2020/1 I 24 knot Friday Seminar Theory on Folklore ) Ic ( - - PowerPoint PPT Presentation

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2020 1 i 24 knot friday seminar theory on folklore ic cl
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2020/1 I 24 knot Friday Seminar Theory on Folklore ) Ic ( - - PowerPoint PPT Presentation

Mar 26, 2023 218 likes •510 views

Unknotting and numbers of spatial embeddings numbers Crossing planar of graph a Akimoto joint Yuta with work University ) Waseda ( Ta ( Waseda University ) Kou ki hiyama 2020/1 I 24 knot Friday Seminar Theory on

slide-1
SLIDE 1

Unknotting

numbers and Crossing numbers
  • f
spatial embeddings
  • f
a planar graph joint work with Yuta Akimoto ( Waseda University ) Kou ki Ta hiyama ( Waseda University)

✓

2020/1 I 24 Friday Seminar
  • n
knot Theory
slide-2
SLIDE 2 L i link UCL ) E Ic CL ) ( Folklore ) Ki nontrivial knot U ( K ) E t ( Clk )
  • I )
Gi planar graph ( Folklore ) SE ( G )

{

fig → 1123 i embedding} t : G → 1123 , t (G) Epix { 03 trivial embedding f E SE ( G ) , Uff ) := d Cf , t ) i minimal number
  • f
crossing changes from f to t
slide-3
SLIDE 3 Q . u If ) E

Ic

If ) ? Ansi ⇒ f E SE ( ⇒ G ) sit . UH ) > I c If ) '

÷÷÷÷÷÷⇒

slide-4
SLIDE 4 . ' . UH , ) > I
slide-5
SLIDE 5 t .
  • n
't . " '

f

. ' . Uff 3) = 2

jonesy

C ( f.) =3

¥

slide-6
SLIDE 6 L i link UCL ) E Ic CL )

I

④

0=00

D Lmirroh
  • f. mirror
D

I

= 00 C C D) = CCL )
  • d
  • IB=c(
L ) ° UCL )Emin{
  • d. B)

Etch

) ,
slide-7
SLIDE 7 generic immersion 9 i G → IR ' is a knotted projection

⇐

it

" Ex .

#

  • s
a ÷÷÷÷÷÷÷ : i e . * . 7
  • "
" " " ' ' ' I

÷÷÷€¥÷÷÷÷÷

. g =
  • T
pp
slide-8
SLIDE 8

⑦

' '

⇐⇐⇒⇒

"

ThnTfm,)=2n

slide-9
SLIDE 9

Sketchpad

  • U ( f
anti ) E 2n

i÷÷÷⇒÷⇐÷÷⇒÷⇐÷÷

:⇒⇒

I t t t 22 22 21
  • "
slide-10
SLIDE 10
  • Ulfznti
) Z 2n ( Case n=4 ) I q region cycle 2 6 5 EE 2/92 n 4 f ESE ( Pg ) Lcf , feet , .es#.e.is.e.HDez4 4 56 7 lift ) :=÷ .lk/flHfCiDlzlf7:='?.lkHH.fcitiDtF..lkHH.fci-s ) Pg lez If ) ÷ .lk/flHfcit2Dt!E.lkHH.fCi-a ) 14 ' t ) E.

lkHH.tk#)t.E.lkftcs.tci.syPgE$2Llfg)=

( 9,0 , 0,0 )
slide-11
SLIDE 11 ÷¥÷i÷÷÷÷i÷÷:÷÷÷÷÷:÷

:

:

:

crossing change between these 2 edges B g = ( 2 , O , O , O ) , ( O , 2 , , O ) , I 0 , O , 2 , O ) , I 0 , O , , 2 ) ( I , I , O , O ) , ( O , I , I , O ) , C O , O , I , I ) ,

÷÷÷÷ :

:

" " " " I ( O , O , I ,
  • 2)
, ( O , O , O , I )
slide-12
SLIDE 12 By a purely combinatorial argument , we can show :

t.si:1#i:::::;=m---IT

. ' . U Cfa )77 , ( 9 , O , O , O ) = 4 ( 2 , O , , O ) t ( I , I , O , O )
  • C o
, I , I , O ) 1- ( O , O , I , I )
  • L
O , O , O , I )
slide-13
SLIDE 13 Probtem Find relations between
  • ff )
and u If ) ! planar graph G is trivial iz able EI G has no knotted projections

÷e¥i÷i÷ii¥Y⇐

  • nline
.

iii.

