=2 : Our aim: the exact asymptotic b eha vior. Our means: analytic com binatorics
  • f
random planar maps. 43
slide-44
SLIDE 44 Diagrams and plana r maps.

R

Prop
  • sition.
Ther e is a
  • ne-to-one
c
  • rr
esp
  • ndenc
e b etwe en
  • r
  • te
d (prime) alternating diagr ams with n no des,
  • r
  • te
d 4-r e gular planar maps (without 2-cut) with n vertic es. Ide a: The
  • v
er-undercrossing structure
  • f
the ro
  • t
v ertex can b e consisten tly propagated to all
  • thers.
44
slide-45
SLIDE 45 Ro
  • ted
diagrams. En umeration. W e ha v e seen that #fro
  • ted
diagrams
  • f
size ng = 2 n+2 3 n n+2
  • 2n
n
  • :
Similarly #fro
  • ted
prime diagrams
  • f
size ng = 4 2n+2 1 2n+1
  • 3n
n
  • :
(pro
  • f
b y ro
  • t
deletion
  • r
bijection with ternary blossom trees). But w e need to tak e yp es in to accoun t. Flyp es act inside Con w a y circles i.e. 4-cuts. 45
slide-46
SLIDE 46 Ro
  • ted
diagrams. Con w a y circle decomp
  • sition.
Lo
  • k
for maximal Con w a y circles. ) they dene a tree lik e decomp
  • sition.
46
slide-47
SLIDE 47 Ro
  • ted
diagrams. Con w a y circle decomp
  • sition.
Lo
  • k
for maximal Con w a y circles. ) they dene a tree lik e decomp
  • sition.
47
slide-48
SLIDE 48 Ro
  • ted
diagrams. Con w a y circle decomp
  • sition.
Lo
  • k
for maximal Con w a y circles. ) they dene a tree lik e decomp
  • sition.
48
slide-49
SLIDE 49 Ro
  • ted
diagrams. Con w a y circle decomp
  • sition.
Lo
  • k
for maximal Con w a y circles. ) they dene a tree lik e decomp
  • sition.
49
slide-50
SLIDE 50 Ro
  • ted
diagrams. T ree-lik e structure. No des
  • f
this tree are 4-regular maps
  • f
t w
  • t
yp e:
  • indecomp
  • sable
(4-cut free),
  • v
ertical
  • r
horizon tal sums. No de degree is n um b er
  • f
circles. Lea v es are the
  • riginal
crossings. 50
slide-51
SLIDE 51 Ro
  • ted
diagrams. Equations for ro
  • ted
diagrams. Let I (z ), V (z ), H (z ) b e GF
  • f
trees according to the ro
  • t
no de (indecomp
  • sable,
v-
  • r
h-sum). Then D (z ) = z + I (z ) + V (z ) + H (z ) I (z ) = X k 3 p k D (z ) k = P (D (z )) V (z ) = H (z ) = (z + I (z ) + V (z )) 2 1
  • (z
+ I (z ) + V (z )) ; P (z ) = P p k z k is GF
  • f
indecomp
  • sable
no des (4-cut free maps). But D (z ) is GF
  • f
ro
  • ted
diagrams (kno wn and algebraic) ) all series (including P ) are algebraic. 51
slide-52
SLIDE 52 Flyp es act
  • n
the decomp
  • sition.
52
slide-53
SLIDE 53 Articulated trees Theo rem (adapted from Sundb erg & Thistleth w aite): Ther e is a
  • ne-to-one
c
  • rr
esp
  • ndenc
e b etwe en
  • r
  • te
d prime alternating links
  • f
size n, and
  • articulate
d tr e es with n le aves. T
  • coun
t articulated tree, put them in normal form ! 53
slide-54
SLIDE 54 Articulated trees. Equations Let ^ I (z ), ^ V (z ), ^ H (z ) b e GF
  • f
articulated trees. Then ^ D (z ) = z + ^ I (z ) + ^ V (z ) + ^ H (z ) ^ I (z ) = P ( ^ D (z )) ^ V (z ) = ^ H (z ) = 1 1z 1 1( ^ I + ^ V )
  • 1
  • z
  • (
^ I + ^ V ): The series P is still the same ) all series (including P ) are algebraic. 54
slide-55
SLIDE 55 Articulated trees. Asymptotic n um b er. Theo rem (Sundb erg & Thistleth w aite, 1998): The asymptotic numb er
  • f
r
  • te
d prime alternating links satises a n
  • c
  • n
n 5=2 with
  • =
101 + p 21001 40
  • 6:15
and c another known algebr aic c
  • nstant.
55
slide-56
SLIDE 56 Prime alternating links. Unro
  • ting
F rom the p
  • in
t
  • f
view
  • f
knot theory , ro
  • ting
is not natural: we r e al ly want the numb er
  • f
unr
  • te
d links Tigh t b
  • unds
  • n
the n um b er
  • f
ro
  • tings
for links giv e: Theo rem (Sundb erg and Thistleth w aite, 1998) The numb er A n
  • f
prime alternating links
  • f
size n satises C 1
  • n
n 7=2
  • A
n
  • C
2
  • n
n 5=2 : Can w e do b etter b y estimating the n um b er
  • f
ro
  • tings
  • f
a random link ? 56
slide-57
SLIDE 57 Prime alternating links. Unro
  • ting
F
  • r
random planar maps, unro
  • ting
is trivial: Theo rem (W
  • rmald,
1994) A r andom planar map with n e dges has almost sur ely 8n r
  • tings
(with exp
  • nential
b
  • unds).
In
  • ther
terms, symmetric maps ar e exp
  • nential
ly ne gligible among lar ge r andom maps. As no w usual, a univ ersal result. But for links the situation is more complicated. 57
slide-58
SLIDE 58 Prime alternating links. Unro
  • ting
There is an in terference b et w een: p
  • ssible
r
  • tings
and yp e e quivalenc e 58
slide-59
SLIDE 59 Prime alternating links. Unro
  • ting
The follo wing steps allo w to circum v en t this dicult y:
  • The
parameter numb er
  • f
r
  • tings
is compatible with the tree-lik e decomp
  • sition.
) marking in GF + singularit y analysis Theorem (S., Kunz-Jacques 2000) The exp e cte d numb er
  • f
r
  • tings
is cn with c
  • nc
entr ation
  • Global
symmetries can b e pro v ed exp
  • nen
tially neglegible. 59
slide-60
SLIDE 60 Prime alternating links. Final result Theo rem(S., Kunz-Jacques 2000) The numb er
  • f
prime alternating links
  • f
size n satises for n going to innity: A n
  • a
n 8cn
  • c
  • n
n 7=2 where
  • =
101 + p 21001 40
  • 6:15
and c = 1 2
  • 371
p 21001
  • 1
  • 0:78:
and c is a known algebr aic c
  • nstant.
Co rolla ries: parameters
  • f
random links. 60