One-Dimensional Computational Topology

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1 Wr rltn so n Stz: Dnn un nur nn, wnn s Sm s Grpn I) ur nn Umsltunsoprton U n BZ-R lrt, II) ur Umrunsoprton wr us r BZ-R ntstt, stllt s Sm nn u r Kullä rlsrrn Grpn r. Dmt st n Gusss Prolm ür n llmnstn Grpn vrtr Ornun löst. [Tus w v t ollown torm: A rp sm rprsnts plnr rp n only I) pplyn t swtn oprton U to t sm prous tr-onon o, n II) pplyn t nvrson oprton to t tr-onon o rovrs t ornl sm. Tus, Guss s prolm s solv or 4-rulr rps.] Mx Dn, Ür Comntors Topolo (1936) Dntons n Rprsnttons Rll tt los urv n t pln s ontnuous unton γ: S 1 2. Prvously w onsr smpl los urvs, wr t unton γ s ntv; or ts ltur w rmov ts rstrton. pts, ontnton, rvrsl Artrry los urvs r nsty orrl tns tt wll t your o; you tout topolost s sn urvs n Pno n Hlrt urvs wr, ust tn o t snnns you n t up to wn t urv s llow to sl-ntrst! Fortuntly, tou, ts s omputtonl topoloy lss, w mns w only n to onsr urvs tt v som nt rprsntton tt n mnpult lortmlly. I ll ous on tr ommon rprsnttons: Polyons Polyons r t smplst (n olst) mol or plnr urvs. Rll tt polyon s pws-lnr los urv. Any polyon s nturlly rprsnt y nt squn o ponts p 0, p 1,..., p n 1 ll ts vrts; t polyon tsl s ompos o t ln smnts p p +1 mo n or nx, ll ts s. W n usully (ut not lwys!) ssum wtout loss o nrlty tt t polyon s n nrl poston, mnn tt ll vrts r stnt, no vrtx ls n t ntror o n, n t most two s ntrst t ny pont Wls n rps Clos urvs n lso rprsnt s los wls n som unrlyn pln rp G. T plnr mn o G my sp tr purly omntorlly (s rotton systm) or pws-lnrly (y spyn oornts or vrtx). Hr w usully nnot m ny nrl postons ssumptons; n prtulr, t sm wl my rvst t sm vrtx multpl tms, or trvrs t sm multpl tms n tr rton. T s o t unrlyn rp G my v wts; t lnt o wl s t sum o t wts o ts s, ount wt pproprt multplty. A vrnt o ts rprsntton onsrs ny urv to l rtrrly los to G rtr tn n G tsl. Rpl vrtx o o G wt smll s n o G wt ron twn t ss o ts nponts. Wtn ts ron rp, w n prtur ny urv γ nto nrl poston, so tt γ ntrsts only wtn t vrtx ss, n only trnsvrsly. Tus, Copyrt 2017 J Erson. Ts wor s lns unr Crtv Commons Lns (ttp://rtvommons.or/lnss/y-n-s/4.0/). Fr struton s stronly nour; ommrl struton s xprssly orn. S ttp://.s.llnos.u/tn/topoloy17/ or t most rnt rvson. 1

