HYPERBOLIC ALTERNATING VIRTUAL LINK GROUPS

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1 HYPEROLIC ALERNAING VIRUAL LINK GROUPS JENS HARLANDER A. W y opoloy n omy of l lnk omplmn n op. W o op fn y Wn pnon of n pm n lnn l lnk CA(0) n ypol. MSC: 57M05, 57M50, 20F65, 20F67. Ky o: Alnn l kno, ypol op, Wn ompl, non-poly q ompl. 1. Inoon l onn omy n opoloy of l lnk omplmn n op. noon of l kno pp f n [11]. M knon n ll kno n. A onqn of on omzon om fo Hkn mnfol, ll kno op fnmnl op of non-poly p n kno omplmn ypol n kno n o no ll kno. A l kno pop mn k : I F I, I = [0, 1] n nl n F omp onl f on ony omponn, n o k(0) n k(1) onn n F (0, 1). If mo n on poply m nl pk of l lnk. Mo nlly, on ol llo f non o mo n on ony omponn n lo m l, ll no o o. A poon of l lnk o op f F {1} 4-l p n F o- n n-on nfomon po. Gn l lnk poon, n off Wn pnon n l y. S Wn pnon no ll on p. A ll on p fn on (no nly onn) p Γ on {,,,...} n y on ll y. o Wn pnon no n on-o-on oponn {,,,...}, n lon n on-o-on oponn. An n Γ n n nn, ll y, lon = n Wn pnon. No Wn pnon om fom l lnk no y ll on p onn omponn of p nl. omponn of p n on-o-on oponn omponn of l lnk. A l lnk op op fn y Wn pnon of l lnk. Ll on p n Wn pnon o f n y of mnonl kno oy. Wn ompl, n 2-ompl l fom Wn pnon, of ll on pn of on omplmn n 4-ll. qon of Wn ompl of ll on pl n nly of ll knon ln o W py on: ompl of n pl 2-ompl pl. Ho o ( [6], [7]) f fn 2-ompl K ompl of 2-ompl 3-fom o pon, n K omooplly 1

2 2 JENS HARLANDER qln o Wn ompl of ll on. Of o, Wn ompl of ll kno (poply m nl n 3-ll) knon o pl, poof of f l on 3-mnfol nq n no omnol poof ol n o ll on n fon. Ep n lnn ( Wnm [18], W [19], n [3], p 220). Fo y of l onnn W on oly [4] n Rook [17]. Rll n lnn kno poon on p ll pm f n m o ony n kno n ly o pon onn ll o no on n no. n lo p n m of ll ompoon of p l o poon, m p of of 4-on, on fo on. kno poon pm f y loop n on klon of ll ompoon l o poon ln l fo. In nloy on n fn pm noon fo lnn f lnk poon. Ho, n F ony o fnon lly. Fo mpl, f l lnk only on omponn, n ony 2-yl n ll ompoon l o poon. W f y onn yl only on of ony. A lnk poon on f F ll lnn f on non o- n n-on lnly on l lon ny omponn. Sn 2-ll n ompoon of F l o poon om fom on, o oppon n onn y o-on. W f o o-on. o o oppon f o n-on. Dfnon 1.1. Gn n lnn lnk poon on f F no ll ompoon l o poon. W y poon pm f y loop onn n F (1) {o-on ony } o n F (1) {n-on ony } ln l fo. Fmo y poon n f y loop n F (1) {o-on ony } o n F (1) {n-on ony } pn non-l lmn n fnmnl op of f F ln l f. No fo n lnn lnk ko olon o on of F mn y poon. On olo opon o yl of n on on l on ony of on (n onlok fon, y), n o olo opon o yl of o on. ko olon mpl 1-klon of ll ompoon l o poon -p p, n o o l o 2-yl o mll loop of ln l 4. So o onon no oo nn. pon of n D. W [19]. No n mpl mo n o n of poon nnn o y nl of f. I on n [5] pm lnn lnk poon on f pl Wn pnon. H on of nl l of l. om 1.2. Gn pm lnn lnk poon on f F. Am poon o no onn 2- n 3-nl ( F 1). n G, op fn y Wn pnon of poon, fnml op of fn p-eln 2-ompl of non-po. Fmo, f lnk poon n n n G ypol. Rmk: A lnk poon on F only on omponn o onn 2- o 3-nl. Con ll ompoon of F l o poon. F l y 4-on n o n ony., fo on, o n no of lny l n fo.

