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

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

Folklore ) Ic ( CL ) UCL L ) link E i I ) U ( K ) E t ( knot Clk nontrivial ) Ki - Folklore ) ( planar graph Gi i embedding } { fig 1123 ( G ) SE → Epix { 03 t (G) t 1123 G → : , embedding trivial t ) d ( G ) Cf f Uff SE ) E := , , t from f to of changes minimal crossing number i

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

, ) . UH > ' I .

t 't " ' f n - . . jonesy 3) Uff 2 ¥ = ' . . ( f.) =3 C

Ic CL ) UCL ) L link E i ④ I 0=00 Lmirroh f. mirror D I D 00 CCL ) = C D) C = L ) -IB=c( od Etch d. B) ) )Emin{ UCL ° ,

i e knotted projection → IR 9 immersion ' G generic is i a ⇐ " it # Ex . s ÷÷÷÷÷÷÷ - a : ÷÷÷€¥÷÷÷÷÷ . * . 7 " ' " " - " ' ' I g . = - T pp

⑦ ' ⇐⇐⇒⇒ ' " ThnTfm,)=2n

Sketchpad ) U ( f E 2n anti o :⇒⇒ i÷÷÷⇒÷⇐÷÷⇒÷⇐÷÷ I t t t 22 " 21 - 22

) ( ) Z n=4 Ulfznti Case 2n o region cycle I q 2/92 EE 6 5 2 4 n f ESE ( Pg ) feet , .es#.e.is.e.HDez4 Lcf , 4 7 :=÷ 56 ) lift lkHH.tk#)t.E.lkftcs.tci.syPgE$2Llfg)= ) ÷ .lk/flHfCiDlzlf7:='?.lkHH.fcitiDtF..lkHH.fci-s ) If ) Pg lez .lk/flHfcit2Dt!E.lkHH.fCi-a ' t ) E. 14 0,0 ) ( 9,0 ,

: : : ÷¥÷i÷÷÷÷i÷÷:÷÷÷÷÷:÷ 2 edges these between crossing change " I 2 ) O ) I 0 O ) O ) O 0 I 0 O ( O 2 B g ( 2 2 0 O O , , , , , , = , , , , , , , , , I ) O ) O ) I C O O ( O ( I I O I , , I , , , , , , , , , , ÷÷÷÷ : : " " " 2) O ( I O - , , , , ) O ( I O O , , ,

a purely combinatorial argument show By : can we , Cfa ) 77 t.si:1#i:::::;=m---IT U ' , . . , O ) , O ) , O ) t ( I I C o I O ) I O 4 ( 2 0 O - , , ( 9 O , O , = , , , , , I ) , I ) O O O L ( O I O 1- , - , , , ,

If ) ! Probtem Find and relations between off ) u iz able EI projections knotted trivial planar graph G has G no is ÷e¥i÷i÷ii¥Y⇐ iii. = has online I = . t.si#i::ai:i:ai::i:i:ise

Cf ) ! Find relations and between off ) u does what it ? mean - all of knots set set X the ok - : - , X f knot K invariant → - i - I ' ] 2ft 't Exe K Di problem-LDecideflkIE.IT → Oia i Alexander Polynomial Ya Hs tri - y =p It ) Pll ) =D = { ' ] 't D ( K ) t 2C PH ) Plt E I , I Alexander )

: K knot fi invariants X Y K g X sets i Y → → i , , , Xx Y , g) K If ' → - KY kn I fl glk ) ) , Problem_2DecideH,g)lK)EXT " fandg " between at ion thgkiskoaiii.IE:1?....CT ' ] ( K ) 43 xD 't { It 13×2 l K ) I U D) n = ⇒ ,

. K Ezo → C ' o CIK - Crossing of Ya K number ) - t Zz ik → U o o Fits of NYK K number unknotting ) i braid ik Zz I → o - o w u braid ( K ) Kri I - bridge K I o I i → - zo U u - I bridge l K ) Kind

:i÷ BBB

I ) ma]KE :* ± braid ( Izaak ¥77 c - . , , ' ⇐ . O 4 O 3 O O O O D O O O O O 2 O D O D I O C o > 5 6 3 4 8 I 2 7 O 9

I ) .cc ' , bridge - T⇐f! bridge 3 bridge - I ^ 5 4 3 O O O O O 2 O O O O O O O I O C o > 5 6 3 4 8 I 2 7 O 9

I ) bridge braid C I o - - , btidge-llklabraid.ie# bridge . , ^ 0 5 4 O O O 3 D D O D D O 2 O O O O D I braid O > o I - 5 3 4 I 2 0

I ) braid - , OCU braid I - ^ D D O O 5 O O O O 4 O O D O O 3 O O D O D O O 2 O O O O O I U O > 0 5 3 4 I 2 O

I ) ( , bridge U o - ? ( not yet ) bridge sure I - ^ O D O O 5 O O O O 4 O O D O O 3 D O D O D O O 2 O O O O D I U D ) 0 5 3 4 I 2 O

÷÷÷÷÷÷÷÷÷÷:÷:÷÷ m } ft CK ) m ) Ck ) y ( a = max I : = ,

Az ) ( K ) 2=0×2 ( E C , . 6 o D knot )=2nt1 5 C ( torus -11 ) ( 2,2N - . 4 O )=nt 0 D knot 3 Az Kzinti ) torus - O D 2 lanky D O O I ° o 0 O O O I - - 2 D - 3 - 4 C 3 4 O I 5 2 6 7

SE (G) (G) SE Zeo Zeo U → c → : : , ZEO U ) (G) ( SE c. → : n÷÷::÷;:m:":iEum* 7- Exe planar graph DO GZ sit Gi . I SECGD # SE I OOD ( a) ( c. a) ( c.

tf (G) off ) ESE 't 2 , e U ) ( SEIG ) ) # I c. a) ( SELOOD ( 1) E C ' c. 2. q . . Gg £00 ⑨ rzED÷÷÷ 2,1 ) ) ) ( UH Clt ) ( = , : c I g) =4 u±Ic UC g) =2 4 ' C = . .

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