2 t ntrston o t urv wt ny ron onssts o sont prlll pts. W otn on t r ow t urv vs ns t vrtx ss, ut nssry, w n ssum wtout loss o nrlty tt ny two urv smnts ns vrtx s ntrst on trnsvrsly or not t ll. T tul rprsntton stors t wl n G, totr wt lt-to-rt orrn o t urv smnts wtn ron. Equvlntly, w n spy urv y ts ntrstons wt t s o t ul mp G ; Ér Coln Vrèr n I nm ts t ross-mtr mol. W ssum tt t rprsnt urv γ vos t vrts o G, tt o G ntrsts γ trnsvrsly t stnt ponts, n (typlly) tt ny two smnts o γ wtn t sm ntrst t most on. T ovrly o G n γ s vrtx or ntrston o γ wt s o G, n n or vry smnt o γ or G twn t ovrly vrts. Tn γ tsl s rprsnt s smpl wl n ts ovrly rp. T lnt o urv γ n t ross-mtr mol s t sum o t wts o t s o G tt γ rosss, ount wt pproprt multplty. Any polyon n onvrt nto wl n ts m rp; onvrsly, t unrlyn rp G s m pws-lnrly (or n t ross-mtr mol, vry smnt wtn s ln smnt), tn vry wl n G s polyon Gnr urvs Fnlly, w n rprsnt suntly n urvs rtly y norn tr omtry ntrly. A sl-ntrston γ(s) = γ(t) o urv γ s trnsvrs, or som ɛ > 0, t supts γ(s ɛ, s+ɛ) n γ(t ɛ, t + ɛ) r omomorp to two ortoonl lns. A los urv s nr vry sl-ntrston s trnsvrs; omptnss mpls tt nr mmrson s nt numr o sl-ntrston ponts. Gnr urvs r somtms ll mmrsons or rulr urvs, ut tt trm mor ommonly rrs to los urvs wt ontnuous non-zro rvtvs [11]. In t, t most ommon wor us to sr ts lss o urvs s urv! T trm nr s ust y t osrvton tt vry los urv s rtrrly los to nr los urv. Mor ormlly: Lmm 2.1. For ny los urv γ: S 1 2 n ny ɛ > 0, tr s nr los urv γ : S 1 2 su tt γ(t) γ (t) < ɛ or ll t S 1. Proo: Fx los urv γ: S 1 2 n rl numr ɛ > 0. By oosn n rtrry spont on t rl, w n rr γ s pro unton γ: 2 wr γ(t) = γ(t + 1) or ll t. Comptnss mpls tt w n ovr γ wt nt numr o lls 0, 1, 2,..., n o rus ɛ/4, wr ll s ntr t som pont γ(t), n n prtulr, 0 s ntr t γ(0) = γ(1). For ny nx, lt B not t ll o rus ɛ/2 wt t sm ntr s. W nutvly n nt squn o rl numrs t 0 < t 1 < < t N n ns 0, 1, 2,..., N s ollows. Frst, lt t 0 = 0 n 0 = 0. Tn or ny nx > 0, n t = mn 1, mn{t > t 1 γ(t) B 1 } I t = 1, w st = 0; otrws, lt ny nx su tt γ(t ). Comptnss mpls tt γ s unormly ontnunuous: Tr s rl numr δ > 0 su tt or ny t n t, γ(t) γ(t ) < ɛ/4, tn t t < δ. It ollows tt t N = 1 or som ntr N 1/δ. T ponts γ(t 0 ), γ(t 1 ),..., γ(t N ) n n n-vrtx polyon γ, w w n prmtrz so tt γ(t ) = γ(t ) or vry nx. By onstruton, w v γ(t) γ(t) < ɛ/2 or ll t S 1. Ts polyon s not nssrly nr; n prtulr, t my ontn rpt vrts. Howvr, wt prolty 1 w n otn nr polyon γ y movn vrtx o γ 2

