l･Seiya Negami at Yokohan訊National univel･sityand Pmfらsor Akim Y;lsuhm･a

一 −

7; ゛ヽ,こ乙一一T7Vご回7yl｀ ｀（ご ｀ pn

二L｡＼心.l,． ‑..‑..:.｀ぐ一二八‑‑‑‥ごｰ‑‑j一一ご ‑L‑j

一一

／ 。

 ･･乙 Ｘ 。

・− ‑ ヽ f j ¶

肖

ノ

一 一

一 一

● ? ゝ

｀ ｙ

ヽ一 一

−‑・

/

≒ −

tx ●

f｀w

･Ｗ

‑ J i

4−

ノ

− 〃

＿、．．Ｘ．．

t d ㎜ = ■ ノ

･J W

●−

Wj‑ ‑ ミ ミ → ･

‑￨

−

､ｙ･Ｘ ゝ．ノ

●− d J ㎜ −

. ･｀、'･･'べ

kノ．．

＝⌒●

  .S

, ． ｒ ｀ ４

．１ ｀｀

  〜 ． ￣ ． ｊ

ゝ ｓ

一 一 ・ ｄ

。･芦

ゝ./

−｀ ゜ ｀ 一

〜 . ＿j   ●I ･ ・

‑

−

ｘｆ

･ r ‑

● ｀ 一 一

W

〃

こ−i  八

● ‑ ＝ ■ ＝

ｋ ｊ 心 無

皿 ‑ ● ･ ‑ ●

／  ･･

● −

−

−

j y ″ ' ゝ ，

心 ． メ

− l , 一匹

J

II□剔H大学審査学位論文（博士）

￣｀ｓ．．へ.・｀X ﾀﾍ   II

｀ゝ

−

ｉｌ･Ｘ,ｊ‑．̲．．̲ｋＪ ̲Ｉ

〃 ヾ 私 心 Ｘ

− ¶ Ｊ ･ ‑ ‑ ･ ¶ −

−

／

ゝlj‑−‑りりゝ･ ″Ｊ／

ｒ − 一 一 −ヽ

9−/ 一一 W

Ｘ

／ふ ／

Ξ

ＫＮＯＴ‑ＩＮＥＶＩＴＡＢＬＥ ＰＲＯＪＥＣＴＩＯＮＳ        ＯＦ

      ＰＬＡＮＡＲ ＧＲＡＰＨＳ

        ＲＹ

ＴＡＴＳＵＹＡ ＴＳＵＫＡＭＯＴＯ

Ａ ＤＩＳＳＥＲＴＡＴＩＯＮ ＳＵＢＭＩＴＴＥＤ ＦＯＲ ＴＨＥ ＤＥＧＲＥＥ ＯＦ  ＤＯＣＴＯＲ ＯＦ ＳＣＩＥＮＣＥ ＡＴ べＶＡＳＥＤＡ ＵＸＩＶＥＲＳＩＴＹ        ＯＣＴＯＢＥＲ Ｄ９り

塙ふｈ ^{ho(j ･･S(jen(,e and} EI面11a,ri11尽 asedaじniversit,v

li鼠 Finally，l wish t,o expl･ess my hearty tlmnks t,(‑)my famny 函r suppo】ling my s(;hoolli鼠

・−

副（）kul珀Shilliuku‑ku

V(!1･S11

ｅ 

摺

ｄｇｅｍｅｎｔｓ

to mv thesis ndvisc･ Proj柚sor Shiyidli Sllzllki al Wi･19da ﾚ'1li

Nnjgoya lllstitutJeof･ Tぼhllolngy. PIlo16sor K()ukド'nuliylma nl lhky,)Wnmal‑1･s 脳i柚n U11iveIISity fnl｀thier gUiding me to the t]mme of my (liS91･tnti011 and h･

紬￨)r()II(=,ssorHit,oshi Mul･a.kmni f1.

剋円1l cll(･oUn胆

nt･ Mljgc】ya jllstjtute ot￣

C11rist,ia,n U11ivel･sit,v柘1

ｊ ｌ

咽ment 'md ndvice.lwislけ(バＲｘｐｎ３ｓ ｌ旦､･印･i 1 Xvs､=仙i LTnivel･sih･.Pmfらsm･ Y(･)sllivuki()

（

l･Seiya Negami at Yokohan訊National univel･sityand Pmfらsor Akim Y;lsuhm･a

f･{lss(1

証,'1'

1,11i。･(:onsistan,guidan(T.(mtlnuragem･ぞnt･ and ａssistanc(1，1 am also in(佃1)lpd to Pro，

雨tlldr hvalnn

)kyo(‑;akugeH｡J11ivellsity follt.hciradvice an(.lhelp t,hrough nly gl･aduate student.

)kyo l削孔8555･ .趾pan

Ｃｏｎｔｅｎｔｓ

lntroduction

１

２

Ｋ

１

１

１

１

１

not‑inevitable projections of Planar Graphs

１

２

Statenlent of result

4 normal form of a pllre braid ‥.

Ｉ〜

ＩＩ

１

２

り乙５

８

13

︸Ｊａｊ ７ １︲ Ｃ５ ＱＩ ¥C1 71 71 cQ 9

44 Ｏ

３

４

Ｉリ

A 11ormal follm ｢･ a diagram of a spatial embedding of a l)lanar graph on inevit,ability

of pUre (nyn)‑braidS●●●●●●I●●･●●II●●●●●●●乱●･●●●●I●●･

Proof of Theorem ］.1.3

n Habiro｀s Cy‑moves and vassiliev invariants

2.1 Slat,emellt,ofr(ヽsult

.2 vassiliev invariant.s and one‑b1･anch tree diagrams

●●●●●●●●●●･･■●●●幽●●●

Bibliography

List of papers by Tatsuya Tsukamoto

lntroduction

 K/hnivm･lm1吼s shown tbe f r)11(‑l,,･il一一1gillt｡,91･est,i11,9 1‑)1ヽ(‑ll'･(lsil.i()11↑,ll,11･s11,･｡,･s ;一一l(lifl･,･11･,･･11,j.｡

bdween klmt, tlheory and spat柚)gT･aph t,heory(PT(jpositjon L.I.2).

Proposition.(Taniyama[25D.  Th.a･,ji,qり｡7T77･ja,7･一句r,カMj9ﾉ･ ,j 7辿M,?/･炉･,り漬

心涌ペノ1｡l,い注四｀?μyz叩ｍｍ晶かz咄｡ 毓:mm2い刀心ams a 画卯mm Q/j,.Ｈ叩川止，

 Fmm t.1ふpropositjon.we（7an nat,urany t,hink about,the fonowillg question

Question.

￡4JC.s哨ぽcC公Ｍa ･re9111ar p rojcctio･n･吋'a7逍ma yygmp/にe,g/｡防, ｢

z2n. 殍"Q7･rl zl ゐ1/泄1万狗7 ひぼ7.1' a7 ｢ ･Mnjf7.¨m:か7･用心zθ7いjr翔｡か1,21.7j,.ｓｎ.