3 HYPEROLIC ALERNAING VIRUAL LINK GROUPS 3 F n 3-nl. In F f p of om (no onnn ypoly) ll knon n ol o n o l o 2- n 3-nl. In f Dn ompl o poon non-poly q ompl fnmnl op G. S fo mpl [3], on on kno. f op of ll on nl n ypol (n n no kno op) f o y Rook [16]. O ypol LO (ll on ) op n l. W ll y mo o n l on n mpl. 2. Almo 3-Mnfol n Spn An lmo 3-mnfol 3-mnonl ll ompl ll lnk f. In on fn lmo 3-mnfol o n lnn f lnk poon, M, W n D n 2-mnonl pn M M, W W n D D lnk omplmn n pn. Gn n lnn f lnk poon on f F no q ll ompoon l o lnk, n fn f o F [0, 1] n mo poply m lnk mp o lnk poon n F [0, 1] F {0}. If mo n opn nooo of lnk 3-mnfol M o ony lo onl f of n 2 + k, n of F n k nm of omponn of lnk. W n n mo lnk nooo n 2-mnonl pn M of M ( [5]). In follon l pon of ll of M. mnfol M l p fom, n n o n non mo. F ollp l [0, 1] of M, n n F, o oponn. W ll ompl n on-o-on oponn o n F. Fo y n F o n n ompl, op n oom. Wnn mo l on q 2-ll fo, oom 2-ll n q n F {0}, op 2-ll n q n F {1} n m q omn fom mo ony nnn lnly lon op n oom. W no 2-ompl y M. In mmy, on n f lnk poon (oponn o 2-ll n o Wn ompl) fn q 2-ll of poon f, yl q 2-ll of pn M, op q, oom q n m q. n po p n F 2. V lnk n M. L n M n no of F. n lnk of n M on of oom l n op l omn fom oom f F {0} n op f F {1}, ply. lo n lnk onnn lnk of

4 4 JENS HARLANDER C 1 C1 C1 A A A C 2 C 2 C 2 F 2. op o o on n lnk poon n oponn 2-ll n Wn ompl. o lo o oponn 2-ll n M. q p q q p p F 3. lf o lnk of n poon f. Sn op n oom on of f onn n M, lnk of n M onn op n oom l. On p of M m p y m q. q on n lnk onn op n oom l. oom l n op l. y om fom on n m q. If α = {p, q} on n q of F onnn pon p n q on of q, n on α = {p, q }, α = {p, q } n α m = {p, q } (o α m = {p, q }) omn fom oponn op, oom n m q. A ypl lnk on n F 3 n 4. If ony of F on ml. op n oom l on on lf n F 4 pl y op n oom. om 2.1. Gn pm lnn lnk poon on f F o no onn 2- n 3-nl ( F 1). n pn M non-poly q ompl. Fmo, H = π 1 (M ) ypol.

5 HYPEROLIC ALERNAING VIRUAL LINK GROUPS 5 q q q p p p F 4. p on lf lnk of n no n M. I on of op n oom l, o onnn l om fom on n m q. p on lnk of m no n W. Poof. W 2-ll n M m of n q n pln. mpon no 2- n 3-nl n poon mpl l fo q op on y no of F. I follo loop n lnk of n M ln l fo ( F 4). Hn M non-poly ( [3]). W ll n o nl on M o no onn n omlly m fl pln. Hypoly of fnmnl op of M ll n follo fom [3], om 3.1, p 459. No f poon f F of M. W nfy F op f n M. A on : M F on y nfyn op n oom onn m, n n nfyn 2-ll o ony ly n nf. Sppo E fl pln n M. If ll ll n E y m n on poon n E Eln pln, l n q fom M. On ol nk of E Vn-Kmpn m nlyn p pln. W ll q n E -, -, o m-q f ll y op, oom o m q, ply. An ẽ n E ll foln f o q of E onn ẽ n ony mpp o m q n F n ompoon of on mp n on. A foln p o q foln. No foln p on of n m-q n - o -q. A foln ln n E o on nly of foln. W f E o onn foln. No f q n F n l lon lly of n q ll n p ony. y lly follo omponn of lnk poon n omponn n n ony. In pl no lo lly n F, y lly n n ony n ln on y nm of q n F. L ρ mml ln of lly of n q n F. No f l lon ny lly of n q n E fo mo ρ q, ll non foln n E, n n non n mp o ony n F n ompoon of on mp n on. N f ẽ foln n o n E onn foln ln. L ṽ of foln ẽ. No on of fo q op