3 stn o ɛ/4 n rnom rton (osn unormly rom S 1 ). Convxty o t Euln norm mpls tt γ (t) γ (t) < ɛ/2 or ll t S 1. Tus, t trnl nqulty mpls γ(t) γ (t) < γ(t) γ(t) + γ(t) γ (t) < ɛ or ll t S 1. Any non-smpl nr urv γ n nturlly rprsnt y ts m rp, w s onnt 4-rulr pln rp, wos vrts r t ponts o prws sl-ntrston.1 Any nr urv γ s rtn nonl Eulr tour o ts m rp; wnvr t tour ntrs vrtx trou rt, t xts tt vrtx trou t oppost rt rv(su(su())). Tus, t rp n rprsnt y t m rp tsl, wt no tonl normton. As usul, t m rp n rprsnt tr omntorlly or omtrlly. Altrntvly, ny non-smpl nr urv n rprsnt y Guss o, n y Guss s ollows. Assn sl-ntrston pont o γ unqu ll; t uss o o γ s t squn o lls nountr y pont movn on roun γ, strtn t n rtrry spont. A sn Guss o lso rors ow t urv rosss tsl t vrtx: postv or rt-to-lt rossns n ntv or lt-to-rt rossns. Fur 2.1. A urv wt sn Guss o Guss Co Plnrty Aroun 1830, Guss [5] s ow to trmn wtr vn Guss o rprsnts plnr urv, or mor suntly, wtr vn Guss o s plnr. Frns sr t rst lortm soluton or sn Guss os, w onsst o ust twlv nstrutons [4]; Crtr ltr sr mu mor nturl n nrl ( somwt lss nt) soluton [1].2 Crtr osrv tt vry sn Guss o orrspons to unqu 4-rulr rp wt unqu rotton systm. Evry 4-rulr rp n t pln wt n vrts s xtly n + 2 s, y Eulr s ormul. Tus, sn Guss o s plnr n only t nu rp mn s xtly n + 2 s.3 Guss tully s mu mor ult quston: W unsn os orrspon to plnr urvs? Ts s ruly t vry rst nontrvl omputtonl topoloy prolm!evry unsn Guss o lso rprsnts unqu 4-rulr rp wt wll-n yl orr o s roun vrtx, ut lvs unsp w rottons r lows or ountrlows. Guss s quston s to trmn wtr t yl orrs n ornt so tt t rsultn mp s plnr. 1T m o smpl nr urv s ovously smpl yl. 2It s possl tt Frns s lortm s tully quvlnt to Crtr s, ut I vn t su n sssmln Frns s unommnt mn o. 3Mor nrlly, ny Guss o nus n mn o 4-rulr rp onto som orntl sur, w ltr utors ll t Crtr sur o t Guss o. 3

4 2.2.1 Guss s Prty Conton Guss osrv wtout proo tt t Guss o o vry plnr urv stss smpl prty onton: Evry sustrn tt strts n ns wt t sm symol s vn lnt, or quvlntly, symol pprs on t n vn nx n on t n o nx. Ts prty onton ws rst prov nssry y Ny [8], ultmtly y ruton to t Jorn urv torm; n lmntry proo ws ltr vn y Rmr n Topltz [9]. Guss lso osrv tt t squns n stsy s prty onton ut nnot rlz y plnr urvs, so t prty onton s not sunt. In t, Ny [8] tully sr omplt lortm to ronstrut plnr urv rom ts Guss o, n tus to trmn wtr Guss o s plnr. Ny s lortm s vry rnt rom t ppro sr low, n t s unlr wtr t n mplmnt to run s quly, so I wll r tl srpton o Ny s ppro to t omwor Dn s Tr-Onon Conton Aout 100 yrs tr Guss, Dn [2] sr s Gusss Prolm r Trt n propos notr lortm to tt plnr Guss os. Dn rst unrosss t vn Guss o y rvrsn vry sustrn twn ntl lttrs, n rtrry orr. I t vn Guss o s plnr, rvrsl orrspons to smootn vrtx o t urv, rvrsn on o t two suurvs tt strt n n t tt vrtx, so tt t ovrll smoot urv rmns onnt. For xmpl, vn t Guss o, Dn mt pro s ollows, rvrsn t sustrns n lptl orr: T rsultn strn nos wly smpl los urv tt tous tsl tnntlly t o t ornl vrts. Bus w n rvrs sustrns n ny orr, t rsultn tou o (n t orrsponn urv) s not unqu. I w rvrs t sustrns on t tm y rut or, t ntr untnln pross rqurs O(n 2 ) tm, ut s w ll s sortly, t s possl to untnl ny Guss o n O(n) tm. T Guss rm o t tou o onssts o yl o 2n vrts, ll y t symols n t tou o n orr, plus s onn pr o ntl symols. Dn prov tt Guss o s plnr, tn t Guss rm o t rsultn tou o s plnr rp; tt s, w n m som o t ors ns t rl n t rst o t ors outs t rl so tt no pr o ors ntrsts. Dn rrr to plnr Guss rms s Bum-Zwl Furn [ tr-onon rms ] n tr orrsponn os s Bum-Zwl Rn [ tr-onon strns ], us ty n lso us to sr tr-otr ompostons o rtrry pln rps. 4