･μ゛7'･ﾌﾉ心印7 m m.晶lain 殄ttp://www.1.i71.

p7.c.s :7｀俑心友71 ｢切戸？

h1 Chapt,er↓，we aduaHy show t,hefollowing (Thenrem 1.1.3)

ＴｈｅｏｒｅＥｎ．    Ｆ併la叩咄 ･沁0辿矧丿句戸唄/j.μ仙.TR3.訪,自TJ一肩a,7面7附7･炉･n,･h.『;

       』‑j      ら.

a7 ｢山‥rりu,函‑戸句cd必ｎＧ⊂Ｒ2心c/口仙Drl四'?/山鯨ｍｍＱ随雨j, 沂ﾒQ川イに四山口71,s a.｀忌山りm7η.哨ａＤ･印7･esc71ts II.

￨ぐ17h山il'o has given a.nec･6sary gnd sufli(jelltcondit,ion for two knot,s to haveth(

sa､nle vassihev invalliantJs of t･ype n lﾌ)y int.rodudng locah･noves fol･ knots. m11(!d C､‑

･･s･s'･‑■yif.  A...j‑mnves. Aduany.....11...、/･1 ...‑̲̲..j.､̲､..̲..‥̲止)‥̲̲.‥−‥ U.･1･         ﾉ･・      1aイ迄‑move does not, chan匹any vassiliev invariant of typ,n， −l 1･fol･

any k】1(止 Howcver. it might, change a. vassniev inval･iant,〔〕ft,yl)e 7J ｢a knot｡111 t.he

(ｍ[／･＝２．Ｍバ]k肛1a h邸 ｢r四dy沁ven the d汀伺･cnce oF on,he vass山ev泊ml詣n↑,s of olldel･2 1T)dwe円付wo knot,対,h計にl川犯t,mnsformed into each othe川)vaC｡−mov(!

in[22]. ln Cλ1)ter2いve show t,he山面re11(･e of the va,ssihev inmrimlt,son)rdelつy

‰汁wccn two kn(面t,h計can b(づ,mnsfomled治to四ch ot,he円乃皿に,−move｡

      [

Chapter 1

Knot‑inevitable projedionsof Pla‑

nar Graphs

 ln tjhischaptel'，we show that for any knotted planar graph 召in R:y there exist a       −

planar graph G and it.sreg111ar projedion G C IR2 stlch that every d向男m￣1obtaincd n｀om G contains a subdiagram t.hat,represents jEjr.

1.1 Statement of result

  Ld G1)e a Hnite graPh. べve dcnot(リhe vcrtex set and the edge set of G by ド(G)and￡(G)respectively.Aq/dds

F(G)and￡(G)respectively.Aq/de is a loop orasimple

a looP orasimple

connected

connected graPh whi('his

graph which is

holneolnorphic to a circle. A 7J ｢みisa simple colmeded graph which is homeomorphic

to aバ･ls'Mrl line segment. A s7 ｢狛.lgmゐ 歸函,g of G is an embedding !7 : G H，IR3 of G into t,he 3‑dim(ヽnsionaIE11clidian space R3. and it.s image Gg ニ,9〔G〕is called a 叩 ･1 ｢9m7法，lf it.(:olllsistsof a single cycle or a disjoint union of cycles， then it,is (｀alled a l:7･ ｢orａ＆以:，A graph is called ld,(m,a,rif it has an embedding into R2. A

re!7 ｢arl pm面ｄ必n of a gral)h is as drawing on the plane whose mulitiple points are o ｢y tinitely many doub1〔M〕oints of edg(ls.lfw(･give over" an(い｀un(ler" infornlation at. each double pointj to a regula.r project.ion of a･ graph， then we obtain a. spatial graph(n ｀ar the plane)and we c･allit a. diagram of a. spatial graph (Figur(?1.1.1).VVe

２

not(ヽ↑hatev(ヽrv di ^ｌｆ卯'anl ilH1ふpap(｀r^ｊ is obtain,･d iIHhis way‥A diagl･alllis ns(ヽfulfor

shl(lyi叫いい可￣)atial gml)h

→

＼  ／

       Figure l.I.1

 Recently.tllel･e al･e mally works on Ramsey‑type theorems for spatial gl･aphs ([131，

[161.[171，[18D.These fono。 thc celebrated paper of ,1.H.Conway and C!. Gol･don ([6]).They showed t,hefollowing theorem.

Theoren1 1.1.1(Collway‑Gordon[6]).  EでenJs･7J ｢紬,1eT油dd,m9 of Ki con,t・,n,9 a n,m,‑tn17ial kn,ot

 Starting wi山this theor(｀lll.S. Negami showed that. every r(｀ctilinear spatial embed‑

dillg of)八'｡.witll su佃(?iently large posit.ive int(゛g ｀rn contains a prescribed knot. type，

wh(!re a spatial graph is y･,?d辿nazr if each ･･its edges is a straight lille segm(･nt,.Hpre note tllat t.his theorenl (loes not hold f()r generahpatial embeddillgs. For examplejf one makes a lo(｀alknot on each edge of a spatial embedding of A',1. then any cyde i11 瓦｡will involve a lmmber of such local kllots. which restricts thc knot t,ypes.

 ln tl函dlapt(｀r. we show a. Ramsey‑type theor(ヽm for a regular pro.iection of a plallal' graph. ltJis well knowll in kllotj t11()ory that for a･ny link diagram， there existJs a sellies of r:1｀ossingchanges whic'h (｀reates a diagllam of the unlink.ln ot.her words.

for ally regular projedion of a disjoint lmion of cy('les，there exists a diagram of th(?

lmlink obtained fl｀om it by giving over and under information t.o each double point

      3

(Fig11re l ｡1.2).This fact is us ･fnlf()rcal('111atillgpolynomial invariants for knots and

1il

fo

lks. Howev(,r.t,his is 11()t.truefor planar graphs.ln fad. K.Taniyama show(ヽd thr llowillg.

Proposition 1･1.2（Taniyamaj25D.   77たere j.s a. rEI!7?ゐｆ戸7加dj っnQりIZゐn 7｀

卯叩ゐ｡Ψtich.岳Q,telje¶' dia9r{lmu obtuined from it co7,ま{11,nsa dia9T{lm of a llo7lf Knk.

  The pla,nar glヽaph shown in Figure l.1.3 is an example of this proposition alld Figure 1.1.4 is a diagram obtained from the projection.

→

Ｉ

Ｏこ○○

Figure 1.1.2

２ ^３

１

３

Figul･,ヽ1.1.3

｜

Figllr(ヽ1.1.4

rhen，we ilsk a question: Does there exi吋,a regular projection of a planal･ graph

such that every diagram obtained from it contains a prcscribcd kn()t type'? Hcl･e we introduced t,heconcept of illevitabilityollspatial graphs.

Dennition. LetJj7 be a spatial graph. A regular projection of a graph is called 召‑ｍａＪ沁z仙?if every diagram obtained from it contjains a dia･gram of j7

 Then，the fonowing is ollrnlain theorem. Here note that knots and Hnks arespatial embeddings of planar graphs.

Ｔｈｅｏｒｅｎ１ １．１．３． ７Ｌｄ召 ＆a‥sl)atialembeddt71.9 of a pl.anargmp11.The11， there e一豆s a71刄‑m,?7沿 ｢心7J？7･ection of a plan.ar9ral)h.