6 6 JENS HARLANDER on ṽ n m-q foln ẽ n of n m-q. lnk lk E (ṽ) of ṽ n E yl of ln fo n lk M (ṽ) = lk M () ( lnk nf n on mp), m of ṽ n on poon. lf of F 4 o lnk. If ony op n oom l n lnk pl y. on of q ṽ n m-q my m o lo of nly 4-yl of fom p q q q q p ( p q no n lnk onnn p o q ). Un ompoon of on mp n on yl mp o pq q q qp. Hn 2 2-mn n ṽ onn ly o foln, oponn o p n n lk E (ṽ), n foln p y non-foln,, y p of o n E p o ṽ. o o ẽ n f of o onnn ẽ mnn fom ṽ o foln. Connn n fon o p o ṽ n onn ẽ foln ln. (In f lopn lolly fom foln p o E l n k o fon f olo m-q n - n -q lk.) No on -nfn lly of n q on ony foln ln. Sn om q n ln lon lly n no n ρ non foln ppnl o o foln ln. ppnl foln ln n E. L ũ of non of o ln. n 2 2-mn of fo q op on ũ onn ppnl foln. o 2 2-mn n E onn ly o foln, n y p of o. W onon. Rmk. W n o f fo pon omnol poof n. O onl poof on on Hypolzon: If M omp, onl, l n ool 3-mnfol Hkn (.. onn poply m nompl f) n π 1 (M) no lly ln, n no of M m ompl ypol. S [10] fo poof. mnfol M = F I {kno lnk} l, onl n ool n kno lnk pm ( ool n non-oo on). In ll n F n lo kno lnk no o kno on y Mno [13]. H poof n op fo nl. ony M n nompl f, o M Hkn. Sn π 1 ( M) op of π 1 (M), n n of M l o, follo π 1 (M) no lly ln. So ll onon fo on Hypolzon f. Sn M o no o ony omponn ypol on no of M n kn on o-omp ( Son 5 of Mon [14] o Kpo [10]), mpl π 1 (M) δ-ypol Wn pn. An lmo 3-mnfol W on fom mnfol M y onn off op f F {1}. W no on pon y n ll W Wn p o f lnk poon. I on n [5] W omooplly qln o Wn ompl of f lnk poon. In pl 2-mnonl pn W of W on lo omooplly qln o Wn ompl. po of ollpn M o M n ppl o W n 3-ompl onn M, op f on off. W ollp 3-ll y pn n q of op f. No only 2-ll o op nl omn fom onn off o on m q n n no o n n-on ony (ll m lnk poon o lnn). W p y

7 HYPEROLIC ALERNAING VIRUAL LINK GROUPS 7 C 1 C 1 A A C 2 C 2 F 5. A on n lnk poon yl o 2-ll n W, on n oom f q. o on nl ono o of nl o on. l n ll n, pon o ly, op lon o mo on m q. ln ll follo. of (oom) f F o on, fom oom f o ˆ onnn of F o on. 2-ll q fom oom f o onl 2-ll omn fom m q n M op on off. S F 5. V lnk n W. pn W on fom M y onn off op f n n ollpn 3-ll o f q. Ey no op no ony of ly o 2-ll, on nl fo n m q (ll f lnk poon m o lnn). W po onl 2-ll y nnlly ollpn on nl o op. op mn l n lnk of op. S F 5. A op n ony of F onn n m q only f n non ( pp fo Dfnon 1.1 fo fnon of o- n n-on ). If o no lon o ny m q n ll pp n on nl ollp. o lnk of n W F (1) {o-on ony }. lnk of n no n W on fom lnk of n M y onn off op l, n ollpn nl y pn n of op l. V of op l ll lny o n n lf off. S lf of F 4. If ony, n on ml, l pl y n nl Dn pn. An lmo 3-mnfol D on fom W y onn off oom f F {0}. W no on pon y n ll D Dn p o f lnk poon. A 2-mnonl pn D on y po nlo o on n n po on, 3-ll ollp o oom f q. W no ollp nl on on ony f n pfom nnl ollp fo. l 2-ompl o, n, n on-o-on oponn n F n on q 2-ll fo y on n poon. S F 6. lnk of n D F (1) {o-on ony } n lnk of