5 Fur 2.2. Lt: Smootn vrtx o plnr urv, rom Dn [2]. Rt: A smoot urv wt Guss tou o. Fur 2.3. Lt: A tr-onon ur, rom Dn [2]. Rt: T plnr Guss rm o. Dn s onton n xprss mor ntly n trms o rnt rp, ll t ntrlv rp o t o. T ntrlv s n vrts, on or stnt symol n t tou o, n ny twn ny two symols tt ntrlv x... y... x... y n t o. A Guss rm s plnr n only ts ntrlv rp s prtt. T ntrlv rp s O(n 2 ) vrts n s, n w n sly prttnss n O(n 2 ) tm. Fur 2.4. T prtt ntrlv rp o. To omplt s lortm, Dn osrv tt w n trnsorm ny Guss rm nto 4-rulr rp y rpln or wt pr o rossn ors wt rossn, s sown n Fur 2.5. Ts rrossn pross yls snl los urv onsstnt wt our ornl Guss o tn, trvlly, t ornl Guss o s plnr. Otrws, t ornl Guss o s not plnr. Dn osrv tt s tr-onon onton s nssry ut not sunt or plnrty onsr t Guss o n s wtr Guss s prty onton n Dn s tr-onon onton r sunt. In t, ts two ontons r sunt, s prov y Dowr n Tstltwt [3] lmost 50 yrs ltr. 5

6 Fur 2.5. Lt: Buln los urv rom tr-onon rm, rom Dn [2]. Rt: A plnr urv onsstnt wt t ornl Guss o, tr Dn [2] n Kumnn [7] Morn Proo Rtr tn rptn Dowr n Tstltwt s rumnt, I ll v smpl sl-ontn rtrzton o plnr Guss os wt omplt proo. Ts rtrzton s ultmtly s on t ontons o Guss n Dn, ut uss mor morn lortm tools sr y Rosnstl n Trn [10]. (Rosnstl n Trn [10] sr n nt mplmntton o Dn s lortm, nlun t son untnln ps, wtout xplotn Guss s prty onton.) W strt wt smpl onsqun o t Jorn urv torm. Lmm 2.2. Evry pr o nr los urvs tt ntrst only trnsvrsly ntrst t n vn numr o ponts. Proo: Lt α n β nr pr o los urvs. Rll t prty tst or n wtr pont s n t ntror o smpl los urv or not. T sm prty tst llows us to olor t s o β ltrntly l n wt, so tt ny two s tt sr n v oppost olors. Now mn pont movn roun α; tm ts pont rosss β, t movs rom wt to l or v vrs. T movn pont strts n ns n t sm, n tror must n olor n vn numr o tms. Now lt X strn o lnt 2n, n w o t n unqu symols pprs tw. Lmm 2.3. I X s t Guss o o plnr urv, tn vry sustrn o X tt strts n ns wt t sm symol s vn lnt. Proo: Lt γ plnr los urv. Smootn γ t ny vrtx prous two suurvs α n β. Up to yl st (rltn n o spont), t Guss o or γ n wrttn s x y, wr strn x nos t vrts lon α n strn y nos t vrts lon β. E sl-ntrston pont o α s no n x tw, n t otr symols o x no t ntrstons twn α n β. W onlu tt x s vn lnt, w omplts t proo. T strn X ns 4-rulr rp G(X ) wos vrts r t n stnt symols n X, n wos s orrspon to (yl) sustrns o lnt 2. Morovr, X ns prtulr Eulr tour o G(X ); t s o ts tour r ltrntly rt orwr n wr. S 6