1.2 A normal form of a pure braid

  XVI･ tr(ヽ沿,a bra,idas a diagram，and so dcfinitions are slightly ditfcrellt from the lls113! mles ((f 団). Let /J･リbe real numbers with 7J ＞り. Xve can a brjd diagram in [0，1]×【9,p】with 71 strands an フトゐｍ一河11/μ?(p,9)if the end l)oints ohヽv,･ry slrandare on lﾒ＝7)andlﾉ＝り.We simply call an ･り‑braid of type (LO)       −

alにμ‑hraid. Xve say that two 7･トbrajds of type （p･り）召｡and召;，areり ｢mZe ｢if

召,js（hjfol･mabh，to召;，by a nnke sequence of Rei（lemcister moves alld an ambi（ヽn↑，

isotf)1)yl｀el.∂([0,1]×[りj]()f R2. An n‑braid is (Jled p,re if it indllces ↑he trivial permutjat,ion in the svmmetric 既ｏ１１１，＆．Ａｐｕｎバyljトゐ7･a㎡isa pllre 71‑braid such

permllt･ation in t】le symmet,ric groul)＆.A pure (nj)‑ゐ1･㎡d is a pllre ?1‑braid sllc]1

that. it.s strands β1，/ゐ，‥‥仙−1.･.j'一j,･＋1，一一・・.jlj｡111｀estraight line s昭m(jnt amLjパroSseS

o ｢y /jl，β2，‥・，功一1(see Fig11re l.2.1 foran example).M7e define the pr 殳zd oft,wo rl‑

braids of type OJ, g)瓦,and耽,as follows. We obtain th(リ'‑braid of type (ρ･(7J十り)/2) by compressing召,1. A11d put that on the top of t.lle71‑b°id of type ((p十9)/2･9) ohtained from 召'by compressing 召;,．

a pure (4,3)‑braid   Figure 1.2.1

The foUowing is already known (cfぺ4)p.149)but,here we give an alternat,iveproof to it｡ へve call a pure n‑braid ddormed a､sfollows a normal form of a Pure μ‑braid

Proposition 1.2 (71，0一石翔,z心(f＝

一 １．

j4M/戸zre n一屏a. ｢can. 陥＆かrm 汾 IQapr ｡d司ju7で  峠

P7て)of. Letj 召be a.pllre ?l‑braid.Xve cal】↑he strands jl‥‥‥,･fj｡from t,he len.Xve asslmMl that. its crossillgs have different l/‑coordina.tes.Let p L)e a real number sllch that there is no crossing ()f･･……j,.onthe halflplane ly ≧7).Xve may a.ssllnlle tllat｣17°1/2.

      (j

Xve show that we can deforln 召so that any （･rossing ofljりul（［ち（j≠n.≠.刈occurin dle half planりﾉと1/2. When訊゛e lnove these cllossings↑hrongh the upp｛｀rcrossillg･

there are 16 casljs as described in Figul･e 1.2.2 to be considered. ln each case. we can easilv find su(:h a deformation as in Figllre 1

Therd()1･e召is the produd ()f召 and a.pure(

︲2  爪 3，1eaving thc details to thc reader.

n･）‑braid，where 召 is a p11rc ?トbraid with straight n‑th strand. By doing such a proccdure illductJively.M e ca,n （｀omplet,e

the l)rootl

・１ｎＤ︲

β/匹

ブ

I‑A

≪

２

ｘＩ Ａ

ノ

1‑A

ヅ

1‑B

だ

ブ

／︲︱

Ｂ

ﾚｿ

1‑C

士

2‑

☆

｀ ，

Ｃ

千で

1‑D

゛ ｀ 1

ｋ

ナ

2‑ Ｄ

寸

 ど｀ヽ，

 1‑E 甘

3‑A

Figure l.2.2

≫づ

ﾚ］

だ

ノ

匹レリ穴

f沁

ど

牛

プＯこ

土

3‑A ﾔﾝ

心

甘

心ﾌﾞ ﾚ]

.ﾌﾟ

Figure l.2.3

き

レソ

1‑H 甘

3‑D

□

１．３ Ａ ｎｏｒｍａｌ ｔｏｒｍ ｏｆ ａ ｄｉａｇｒａｍ ｏｆ ａ ｓＰａｔｉａｌｅｍ‑

ｂｅｄｄｉｎｇ ｏｆ ａ ｐｌａｎａｒｇｒａｐｈ

   Ld，G be a planar graph， alld ，S'a subgraph of G.Let yj : G 升y be an elnl)edding，where S2 is a 2‑sphere in IR孔 For 。 EU(S)and e E￡(S)，we take a disk neighborhood 7),バ)f,以内and a band neighborhood ｣臥of∫(e)in S2.Then (U･JEIりjS)μ')U(Uべ≡E(s)瓦)iS Called a 山武/硲ndn哺7ゐ.borhood oH(S)Mld denoted

byじ(fS)(Figure l.3.1).lf S is a spanning tree of G， then we ca11 rげ．Ｓ)∩。以Ｇ)ａｎ ａＭｎ直心spa7min･q tree oj･/(G).Here we also call びげ，S)a disk/band neighborhood

ofl t.he extJended spanning tree. Moreovcr. let 夕：Ｇ冊R3 be a,spatial embcddillg.

Ther回7oj−](71)is called an extended spanning trce of !7(G)，whereTis an extended 印anning tree o仁以G)(Figure 1.3.2).Let ･7 be an extended spanning tree of･八Ｇ)，

andびa disk/band neighborhood of 71. A pair of･r and びis called a, &a5icn?dan171e ofμG)ifびcoincides witll the rectangle [0,11×[2.31 and each point of rn∂びison the line μ＝2.Then /(G)is called a ln脚al戸j7でμ ･r印r,?se ｢a12･7z of G of type o if it has a.basic rectangle， /(G)∩([0.1]×[0. 2D consist.s of vertical straight. lines an(1

5(G)＼∫(G)∩([0,1]xlO,3D is under the J‑axis (Figure 1.3.3)･

μ(刃ぽ陶 ^Ｕ(ｆ,ｓ)

Figl11･e 1.3.1

   S

an extended spanning tree o冒(Ｇ)

言

肩印

１

ご）

戸別

Figure 1.3.2

４

   ６ びＵ方図

Figurc 1.3.3

an extended spanning tree oｌｇ(Ｇ)

a trivialpure plat represenlation of G oftype o

 L jt･./･(G)be a. trivial pure plat represent･ation of G of type jl，‥・，心.we detine a trivial pure plat, representation of G

n）

O. For posltlve lntegcrs of tvpe a/1、‥‥ぬj as

，‥‥心，we detille a. t.rivia]pure plat representation of G of type (dl，‥｡，d｡)as Hows.where 2･,l＝げ∩∂じ￨.Let,/I‥‥j2｡be the vertical lines of ｡μG)ill[Oj]×

{0,21.Letり ら∩佃＝２}

‥・，らbe the edges of G， where at least.one of the two points in

is on tjle left of e

J･∩{μ゜2}.XVe assume that らand /1･(j＜1')are

subarcs ofり.Thell，delcte the subarcofGullder t,he jr‑axis. P11t 2必vertical straight，

lines of lallgth l betwccn /j and ら＋linO≦ﾒﾉ≦1. Connect /j. these v('rticallines and /1.with arcs in l/ くO or in [Oj]×圃2]altcl｀nat.ely from thc left so tllat we oMaina11

９

eml)eddhlg ｢･the edg(リ｡. Apply su('hadd()rmation tn t11 =｀edg(｀sり l.3.非

ｙ ３ ２ １

ｋ

ズスヘ

 一

に

０ し」しと＝ﾘ１ ７

ヨ  →

a trivial pure plat representation of G oftype o

Ｅ  →

Figure 1.3.4

a trivial pure plat representation of

G oftype (2,1,2)

らj（Figl11･e

 A(liagram jP of a spa↑ial embedding of a planar gral〕his caned a pure plat repre‑

sentation if there exists a trivial pure pla.t represent41tionToftype(尚，‥‥心)and a pure ?n‑braid 召(m＝271＋2r11十‥・十2心)such that に)is obtained from Tby replac‑

ing匪1]×[Oj]with召.VVe remark here that a pure l)lat representation of a spatial graph is homotopic to a trivial p11re plat.represent.ation. Thus ptlre plat representions aredefin jd only for planar graphs. Then we obhlin t.he following proposition.

