Teoria dei nodi9 449

f M IR Morse if ell m't lots are won defevery f M R becomes Man after smallpertaabetiece

o

M 2 CATE TOP DIFF PL

Jordan Soho.inbesfpwbleuif SCS S 2 8

7 0 S'M B 013,05 with BJE DmFB

Answers YES n 2 CAT _TOP

NO m 3 CAT Top

Alexander's horned sphere

ther YES m 3 CAT DIFF

YES n 3 CAT PL similar proof

PIoof fact 8315 has 2 components

IAST

Claim enough to show that Ep S C IRbonds B with BED Enough because

switching x c B with get other Drs

Chaim there exists a height function f B Rs.t flag Morse height L projection on

parametrized lice or simply

f no critical ft FINERY

Gdeat take any 1 projection and slightly perturbit so that reticle fo S teams Moore

2 take Gauss mop N S Iaced choose re S s f V V are

regular relics of N j take l t projectionon Spaid not

f

TA 9

a

0

9

T.TV In Tparlu

Fi f

0Have f lR R height h flag Morse

wlog can assume that critical values Cosas canall have one preimage Choon Cosa.cc c cancers

notice that h aj collection of circles inpixlailettai

actually h Tai E aite collection ofghitas

1

Gastead h ki E cite collection

of cylinders

with 1 local exceptionsigns eigenvaluesof Hk

t

Perform surgery on along each circle

of f aj individually

nixiaisf.fr YE

If the cinder in b la are vested I

arrange the heists of surgeries so thatthe inner owes take place father fun 024141

EF.tt Y

Result smooth closed surface s.tn eachcomponentconsists of at worst the following

one vertical tube on a Horse piececowhaing only one aidud point

some cops added doing surgery

Claim each component of the sugared spherebounds B with BED D's ignoring othercomponents Reason i

1 tube cops 1

a local minImax taps

7

3 saddle cops

y Bottom

DIC IIas

EFF

Sgune

CET

www.tC

t.f.EE

li

3 cops at different heights

0 does bound bond 3 disc

Notice never used so for the fact that SES

Claim the property that each individual cowpatof the surface bends a closed D3 revives

on I mudo surgeries one at a time

O

Rent

Gf dg tha N no separating after needingthis surgery D remains non separating often

undoing hath surgeries non separating o

S E 82 impossible by ID Schau flieseasy in DIFF category

Now 9 213 CE 2132 BI E Dmust show C 813 BED

Three cases

Cp C

B

F BpBz

Ps CzCz G

CRIB VBI compact

t2

BE BiBBED BED KII

o

Bridge index

LCS's link

b L minfk ISE S CS's s tIPL

5 2130213 and

Bj hABjE EINEN

Rey b L E overancs in a diagram

How to visualize bridge intex k links

Sare trivial bells1

I with tangle of kstrands

Take two parallel copies of S

S

rn n

p

7 fmi.im

Mette sin

round composition of 4 mops given 841010why orient presving mapping 4,2 pH totheciselresTitis therefore isotopic to it

through mops that need not preserve It 12kt

n nbridge index Klink is

ma IEToadgsfEIfoegq

U r U64 braid

theorem TCS's TED tonusthen 2k knot s t T Z K

Iroof Clavin there exists a sphere S s.t

SAT and some component of Sht does

not bound a disc on T

Puff of Claiat take height function fs.lah f It is Morse

Suppose aibad relax one z c Em i take2 C W C te r Em Wm

Tj h C a wj3

Noaa tIi

Property of Tj being woo of puncturedspheres

true fan f Ifalse for f m Tm T team

Take swellest m s k Tom has property Tnt doesn't

4 types of transitions depending on nature ofcritical value 2 on

II Elliff

still holds

II

ashed holds

3 IIstill holds

a It.EE

Ay Oz in differentagentsD still holds

transition is of type 4 with Milk insauce compound

Hence Na is non sep on Timeremains non Sep in T

Chain 4 pored with S h Wm

qtm.euTake N'C AT innermost on 8

If V bends a disc on T

tdefasshae.sit.EE

DUDE S'tDa l B DOD Bo

8Bs

can isotope T across Be removing it in

the intersectionnew T

s

s a

Continue rude find V innermost ou S s Z 2bDC S1 V does not bond disc on T

Swigert along D

T BiT

Ti B

N does not land disc on T

I sphere bounds discson both sides

Take B

then K x up than T 201K

T

o

Def if Ko Ki C S are oriented knotsI define their connectedsume Ko KeIsotone Ko K s t F SED 5 2130 213

with KjcBjtake QE oil x oil oriented s t

Q n Kj jl x 6,13 with orientation of IQmotchy that of Kj

and Q n 5 9421 10.13

Ko K Ko UK o.mx to 17

Ut Tom xlo.it

RIP Ko Ka is well definedregardless of all choices made

For unoriented knots Ko K is one ofthe at most two unoriented knotsobtained on TI KT using arbitrary eieh.huon ko ki

At most two C K'd f Ko Ko K

Peacock Koe ko tha ko K well deffor unoniented