= has = I

t.si#i::ai:i:ai::i:i:ise

slide-14
SLIDE 14 Find relations between
  • ff )
and u Cf ) !
  • what
does it mean ?
  • k
: the set
  • f
all

knots

, X
  • set
f i K → X
  • knot
invariant I

problem-LDecideflkIE.IT

Exe Di K → 2ft 't ' ] Ya tri Oia Hs i Alexander Polynomial D ( K ) = { Plt ) E 2C t 't ' ] I PH
  • y =p It )
, Pll ) =D I Alexander )
slide-15
SLIDE 15 X , Y i sets , fi K → X , g : K → Y i knot invariants If , g) '
  • K
→ Xx Y kn I fl

KY

, glk ))

Problem_2DecideH,g)lK)EXT

" at ion between fandg "

thgkiskoaiii.IE:1?....CT

⇒ I U , D) l K ) n { 13×2 It 't ' ] = 43 xD ( K )
slide-16
SLIDE 16
  • C
' . K → Ezo Ya t CIK )
  • Crossing
number
  • f
K
  • U
ik → Zz
  • Fits
NYK ) i unknotting number
  • f
K
  • braid
  • I
ik → Zz
  • u
w Kri braid ( K )
  • I
  • bridge
  • I
i K → I zo U u Kind bridge l K )
  • I
slide-17
SLIDE 17 BBB

:i÷

slide-18
SLIDE 18 . ( c , braid
  • I ) ma]KE

± :*

.

¥77

' ⇐

Izaak

, 4 O O 3 O O O O 2 D O O O O O I O D O D
  • O
> C O I 2 3 4 5 6 7 8 9
slide-19
SLIDE 19 .cc , bridge
  • I )
'

T⇐f!

bridge 3 bridge
  • I
5 ^ 4 3 O 2 O O O O I O O O O O O O
  • O
> C O I 2 3 4 5 6 7 8 9
slide-20
SLIDE 20
  • C
braid
  • I
, bridge
  • I )
bridge . ,

btidge-llklabraid.ie#

5 ^ 4 O O 3 D D O 2 O D O D I O O O O D
  • O
> braid
  • I
I 2 3 4 5
slide-21
SLIDE 21

OCU

, braid
  • I )
braid
  • I
5 ^ D D O O O 4 O O O O O 3 D O O O O 2 D O D O O I O O O O O O > U O I 2 3 4 5
slide-22
SLIDE 22
  • (
U , bridge
  • I )
bridge
  • I

? ( not

sure yet) 5 ^ O D O O O 4 O O O O O 3 D O D O O 2 D O D O O I O O O O D D ) U O I 2 3 4 5
slide-23
SLIDE 23

÷÷÷÷÷÷÷÷÷÷:÷:÷÷

y ( m ) : = max

ft

Ck ) I a CK ) = m } ,
slide-24
SLIDE 24 ( C , Az ) ( K ) E 2=0×2 6 .
  • 5
D . C ( ( 2,2N
  • 11)
  • torus
knot )=2nt1 4 O 3 D Az Kzinti )
  • torus
knot )=nt 2 O D I O O D °
  • lanky
  • I
O O O
  • 2
D
  • 3
  • 4
C O I 2 3 4 5 6 7
slide-25
SLIDE 25 c : SE (G) → Zeo , U : SE (G) → Zeo ( c. U ) : SE (G) → ZEO

n÷÷::÷;:m:":iEum*

Exe 7- Gi planar graph GZ DO sit . ( c. a) I SECGD # ( c. a) ( SE I OOD
slide-26
SLIDE 26 tf ESE (G) ,
  • ff )
't 2 e Gg q . ' . ( c. U ) ( SEIG )) # I 2. 1) E C
  • c. a) ( SELOOD
⑨ £00 ( Clt ) , UH ) ) = ( 2,1 )

rzED÷÷÷

:

c I g) =4 u±Ic UC g) =2 . ' . C = 4
slide-27
SLIDE 27 Thank you very much for your

listening

.