Proposition l.3.1  yln,1/即 殃 ｢，?77,ゐ 歸m!7司l(m!J pl{17MIr 9Tq}h has (1pu｡･eμ n17J7･ぎりj弛 ｢ ｢Zθyj.｡

P7･oof， Let. G be a planar graph， y : G →S2 anembedding and g : G →R3 a.

spatial embedding. Specify a spanning tree of G， and t,akean extended spanning

tree ofがG)and a,basic redangle of respectively. Shri】lking7 so as t,o be

it whi(ヽh we denote bv Tan(I召

respectivejy. Shrinking 7 so as to be in a neighborhood of a vertex for がG)，we (･an make 7' have no crossings(Figure 1.3.5).Using an ambient, isotopy.we make ･7

coinci(le wit,hanext.ellded spanning tl'eei117?. andがG)−Tbe under the line･!j＝2.

Ｌ(ヽｔｎ

ら1)e the ed卯s not containcd in the sl)anning tree ot'G so that,Ghas a

       川

left‑゜ol｀eilltel｀sedion ゛'ith!/゜2 thall t.hose ofり(i＜j･).Let尚be the left‑most illtersedion ofQand !ノニ2. L゛Lheljrom唱，a11 1ocj minimal and maximal points of

Q i11タ(G)by鳴・‥．ｊ;が〕−I･ and label tlhe right￣most intIEsedion of Q 311d !/ ゜ 2 by

当年） Modifly ei so that local ma･ximal points 嶋･ぬ･‥．ｊな間‑2 111ay be on !ﾉ゜2 near

the right of心and ei ajlwaysclockwise touch l/'2(Figure 3.6).Then、modifyらso that the locj minimal point d; may be on !/＝−l and have itEr coordinate between

those of d;‑1 and dyl， and t.hen e,･clockwise touch l/ ＝−1(･j＝1,3,‥.，21/(1)−1)

１

亙

だ ら

３

２

μ印

４

Ｉ

Figur(!1.3.5

  11

ｼﾞ旬

a basic rectangle of μ印

づ

１

 心 白 白⁚ にり

ベジ

心 爪□□

めム⁚ にり

奸匈

2y＝−1

Figure l.3.6

ｙ ^一_一 ‑1

lf there is an edge which has intJersections wit.h l/ ゜ 2 betweeen those ofらthen modify the subarc in under the z￣axiswhich contains 嗚邨)−1so that, ・ coordinates

of these int,ersedions may be between those of the subarc and jr‑axis(Figurel.37)

ﾚ付言〜

Figure 1.3.7

レそ ｰ．

言

By an ambient isotopic modification、we make the braid part in 7 ×7.Thus we

obtjn a l)u1'el) t.rePresentation of ダＧ) □

 Noy since a part. in (Oj)×(0,1)of the pure plat represent.ation 〔〕fj7(G)is a pure braid. it has 11ormal fornl (Proposition l.2.1).Thus，we ca]l the representation of

!7(G)with its braid part. deformed int.o a normaHlorm， a normal form of g(G).

］２

1.4 0n inevitability of pure (陽痢爽braids

ln t1ふsedion， we show tJhekey lemma. At t.hebeginning， 訊'eshow a special case.

Ｌｅｍｍａ １．４．１  Ｇ２,Ｊび｡ap,177回71，7j.)一石ｍ ｢召バ尨7‑,り,9aμanar鰐卯ｶ.G一法仙I

，a･nμ1刀心ねs，恥}c l/(G)s 殃禎/in!μ尨かふ一回c四d雨｡r,.s.

円Th,ere is（m eyrlbeddiyl,9f；G−･s，[O川×匪1にR2 ac幻み ｢   y隔]＝(た/(n十1)，1)匹df(6)＝(た/(n十1)，o)(た＝１，‥・パo.

(2)7‰rdsQ CQnlm71Qus m･叩ｇ：Ｇ→匪1]×匪1]C R2 11函咄咄,心c6a7河辺a.r   p7司 ediQn司IG such that:

  (ａ)凶4)＝y叫Ｄｎむ7㈲)＝阻k)(た＝1，…，nDj ｢

       〜      ．. ‑.

  (b)For（m!J（Ra9r（1m G obta伍dfTヽom9(G)バlzere心□z juM2卯ram S C Ｇ ｓｌ,ｃｈ，

     〜

   話.alSie一如αle ｢仙石．

Proof..Let召beapure (n,n)‑braid.Firstwe construd G，y and g so that they sat.isfyconditions l a,nd2(a,).Secondly訊゛eshow tllat they also sat,isfycondition 2(昨

ら

For a pair of an even number P and an odd number 9，we ddne a l)lanar graph

り ゛hich has an embeddillg /' : G｡,9そ(Oj)×(Oj)C R2 as follo゛s. P11t p cycles   ・゛も(i°1･‥‥p)concentric311y. Thell idel!tify lり゛'it.htぐ1iりand j have 7yD'4 ・ lも(i°1･‥‥7))concentricajly･ Then identify llJjwit hり+1 if i and j have

different parities (i＝1，‥.，jフーlj＝1，‥・，29)(see Figure l.4.1 for an examplc)･

Thus□/((み,9)￨゜(p十1)9 and l￡(G7･,9)￨゜2周.Here we put p十1 auxilia.ry concentjric

circlcsべ〕、

ら…

^{4 and} g auxiliary radial straight lines べｊ;，‥．ｊいn(0,1)×(0,1)

as Figure 1.4.2. We den(鑓e by c; the uniquevert.exa.t intersect,ion of s; and らｆｒｏｍ ｎｏｗｏｎ(ｉ＝０，．‥，ｐｊ＝１，‥．，９)．

１３

貼弓削十Åヤ

︲n Ｖ １４ ｙ

Ｖ

Ｉ﹃４

 Ｕｒ

→

Figure 1.4.1

ＯＩ Ｃ ／１ ／

バG4淘 ／

Figure l.4.2

Let 11 be the number of crossings in 召 and l)ut j7

バG4淘 ／

／り￡ 〜０３

ＦＣ＼ ／１Ｊ

 ｊｉ

＝27巾2,十のand y＝4n−1

VVe dcfine t,hegraph G as a planar graph which contains GM,A･ as its subgraph and

satisfies the flollowing three conditions (see Figure 1.4.3 for an example)