7 rntly, G(X ) s rt rp wt s x x +1 n x x 1 mo 2n or vry vn nx. For xmpl, X =, t rp G(X ) onssts o t ollown s: S Fur 2.6. Fur 2.6. A urv wt ltrntly ornt smnts. Torm 2.4. X s plnr Guss o n only G(X ) s plnr mn tt ontns wly smpl Eulr tour. Proo: Frst, suppos X s t Guss o o plnr urv γ. Expt or t ornttons, G(X ) s t m rp o γ. Lmm 2.3 mpls tt vry vrtx o G(X ) s n-r 2 n out-r 2, so G(X ) s n Eulr tour. Morovr, t s nnt to vrtx o G(X ) ltrnt n, out, n, out. It ollows tt vry Eulr tour o G(X ) s wly smpl. Convrsly, suppos G(X ) s plnr mn tt ontns wly smpl Eulr tour T. Wnn numr rumnts mply tt t s nnt to vrtx o G(X ) ltrnt n, out, n, out. T ornl strn X ns n unrt Eulr tour U o G(X ), w trvrss s ltrntly orwr n wr. U rosss tsl t vry vrtx o G(X ). It ollows mmtly tt U s los urv wt Guss o X. 2.3 T Pl o Twn Sts Torm 2.4 vs omplt rtrzton o plnr Guss os, ut t lvs opn t quston o ow to plnrty lortmlly. Most o t lortm s strtorwr; w n ul t rt rp G(X ) n omput n Eulr tour T n O(n) tm, n on w v plnr mn o G(X ) tt ms T wly smpl, w n xtrt t urv U n O(n) tm. T r prt s n wtr T s plnr tou urv, or quvlntly, wtr t ntrl rp o T s prtt. W v lry sn ow to nswr ts quston n O(n 2 ) tm. In t rmnr o ts no, I ll sr n lortm o Rosnstl n Trn [10] tt solvs ts plnrty prolm n O(n) tm.⁴ Frst w opt vrnt notton propos npnntly y Dowr n Tstltwt [3]. W no t Guss o n n rry t[0.. 2n 1] tt nos or nx t nx o t otr ourrn o symol T[]. Tus, or nx, w v T[t[]] = T[] ut t[]. For xmpl, t Guss o ns t rry [12, 6, 21, 20, 18, 8, 1, 11, 5, 17, 13, 7, 0, 10, 16, 19, 14, 9, 4, 15, 3, 2] ⁴Ts s not t rst lortm to solv ts plnrty prolm n lnr tm. Aout tn yrs rlr, Hoprot n Trn [6] sr t rst lnr-tm lortm to n rtrry rp s plnr. At vry lvl, Hoprot n Trn s lortm rsmls Ny s 1927 lortm or n urv plnrty. 7

8 For t s o llustrton, w ornt ts nx rry vrtlly, n w pross t ns n t rry rom t top own. E stnt symol n t o rprsnts or n t Guss rm. W wnt to lssy ts ors nto two lsss lt n rt w n rsptvly m ns n outs t mn rl o t Guss rm wtout ntrstn. I w nw t lt-rt lsston n vn, w oul vry tt no two ors ntrst n O(n) tm usn two sts, s ollows. Hr w ssum t or symols r t ntrs 1 trou n, n t ooln rry IsLt nts w symols orrspon to lt ors. VryLR(T[0.. 2n 1], t[0.. 2n 1], IsLt[1.. n]): L nw st R nw st or 0 to 2n 1 t[] > rst npont IsLt[T[]] Pus(L, T[]) ls Pus(R, T[]) ls son npont IsLt[T[]] x Pop(L) ls x Pop(R) x T[] rturn Fls rturn Tru O ours w on t now t lt-rt lsston n vn; tt s wt w n to omput! Rosnstl n Trn s lortm uls ts lsston usn st o prs o sts; to lp vo onuson, t outr st s ll pl. E pr o sts on t pl onssts o lt st L n rt st R. At t n o t t trton o t nnr loop, t pl o sts ontns ll vlus t[] su tt < n t[]. In prtulr, t t strt n n o t lortm, t pl s mpty. E trton o t lortm mntns t ollown t strutur nvrnts: For lvl, tr L or R s non-mpty. For lvl, ll ors n L R l n t sm omponnt o t ntrlv rp. For ll lvl, no or n L R s ntrlv wt ny or n L R. For lvl, t sts L or R r sort n nrsn orr (tt s, ns r pus n rsn orr). T pl tsl s lso sort n nrsn orr: For lvl, ll ns n L R r smllr tn ll ns n L +1 R +1. Intutvly, t sts L n R n pr ontn ors o t Guss rm, rprsnt y tr r-numr nponts, tt must l on oppost ss o t mn rl. Equvlntly, t ntrlv rp ontns s twn vry nx n L n vry nx n R. Morovr, t ors n st r proprly nst. Howvr, t ors t lvl n npnntly m on tr s, tr wt L on t lt n R on t rt, or L on t rt n R 8