8 8 JENS HARLANDER C 1 A C 2 F 6. A on n lnk poon q 2-ll n D. F (1) {n-on ony }. In [5] pn o po pm lnn f lnk poon pl Wn ompl. 3. omy of Wn pn om 3.1. Gn pm lnn lnk poon on f F o no onn 2- o 3-nl ( F 1). n 2-ompl W n m no non-poly p-eln ompl. If f m poon n n G = π 1 (W ) ( omop o op fn y Wn pnon of lnk poon) ypol. Poof. W q 2-ll n W m of n q n pln, n pl ll on nl π/2. E onl 2-ll mz pln on, nl o - n π/2, fo mnn nl n 3π/4. Alnn mpl lnk of n W F (1) {o-on ony }. Pm mpl no loop of ln l fo n F (1) {o-on ony }, o loop n lnk ln l 2π. N on lnk of. In no, lnk l on off ( of F 4). Sn poon o no onn 2-o 3-nl, l onn l fo. loop n lnk p of l onn l fo l n n ln l 2π. A loop n lnk onn on (n no p of l) onn n n nm of on, of ln 3π/4, n l on l n n lo ln l 2π. If ony on ml, l pl y n nl. W no loop o 2π n lnk n n W n non-poly [3]. No l m f lnk poon n n F n lnk poon l o omponn, o n of F l o. In o o o G ypol n o po nl on W o no onn fl pln ( [3], om 3.1, p 459). Sppo fl E n W. n E l mz 2-ll of W, fl q n on of W ( F 7). lnk n lon loop n lnk of W of ln 2π. If lon onn

9 HYPEROLIC ALERNAING VIRUAL LINK GROUPS 9 F 7. A lon of pln m 2-ll of W. o ln n op. pn n W ( no n pn W ). lnk n loop of ln fo n 1-klon of op f, n ony of 2-ll of op f ( p of M ). ll, n fo on pl on n lnk n loop of ln fo n 1-klon of op f. Dnn mpl loop pn l lmn n π 1 (F ). In f, loop on 2-ll n op f. n n follo. Sn loop omooplly l, n l Vn Kmpn l q 2-ll of F o ony loop of ln fo n onon. Sn F lf non-poly q ompl ll. All q n Vn Kmpn zo n n no l o ql o zo. I follo fom omnol G-onn foml ( fo mpl MCmmon[12], p 41) ony π/2, o Vn Kmpn on of nl q. W n off fo nl on - n pl y q 2-ll of op f. In y l lon of Eln pln y q 2-ll of M n lnk n lon loop n om lnk of M of ln 2π. opololly p, pll fl pln E W W off p 1 ( ), p : W W on poon, n on fl pln Ē M, M n nm on of M (ll M M onn n W.) fl pln lf o fl pln n M. on f, n H = π 1 (M ) ypol y om 2.1, no fl n M ( [3], om 3.1, p 459). An nlo l ol fo Dn ompl. om 3.2. Gn pm lnn lnk poon on f F. n 2-ompl D non-poly q ompl. If f m lnk poon o no onn 2- o 3-nl n poon n, n Q ypol. Poof. W 2-ll n D m of n q n pln, n pl ll on nl π/2. Alnn y lnk of n F (1) {o-on ony } n F (1) {n-on ony }, ply, n pm y o loop n lnk ln l fo. mpl D non-poly