(1)U(G)

り一ぐ

）拉Ｇ）

＝{ら・

＝び,lcl，

７ a

･γ ? 61，‥・，恥}ＵＦ((ゐUv)，

汗訣(−ﾄ1一回

貼‑1回巾‑１)}Ｕ￡(ＧＭぶｙ

 Ｃ Ｉ６ _０

２ ７ １ ９

(7 7ぷふ･＋1 al弓０十申

(3)There is an embedding y : G 冊[Ojlx[0川c R2 such that

伺

メｊａ

ぐ ｙうｂ

ぐ

レ

, Ｍ

Ｉ ． へ

(貼 ^）

一

一

一

一

几

(貯(n十1)

，ID7(f(岡＝(回叫十旺o)(1＝1 O and

NotethtlU(G)半(2八(n十の十1)(礼−1ト2n and l￡(G)￨＝2×屈(八十の(礼一万十％

These G and .farethe desired planar graph and its embedding

     ♭、 ♭1万 an

embeddingμGJ of

G in

lhe case

lhal l

＝211dd＝2

Figure 1.4.3

No゛'゛'e gi｀'ea regular projection y(Gy9)･s the image of ,/･'(G,.J 1!nder the 2‑fold L)randled covering map of R2 brandled over t･he center of concentric circles. where ダ：(み､9H'(0､ 1)×(0･1)is a continuous map (see Figul｀e114 4 for an example)

concretdy， we constrad y(G四)as follo゛'s･ Putづn(Oj)×(O･

an example).More 1)、P十1 auxiliary

co1!cent4 ic cil｀clesso，sl‥‥ りand 9 11｀ili゛｀yr8dial half lines /1，ら‥‥ﾘ9 ends

on t･hecenter of恥(Figure 1.4

^5).Put the vcrtex りon the int.ergction of s，and り       15

回 ^‑一

() 7)J ^{=１、‥．}

g).connedG

^{ａｎ(ｈＳ．ゴ}

and tllesubscl･ipt j is considered modulo 9

・ − ■ j l

1 5

｡ 。 j

/4

r︲

/l

μG2,幻 ／

.4'･

J2 ５にjo

/3

ll cj4  → ｌ

→

Figure 1,4.4

り l

Ｎｖｌｙ Ｓ

 ＣＩ２４

ふ:l

．・

０４ Ｃ ＱＩＵ

・φ⁝⁝・・﹃ １⁝⁝⁝ｊ・．

Ｃ

   ・・・．ｊ・｀″４２

０１⁝ ⁝

１１.2１ ＪＣ・・

2 匍

･ , ． ３

a】ldconned cい1nd r甘いwhere l ≠j]

/rG2,幻

ぷ

ｅＣ．

に り

︑・Ｃ．⁚

Figure 1.4.5

→

g'rG2,5J

 Usil!g y and y(Gjり')り｀'e(lefille ｡ colltil111olls1113p 17 : G →匪1)×[Oj]C R2 which induces a regular projection of G and satisfies the following six conditjons (see

Figure1.4.6foran (ヽxample).

 (1)j/￨らにﾆﾆﾀﾞ

 (2づ(絹＝汗/吊十1)パ)皿か(脳＝

んぐ

/叫十牡Oバん

^一一

 角

︱

廊

１６

− W ‑

(3)For any edge いΞ￡(G)d7(e)is an embed(1edarc.

(4)g(匈ぺ‰十1‑JJhas no intersectiolls(だ１に‥・剛

(5)j7〔αμ･し1+&〕)hasan exactly one intersedion with g(Q'ld･1申J(だ１に‥'･り￣1) (6)!7(αJr4L＋1)hls exadly 71￣1 intersectiolls゛'ithg(りぺにμ))(だ1･‥ ・"￣1)

 Since it is clear that G is a planar gra,ph which sat,isfiesconditions l and 2〔a〕of the lemmajt is sufncient to show that we can find a subdiagram satisfying condit,ion 2(b)of t,helemma in a,ny diagra,m obtained from g(G).

a regularprojectionがGJ ofGinlhecasethat刀＝2 and ,y＝2

Figurc 1.4.6

  17

bn bn−l

・参・Ｓ ・・・・

●

●

●

● 畢

●

●

●

/ば2/3/415

●

●

●

● ●

●

●

● 個

●

●

●

わ１ ａｌ ａ２

 ・一一・

一−ＩＩＩ・一一﹄一−Ｉ一ＩＩ一一一ＩＩ一一

 ・・一着

12;1

・９ゆＩ ・ＩＩＩｌｌ・；ｉｌＩＩＩＩＩＩＩＩＩＩ１一

 ・・一一

−一ゆ１ゆ一一ＦＩＩｉａｌｌｌｌＩＩ１ＩＩＩＩ

 ・・・自

Φ一一ｆ１・ａ一一・｀ａ一ｔａ一巻ｌｔｌＩｆｌ

 一一・・

● ・

● ●

●

●

● ● ●

幽

趣

●

1 ● ●

●

Figure 1.4.7

●

●

a.zl‑2 a,1−￨αΓ1

ＩＩ一Ｆｄ一ＦＩＦＩ一ｉ！ｌｌＦＩＩ

●

●

●

●

●

・・

個 一 睡 ｊ

一争・

：⁝・一月

  ４１■■ ／

●

一一

●

・一・一   − ／

｡ fN  ． ｔ

⁝⁝⁝４  ″１

  ４

JO n

52 励 J4

ＪＭ，２ ぶμ‑１ ５Ｍ

Here Figure 1.4.7 shows a figure obtained from !7(G)by cutting along a half‑line/1

Weuseit to show the exist,ence of the subdiagram a,s above.lf we identifv the left，

      −

end and the right end in that ng111reけhellwe obt・11 j7(GM,N).The subscriptl j of ら

is collsidered modulo 4n − 1       〜Now we wiH nnd a subdiagram S satisfying condition

２(ｂ)･

 Ｌｅｔ(ｦ?be a diagram obtained from ｡9(G)        −Xve take a subdiagram of G which is

equivalent to jg as fonows: ln Step O，we take 2n pat,hs which wm be expanded st,ep

by stJep. ln St可)sland2

^{we expand} the paths eliminating the crossings which t,hcy have. ln Steps 3 and 4，we further expand them so that we realize the crossings of 召.ln Step 5，we make t,hem n paths which is the desired n‑braid

〈SteP O〉 We define 27川)aths(昇

１ ＯんＱ

机

Ｏｎ八 ７り ＝

一 一

一

一

人 弓（77十Ｉべ）

叫弓○十Å･）

a7yj71＋1

Ｏｎ⁚Ｑ

(た＝1、

I  5 昨

(た＝1、… ^7?.一汗

訪

  ！ＯＩ召      〜

，j?2 in G as follws

べ

1ve denote by QI（resp.捌･）a path with its end points at,ら（resp.ら）and

Thell we cal1 わj and a･j｡斥r 殞dp ･ ｢s and caII べ.a.?ljQn‑j& 7召押z7j. We

simply denote by Qj (resp. jちj a patjh with it.s fixed end poillt at 6j (resp. a迂

･we denotc by￡QR(1)the ordered set of the non‑fixed end points of (2n−1)paths Q:｡…，QX，罵，…バ?に1. we say that ￡QjR｢Oisina卯 汕齡@仙nif￡Q｣1拒)＝

{ふべ+21‥・5ぺ十2(2･―聊ぺ十2C2n‑2)}for SOme lμlhat iSけlhey are eVery otlher Verti(･eS

on the circle s,･in this order.  Here，we note t,hat｣1?a crosses 杞仁1，7?2＿2，‥.召?

individually exactly once in this order from it,sfixed end point and that ￡Q7?(0)is in a good position.