9 on t lt. Tus, t pl ontns lvls, tr r (t lst) 2 onsstnt mns o t ors n t pl. Stll n to tully sr t lortm. T[] t[] Pl o twn sts Oprtons Intrlvs oun 0 12 [12 ] nw pr, pus lt 1 6 [6 ], [12 ] nw pr, pus lt 2 21 [6, 12 21] ml, pus rt, 3 20 [6, 12 20, 21] pus rt 4 18 [6, 12 18, 20, 21] pus rt 5 8 [6, 12 8, 18, 20, 21] pus rt 6 1 [12 8, 18, 20, 21] pop lt 7 11 [11, 12 8, 18, 20, 21] pus lt 8 5 [11, 12 18, 20, 21] pop rt 9 17 [11, 12 17, 18, 20, 21] pus rt [11, 12 13, 17, 18, 20, 21] pus rt 11 7 [12 13, 17, 18, 20, 21] pop lt 12 0 [ 13, 17, 18, 20, 21] pop lt [ 17, 18, 20, 21] pop rt [16 ], [ 17, 18, 20, 21] nw pr, pus lt [19 16, 17, 18, 20, 21] swp top pr, ml, pus rt, [19 17, 18, 20, 21] pop rt 17 9 [19 18, 20, 21] pop rt 18 4 [19 20, 21] pop rt [ 20, 21] pop lt 20 3 [ 21] pop rt 21 2 pop rt, pop mpty pr Fur 2.7. Rosnstl n Trn s pl o twn sts lortm runnn on t strn 9

10 Rrns [1] J. Sott Crtr. Clssyn mmrs urvs. Pro. Amr. Mt. So. 111(1): , [2] Mx Dn. Ür omntors Topolo. At Mt. 67: , [3] Clor H. Dowr n Morwn B. Tstltwt. Clsston o not protons. Topoloy Appl. 16(1):19 31, [4] Gor K. Frns. Null nus rlzlty rtron or strt ntrston squns. J. Com. Tory 7(4): , [5] Crl Frr Guß. Nlss. I. Zur Gomtr stus. Wr, vol. 8, , Tunr. Ornlly wrttn twn 1823 n [6] Jon Hoprot n Rort E. Trn. Ent plnrty tstn. J. Asso. Comput. M. 21(4): , [7] Lous H. Kumn. Vrtul not tory. Europ. J. Comn. 20(7): , rxv:mt/ [8] Julus v. Sz. Ny. Ür n topoloss Prolm von Guß. Mt. Z. 26(1): , [9] Hns Rmr n Otto Topltz. On los sl-ntrstn urvs. T Enoymnt o Mtmts: Sltons rom Mtmts or t Amtur, ptr 10, 61 66, Dovr Pul. Ornlly puls y Prnton Unv. Prss, [10] Prr Rosnstl n Rort E. Trn. Guss os, plnr Hmltonn rps, n st-sortl prmuttons. J. Alortms 5(3): , [11] Hsslr Wtny. On rulr los urvs n t pln. Composto Mt. 4: , Copyrt 2017 J Erson. Ts wor s lns unr Crtv Commons Lns (ttp://rtvommons.or/lnss/y-n-s/4.0/). Fr struton s stronly nour; ommrl struton s xprssly orn. S ttp://.s.llnos.u/tn/topoloy17/ or t most rnt rvson. 10

The University of Sydney MATH 2009

The University of Sydney MATH 2009 T Unvrsty o Syny MATH 2009 APH THEOY Tutorl 7 Solutons 2004 1. Lt t sonnt plnr rp sown. Drw ts ul, n t ul o t ul ( ). Sow tt s sonnt plnr rp, tn s onnt. Du tt ( ) s not somorp to. ( ) A onnt rp s on n