10 10 JENS HARLANDER f f k k f k F 8. lf o f lnk poon A(3). On poon f, pn o, no q ll ompoon l o lnk poon. ony y m ll o nf.. Dn mpl no fl n nl on of D. n n y m mn n n poof of om Empl W l fom po on o on l lnk op CA(0), fnmnl op of non-poly p-eln ompl, n ypol. O mpl n l, f on Rook [16]. pp of y-n onn LO op CA(0) n ypol n f-y-yl. mpl mpl f o onon follon: Empl 4.1. Con lnk poon on, on omponn nnn o, o nnn no o o, o-on - omponn. Clly, poon o no onn 2- o 3-nl n pm. In f, F on of nl q. G = π 1 (M) = π 1 (W ) = π 1 (D) ypol. Wn pnon of poon,, =, o G f of nk 2. Empl 4.2. Con q [0, n] [0, n], n n o nm n N. D ln fom (0, ) o (n, + 1), = 0,..., n 1 n lo fom (, n) o ( + 1, 0), = 0,..., n 1. Sn (0, 0), l lon ln fom lf o, mk f on n o-on, n n n-on n o on. No off fo on n Infy oppo on n lnn lnk poon A(n) o omponn on pn o. S F 8. No A(n) o no onn 2- o 3-nl. f n o n omponn of lnk lnn n fo n 5 poon n. o loop n F (1) { o o-on ony } n n F (1) { o n-on ony } (F

11 HYPEROLIC ALERNAING VIRUAL LINK GROUPS 11 n poon f,.. pn o), o no pn l lmn of π 1 (F ) ln n + 1. O om mpl ll op π 1 (M), π 1 (W ) n π 1 (D) CA(0) n ypol. Rfn [1] J. n, N. y, Doon of f op n CA(0) f-y-yl op, Gom. D 120 (2006), [2] M. n, N. y, Mo oy n fnn pop of op, Inn. M. 129 (1997), no. 3, [3] M.R. on, A. Hfl, M Sp of Non-Po C, Gnln mmn Wnfn Volm 319, Spn-Vl [4] W.A. oly, J.H.C. W py qon, n o-mnonl Homoopy n Comnol Gop oy, y C. Ho-Anlon, W. Mzl, A.J. Sk, LMS L No S 197, CUP [5] J. Hln, S. Rook, Gnlz kno omplmn n om pl on omplmn, Jonl of Kno oy n I Rmfon, Vol 12, No. 7 (2003) [6] J. Ho, Som mk on polm of J.H.C. W, opoloy 22 (1983) [7] J. Ho, On py of on omplmn, n A.M.S. 289(1) (1985) [8] G. Hk, S. Rook, Spl m n ll on, Po. of En M. Soy 137A (2007) [9] G. Hk, S. Rook, Apl lll on Po. of En M. Soy (44) (2001) [10] M. Kpo, Hypol Mnfol n D Gop: No on on Hypolzon, Po n Mm 183, k, l [11] L.H. Kffmn, Vl Kno oy, Eop. J. Comno (1999) 20, [12] J. MCmmon, Conn non-poly p n op, Sy l, LMS L No S 358 (2009), [13] W. Mno, Clo nompl f n lnn kno n lnk omplmn, opoloy Vol. 23 No. 1 (1984), [14] J. Mon, On on nfomzon om fo -mnonl mnfol, n Sm Con,. n Mon, P n Appl. M. 112, Am P (1984), [15] D. Rolfn, Kno n Lnk, Pl o P (1976). [16] S. Rook, On lzon of Wn pnon kno op, Jonl of Kno oy n I Rmfon, Vol 3 (1994) [17] S. Rook, W Con - n o, Sn Elon Mml Rpo 4 (2007), [18] C. M. Wnm, o n ony polm fo kno op of ny m pm lnn kno, Po. Am. M. So. 22 (1971), [19] D.. W, Non-poly q ompl, po ln, n non-lly fn op, P. D., Pnon Uny, Dpmn of Mm o S Uny 1910 Uny D, o, ID nln@o.

A simple 2-D interpolation model for analysis of nonlinear data

A simple 2-D interpolation model for analysis of nonlinear data Vol No - p://oog//n Nl Sn A mpl -D npolon mol o nl o nonln M Zmn Dpmn o Cvl Engnng Fl o nolog n Engnng Yo Unv Yo In; m@ml Rv M ; v Apl ; p M ABSRAC o mnon volm n wg o nonnom o n o po vlon o mnng n o ng