      ･‑J

〈Step 1〉 We assume t,hat祀いsover7?? atthe crossing point in G. Then，we denote

by x the crossing point of edges d"￣‰rl and cl'1￣lcP，and take 2n paths according to the following two cases. Here･ note that tlhe non nxed end point of j?? is ぺに＋申

回d言‑1＝べにトぐ二首⊇ぐndャー1＝ぺよ≒レ）

(case 1‑1) lf the edge cil￣1cP is over ぷ1￣ld" at x、 then we t､ake 27リ〕aths as follows(see the left of Figure 1.4.8)、wllere ･β(1r)＝2(n十１−た)andψ(だ)＝２(ｎ十奸

仙 １

Ｑ²⁷¹

ん･

加

い

召

枇

｀砥Uc‰･げ沁)+1 ･・' ぐぷ｡〔2,1‑1〕峻峰ゅl     (λ･＝1， ･･‥n)，

゜司Uc‰げ脳)+1 ・'・ぐ函い2，1げ旨)士2.  (た＝1‥‥，n−1)，

‑

‑

坦

^ＵべJ, _1＋1 ^4, 1十2 cg,1＋3 ・ ・ぺ ￣l d71

crl￣l crl

case 1‑2) lf the edge cy￣1(す帽sover cy￣1cy at x、 then we take 27z Paths as ollowosee the 贈ht of Figure L4.8)、whereφ(ん)andい㈹are sanle a､sabove:

19

１ りん０ むｇ召 ︲りｎ召

一

一

一

一

一 一

尉 司

坦

Ｕぐ円く(○一一!

Ｕぐ㈲回付)パ

Ｕこ仏十lcLc

e2 el＆尺1 /?2

２ ２ｎ−］

一３ Ｃ

c!%

 Ｃ

ぐ ^91一

一則

一句釦司 Ｃ

1

−(27･−1)

1

−(27･−1)

９乙 一

 ２ ７−２３

 Ｃ

示

ぶ固

 乙

岫一如

ぐjy ト2･･

−1

……●･･ｌ

ヤ

(か (ん

Ｉ ＝知３ Ｃ

知２ Ｃ

Figure l.4.8

一 一

１ ○

づー↓）

Q2 el＆別尺2

 ln both casesjt. is clear by constmctions that ￡Q7?(271□s in a good position and that we can eliminate crossings of｣附"and j??t".vve can similarly tjreat the case that 椚isover 召2.Thus，v1'e obtain the following cla.im.

claim l.   suppose that ￡Qj?(j)is in a good position and thatt.he non‑fixed end

points of召:，and｣弓areべ.and c;＋1 respecti｀'ely we ca.n t.ake 2?l paths in (ﾗ，Qr(り，‥・，Ｑが)，j7r(

(1)Qy)⊃Qいnd愕ぃ･)⊃机(た＝1､…、昿 (2)￡QjR(り(i))isin a good position、 and

(3)Uレ

ovel≒

or some f. Let,9(;r)＝jr十2n.Tlhen，

(Q円U7?び))has

one more crossings than ULバ砥Ｕ机)does. More‑

thc only crossing point. is of 凡l and j?j and゛'e can specify ゛｢hether瓦js over 」叫atjthe crossing pointJor not.

20

〈Step 2〉  By using Claim l, we can eliminate c･rossings of 翔t and j?g，7?‰‥・，｣附‑1 by turns in this order as Step l. Then, we take 2n pat.hs Q六万‥，Q2(犬刈j(宍‥・，瓦吋) such t,h計

(Uﾚ(引(司Ｕ枇ﾌﾞ))Ｕ脳中+!)+Ｕ−１ｃと昌s amL)ientisotoPic ｢

∂([Oj]×【Oj】XR)to a graph illustrated in Figure1.4.9,whereη(71)＝2n(yl−1)and

−      ‑‑       〜

G2･1〔d+1〕十1,4n‑lis tjhe sub(hJgra111 of G ゛'hich is the part inside S･,㈲of G. Mor(｀ovel'･

ＥＱＲ(η(n))isin a good position,and non‑hed end points of司ﾂand川(刎are

and回㈲resPectively for some f. Thus， ￡(ﾘ?(η(η･))ニ{cげ1ﾚﾄ3，

ｙｌ

０

αＩ ａ２・‥αΓ１‑ ２ αΓ１‑１ αΓ１

Ｆヽヽ一  Ｇ

一   ．・  ぐじ⁝⁝ソ

   ･ ･ ● I ･ ● ㎜ ● . ･ ･ I

● ● ● ● ● ●

ｂｌ ｂｌｌ…１｡１．２ Ｚ｡１．１

１｡，１

Figure l･4.9

c7yj l＋い

ぐ）

頃司 ^7肋）

１ ら二3 ， Ｑ二1

ズ

くStep3) Letβい‥，ふbe strands of our 71‑braid 召from tlle left｡Remark t,hat every crossing in 召consists ofふand叫(1≦je≦n−1).Wecan assurne t,hat a11 crossings have difrerent !/‑coordinates. Thus let ;rl，;1･2，‥｡，勾be crossings in 召，

where恥has lower 2;‑coordinate t,han恥＿1.

 Here we consider a crossing ;rl. lt consists of汰andゐ＿1、We assume that ふis o｀'el'βn‑1 t JI' Thel!･ ゛e dellote by y the crossing pointl of edges げ7;ｺﾞｰ1〕cな;｡‑l and 引に

1引足＿1)、where咄the integer at St叩 2. Note tha回応仁＝戸(□問:帽･

Xve also take 27z paths according to the following two cases｡

       21

士ノ

㈲sべに) lf the(雨eヤ二言)聯レns o゛eに仏工聯レノ

2n p計hsRs follows (se(付he le住of Figule 1.4.10)，where 匹拍＋３−２(ん−1)andい(ん)＝匹2八十３十２(か一汗

１ 釦

．ｒＱ

峠2

一

一

一 一

恍 (")しJどト惚円

7/(○刊277−1) 7面)十2n

at yへthen we take 2尚大壮φ㈲＝

り(頑ニ２

φ園

順フレ‥心江口;け雷にご(に1､‥･パ)、

 7面)十1 り豺卜1

j面汗(271−1)

‰㈲づ27･−1) ^Ｃ

;(,jTtこ(た＝レ〜ロベ)，

_φ㈲−2n

司゛＝坦々)Ｕ々("‰だけ1‥･ぐ沁し1げ昌｡ぐに‰)ぐに4.1)･

(clse3￣2)lf the edge ﾔﾚﾊぐ沁川is o゛eに混よ1)ヤレ1 at y，then wc take 2n

paths as follows (sec the right of Figure l.4.10) as a,bove:

１

ドＱ

峠2

柏ヅ

一

一

一

一

一 一

・／ whcrc０礼φ(O and 回腸) are sanle

ｕどト吻円‥べ把tt一万 瑕

枇

φ肉

ｏしレ匹

り團十1

 η㈲十1

％収)十1

｀'φ㈲＋や7‑リ  ･か)十(271−1)

‰(○十(2･1−1)

がｎ)十2n φ圃十2n

Ｃ貢ﾊ十2n ｾﾞ縁)十2n

01) U回("り吻)ｏ‥ぺに昌1ド回昌ぴ昌0‑I)

＆1 ２ jも１．１

r……●………●････. 52rl

ゐ2

‥ ‥ ‥ ‥●･ 一一一 一

馬1

Figurel.4.10

２ ﾉぞyl。1/むl

  (λ,＝1，‥。n)，

(λ:＝1，‥｡，n−1)，

ぐ･゜…… 52zl(z･‑1)

………●…･･

1….?ゝ‥･…….

 ln both cases，we take only one (2rossing point, and￡Qj?(2712)is in a.good position.

