Introduction to  Lonant-Schneiderman-Teichner's filtration on (string) links (Contents & 1. Def, F  lwhitney tower of order n &  ids filtration on links g 2. Classification of the filtration 33. Relationships  ωīth Filtratīons on strīng līnks & homology cylīnders C-S-T G1.( Vhitney  40wer F.Itration by derwed Series (vhitney  tower  p & - heish+ : Symmetr c (wt) : acymmetric    louser centrol segaes |ouder 2n oorder DoF ・いわずof order o でもς  a  collection ful? of  properly  immersed  surfaces in  a  4-mosfold  W. o rder (W:):= o , order of an wrtersection point in Wh nWz = order(Wr)+order(Wz) a  lwhītney dīsk  lyhīch pañrs intersection pañrs el order h  =  n+ '. “ •WT of syder n = WT of srder n-i cc+. all intersection points of order<n are pared. _ = P n +e  oriented  unitrivalent tree • univalen+ vertices are labeled from [i.e.m] . Her rooted tree , root is unlabeled vertey) r order of tree = n ùmber of trivaleıt vertices . Eur-Races (whaney disks))そしa rooled troe order  o intorsectim boids � unrosked Hrees o rder 1. ( Tinductively ) m-comp. I l-ink" =  ordered , oriented , "Framed" link , Dof  .  L bounds an order n WT ... ' N, C IW, C IW, C IL = L inks ' （うしをIWn iP ヨorder n WTCB* whose order o surfaces are timinarsed clisks ,bownded by Lo, Li are order n .WT ConCordan+そう3 order n WTcs'xI whose order 0 surfaces immersed onnuli cobourded by L U - L , W" : =  WW a / order n+1  WT cont. : Groyp, under bandcum bounds o rder 1 WT. < Hilhor  Inv - higher-order Sato-Le vine inv. '' highor-order Arf Trv. l ~  L.  ←  L. # -L.  bounds  order n WT Lo # L ,  in IL / conc.  depends or  b But in LL/WT cone , Ts indep, on b Simooth & locally Flat f;l+ratims  equod ) $? Milror invariants मे longHtides of L  are  in  π,(s³-L)n+1   ,-E)  F = π,(d²- ξι,….m}) -Hunヘ π (s -L mu/ L  /11/1(s²_c)n+2 L ie algobra Lie algebra gen by frogen ) Lengidudeg k athous  any  impsed  relation  f  deg  om  スパー(L)  <— ( Aur(L) =  ∑  X ; ® A )  ∈  L ; ® Ln +1 E D :ニ Eor { L,& Ln+, → Ln+2}  . Free  b . Thm . An：W一→Do (i.e. L. bounds order in WT ⇒ long-tndes E π.(s-L)n+1 ) L ～ under E L  boands erder nti WT  → longstudes E π; (s → L) nte のでh: Quen, Jun 3s 1-1 ( ⇒ Win Can be detected by Lin ) ex. Hongitudes e m(s³-心● For n: メル, メル, メル s  onloしのレッAli(Br) =  x, 0 X, x,コ+X2 @CX3,X,I SL (Ker ( v * ) 参 D *  m  z  OL  +  , h igher Sato-Levine inv . t X, @Cx, X, Z Un , SL (⇒ W4t-1 , Can         " O Fix  faeven  f SL *シン・ラfr : odd  ., Z ®Lt →  Kor(ust-x) → Z ®Lt+1 ( - W → ++ N  + A n , SL , A S , 2.5 Wi (lccr9 Kerts o 3 lvieak Cony Arf + o ? (⇒ d : c ) ( Unrooted ) ( , == 22 Corder n trees ) EWT of erder n,d=livks/AS ,IHX , K R シN . using Cochrau's THerated Bing-double Simple clasper surgery along trees" (Goursamv,Habīra) Lounna  Levine's  Carj  Lyon level * S  D  * (= Ker i l,' @Lnt,—Lnt,2 ) Ln free iques -lie algebra X : 80 2 t=i-1よdegreen Lv ,XX=o (reduced version) → Dn [x,Y]=~CY,Y] #" " ↑ Wはコ�⇒co]. D n:4k-1 n Dien 2うh :48–75 Relatinships いぶHh  Altintims  P etric  l-mbs ck ( link, meridianal , disk ) S +r cr s homology cylinder = 3-mfd , homology cobselsm Permelessi - cylonder Tink botween +hso surfaces 2, ex.  Mapping  class grap引SW , Strus, Iznk, in homelesu Artin repミル2 human大 Levine f 1993 , Lenve Gs H  ( Pinite  type inv. F ショットY-Filt. h : even : c) writh D´ n : 4k-1  :  5)  Goith  %.  : m:4kd :ヨし→記の包cor; → G