Ivc can similar】y trea,t the case t.hatツ∂,1＿lis ovel･ β｡at zl.Thus，we obt,ain t.he 柘Howhlg claim.

       22

claim 2  sul)pose tha.t.EQR(j)is in a good position and the non‑fixed end points of凡and罵are c; and cに1 respedively for some f.Letぅ](利＝j:十2n.The11， we can take 271 paths in j、 Qr(り‥‥、Qがり、7八重)ぃ‥、7だぃ･)such that:

(1)切巾〕⊃Qいnd可( )⊃机(た＝1､…、吐

り﹄

ぐ

）￡QR泌国）is in a good position，and

(3)Uﾚ1(尉(oU可つhas one more cross≒thanUL1(借Ｕ机).Moreover，

the only crossing point is of j?n and ｣1ち･，and we can specify ゛'hdher j?･ is over 沢j at t.he crossing or not｣･

〈Step 4〉 XVeassunle that we have chosen crossings ヱ1‥‥ｓ ｌｋ.lfらconsists of strands瓦and馬けhen zJ･＋lconsists ofゐandβj＋lor馮and鳥−l since 召is a pure (n,n)‑braid.ln both cases，lwe can also choose a crossing 24＋1 by using Claim l or 2.

Thus，we can chooseall crossings in 召．

〈Step 5〉  No゛'･￡Q｣1?(7v)is in a good position， ゛1nd non‑fi゛ed end points of 凧?し1 alld jぐareぷ‑1 and ぶfor some がrespectively, where 7V＝2ηり十n−1).Thus，

ＥＱＲ(7V)゜{ぶしむ,十3，ぶし如＋5，‥・ ，φし3，φし1}. Then， we t.ake a subdiagram

〜        〜

S＝ULI八in G as fbllows, whereφ(だ)＝が一如十３十２(ｎ−１)十2λ･＝が−2n＋2を十1，

Ｎ ote that cjり一肢十2二₋ ぷ‑2n−2ゐ+3°c匙4｡＋3＋珈−n is the non￣nxed end point of Qg:

    八   ｘ  八十へ     Ｎ十（2匹3） 八十叫‑2） x＋（2た−1） 八十叫−1）−1 八二荊．Ｕｃφ防）cφ㈲−1 ・ ‰㈲−（2だ‑3）‰㈲−（2い2）cφ㈲−（2匹1）‰㈲一勁−1）−1

暗昌二仁

 /V十(2匹1)−(2匹2) Cφ圃−(2匹1ト(2匹2)

几＝舒しレトご爪八Jぐ

ぐ4)一如+2UQg(た＝1,・‥．ｎ. − 1)

hl t,hisstep， we take no crossings.lt is clear that the result,ing condil,ion 2(b)of t,helemma. Thereforexve complete the proof，

Now， we prove the key lemma. 11sing Lenln!a 1.4.1

       23

S〜 satisfies the

      □

Ｌｅｎｌｌｎａ１．４．２． Ｇ仙四ａ一匹?(71，s)‑ゐ翔, ｢召好印戸べＲ9)バ辰7･e以ａμ皿aり7ra禎 Ｇ前哨佃１，‥。，ａ｡,，￡ｈ．‥，６ｊＣＦ(Ｇ)ｓα出力μ≒μ辰か7ZQ必n!り?∂7一泊θns.

川

ｊ

り乙

ぐ

77回｀Osne ｢ 泱ﾊリ./ :G⇒匪1]〉り９Ｊ]ＣＲりＥＵ晶月c隔)＝(だ/(n･十 1)、F)a ｢y叫)＝(kﾉ(n十１)、９)(た＝1、…、n).

77?er'りs a. co7tt紬､ｕｏｕｓｍ(ｌｐ９：Ｇ一り0､1)xl9､p]CR2 ｢治2た細面ｃ６ａ托卯Zar

1)Tolection ofG such that:

(aD㈲)＝y隔)回向(４)＝川4)(だ＝1−‥，71)α ｢

       〜       〜  〜 (b)FOr（1n!J dmgmm G obtdned 斤ｏｍ９(Ｇ)，th.ere ts a sub(hagram SＣGo釦

       〜

su＆that S t,s egu2り ｢e ｢必召

？ro好Let Gp､9、 y'･ べbe as Proof of Lemma l 4 lexcept tlhat y' : G゛'9

￣47×【q､p】

Leいj be the number of crossingsin 召.We denne a,graph G a,saplanargraph which

collt￡11s G詞s＋の,4･‑l as itjs sd)graph alld s tisnes the followillg three colldit.ions (see Figure l.4.11).

(1)F(G)゜{α1，‥・，ら.，ゐ1，‥‥如}UF(G2s(s+dMs‑1)，

(2)￡(G)二{ゐsd･‥'・6d(･十1一昨‥・・61べ1･･a･μt+1･ ゜lc1十2･‥・9αμ‰+昨‥・・

  ら‑lcg山一印as十16s十h Qs+26s+2･‥・・αjjU￡(G2a(j+d).4s‑1)･

(3)There is an embedding y : G ‑4 7 ×【9,7】]cIR2 such t,hat:

  (a)/IG2ﾀ(s十d),4s‑1＝yへ

  (ｂ)．汽扁＝(た/(n十1)，p)and y(剛＝(た/(n十1)，9)(た＝1，‥・，s)，

Thenjt is clear from Len11na 1.4.1 t.hat G is a,planar graph satisfying conditions

of the lemma.

2,1

□

1.5

ｙ μ

Ｑ

０

ＱＩ 山︒︱

ｱ(G2

  ︱・ｌｔＩ

わ1 幻

● 一 個 麹 菌

㎜■

● ■ ●

･‥‥恥.￨

‥.anl

ｊ＋｜‥・ゐrl

Figure 1.4.11

Proof of Theorem 1.1.3

Proof of Theorem 1.1.3. 心

１ λ :

deform j7 int.o a normal form， and denote it again

bv亙.Let n be the lmmber of strands of the braid part of 亙.XVe can assume t,11at eaCh pUre (nj)‑braid iS Of type (p(社抑))(2≦i≦吐Where p(o＝(ln−1十1)/(n−1) and 9(1)＝叫−o/(n−1).For each pure (71，0‑braid 3， 0f t･ype(p(o，9(i))in瓦 there is a planar graph Gi with 仙七‥・，心,Ｍ，‥・，仙}c l/(Gi)sat.isfyingthe following conditions from Lemma l.4.2.

 (1)There is an embedding y11: G, 升 ｊｘ[９(昨斑o)C R2 such that    μ心)＝(た/(n十1)，齢))andμ帽＝(た/(n十１)，φ))(た＝１，…昨

 (2)There is a cont,inuous map ■ : Gi ‑4 7 ×[9(,･)，p(i)]c R2 whi(･h induces a

I'eg111arprojection of Gi such that

㈲剣貯）

い

九 ＝

t)and l(帽＝翔九)吠＝1、…峠

       〜       〜   〜 (b)For any diagram Gi obtained from 以Ｇよthere is asubdiagramSCG

       〜

  sl!ch that‥S･ is equivalent to 瓦．

25

 Ld ，7be the pure braid part of亙 on 7×7.21nd let 7 be t.hespanning tree hl the basic lledangle ofj7. Then， we define a graph G as a plmlr graph sucohat there is an embedding ｡飴Ｇ⇒Ｒりuch tha日(G)＝(U‰剤妬))Ｕ田−j)(see

Figure 1.5.1).Note that ly(G)￨＝Σ旨ＩＦ(Ｑ)￨−ｎ(ｎ−２)十IF(7)￨，and l￡(G)￨＝

Σ二2圃(伺￨十￨石(Ｔ)￨十n/2＋21石(煩一石(ｎ＋(ｎ−２田(用一万(T)￨)/2. Thenjt is c]ear t,hatG has an 刄‑inevitableprojedion.

ｙ ｌ

０

匹

a ･ ■■'￨

μ

λ

Figure 1.5.1

26

折句

Ｃｈａｐｔｅｒ ２

Ｏｎ Ｈａｂｉｒｏ゛ｓ Ｃｎ‑ｍｏｖｅｓ ａｎｄ ｖａｓｓｉｌｉｅｖ

ｌｎｖａｒｌａｎｔｓ

 Recently it has been Proved by Habiro t,hat,two knot,sKl and ん2 have the sarne vassiliev invariants of order less than or cqual to n if and only if Kl and A'2 can be t,ransformed int,o ea,ch other by a finite se(1uence of(八十1−moves.ln this chaPter，

we show that the difference of t,he vassiliev invariant,s of order 7川)etween two knots that can be t,ransformed int,o eEh other by a C≒−move is equal to the vahle of the vassmev invariant for a onej)ranch tree diagram of order n.

2.1 Statenlent of result

 II1 1990, V. A. vassiliev defined a sequence of knot invariants ([26D.After that some llesults from t.he point of view of knot diagrams have been obtained ([7][19]【20H21D.

Recently K. Habiro has defined a family of local nloves called こいnloves and an e(luivalen(:erelation called (八−equivalence for knots. Habiro showed that two knots have the same vassiliev invariantjs of ordell less tjlan or equal to 7lif and on】yifthey a,re G,+1 − equivalent in [91，From the result of[8]，a family of local moves called C゛?1￣lnoves canbe realized by the moves in Fig. 2.1.1 (we cal

Namely， Habiro showed the 柘Howing･

７１

り﹄

11 this move （几−move）

p M 叫 ,で り り 鴫￨

Ｔｈｅｏｒｅｍ２．１．１ぶ8][9DLeh･1 ＞1＆皿泌＆卯r and Zel 瓦，mj A'リj,り,:n ｢s.77,en ぴ1.eか11Q一n!μIJ CQnd伍∂n,9Qnい?卯咄aZe ｢.

(1)K(m,dK'h(1t)ethe same vassihetH,mJMj,(mts o./Qr＆「 Z,E｣ぶりﾉlanQr e9皿aQ?z･ ， (2)K（md K c（mbetTヽ（msJormed仙,to each otheTb!j（1 f1711tesequence of C7,十1‑mQ?J&9.

‑

‑

〃

‑ 一 一

ら

n＋l n ^{３ ２ １}

白

n十1

Figure 2.1.1

ド・

‑

‑

‑

‑ 一 一

３ ２ ^ｌ

From Theorem 2.1.1jf we perform a（::ﾌ｡一lnove fora knoちwe cannot challge the vassiliev invariants of order less than n of it｡However，t,he vassiliev invariants of order n may be changed by a Cn−lnove. ln this chapter，we consider the value of the vassiliev invariant of order n that, a C ,1−nlove maychange and show tjhe following theorem，

Ｔｈｅｏｒｅｍ ２．１．２．￡ｄ‰(瓦)ゐ,り/le l忌＆9逍etパn皿r仙, ｢げり ｢ｅｒｎかraた71, ･瓦。がａ kylot Kl i8 obtained知ｍＫ ｂ!ｊａＣ。−motJe，び,.enむ｡(瓦)−t･,I(瓦゛)＝士7･。(7こ)， ｢l.en9 恥一a Qne‑br{mch tree &.a9ram that t,s a cerl ･n l;i7一好Cゐ,哨,6eC‰l.a.der￡)仙,gram，

 Xve will show in the proof that t,he sign 士and the one branch tree diagram 刄 in Theorcm 2.1.2 depend only on the cyclic order of the 2(71.十1)end l)oint,sin Fig.

       28

2,1.1 on the knot， A"，A C9‑move is the sarne nlove as the local move caned a 1       −

1111kllottillgoperation ([15D and a C3−move is called t.hc clasp‑pass move in l101.1t is

easy to obtain the following corol】ariesfrom Theol｀em 2.1.2 by writing a one‑branch t｣ree dia･gram as a signed suln of chord diagranls with STU‑relat.ion.ln the next

scction，achord diagram and a one‑branch tree diagram will be defined

Ｃｏｒｏｌｌａ‑ｒｙ２．１．３．lf a hot K i8 obt（1aed from K b!J a△‑un㎞ottin9 operation， thell

ヅ和ぺ･2(y) ＝士晦(①)

，1訪erg ?,･2らがze l/一jZiaパn,m.r紬 ｢rｻﾞＱｒ＆ｒ２

Corollary 2.1.4.がαｈ ･瓦りｓＱＭｉｎ 氓凾窒pｍ瓦6!/α＆l即一卯,9s m,Qw，M.,?72 1･3(A')−

恂(A")

＝o心土剛⑤）

^， ｢lereむ31slゐ,e l/αs一ZiaパｎむαΓjα ｢げ∂r＆r3

 lf 71 ＝2， we have by Coronary 2.1.3 t,ha,ta C2‑move (A‑unknott,ing operation) always changes the knot types([22D. HOMrever if n ≧3，there exists a one‑brallch tjlleediagram 瓦such that t',7(7;,)＝0，that isけhere is a case t.hat a (瓦‑move cannot change the order ?I vassiliev invaria蛍.ln the last sedion we sholv an exalnple for any knot t.hat a Cn‑move does not change t,he knot t,ype for 71 ≧3.

2.2 vassiliev invariants and one‑branch tree dia‑

grams

  The definit,o】lof t,he va,ssiliev invariant,s fol]ows t,he Birman‑Lin axioms in [3]or D.

Bar‑Natanl11.VVhencver we have an invariant of knots which t,akes vah!es in sonle abelian group， we can define an invariant of singular knots by the vassiliev skein

凹