BRANCHED TWIST SPINS AND KNOT DETERMINANTS. Osaka Journal of Mathematics. 54(4) P.679-P.688

Transcription

1 Ttle BRANCHED TWIST SPINS AND KNOT DETERMINANTS Author(s Fukuda Mzuk Ctaton Osaka Journal of Mathematcs 54(4 P679-P688 Issue Date 07-0 Text Verson publsher URL DOI 0890/67007 rghts

2 Fukuda M Osaka J Math 54 ( BRANCHED TWIST SPINS AND KNOT DETERMINANTS Mzuk FUKUDA (Receved May 5 06 revsed August 3 06 Abstract A branched twst spn s a generalzaton of twst spun knots whch appeared n the study of locally smooth crcle actons on the 4-sphere due to Montgomery Yang Fntushel and Pao In ths paper we gve a suffcent condton to dstngush non-equvalent non-trval branched twst spns by usng knot determnants To prove the asserton we gve a presentaton of the fundamental group of the complement of a branched twst spn whch generalzes a presentaton of Plotnck calculate the frst elementary deals and obtan the condton of the knot determnants by substtutng for the ndetermnate Introducton An n-knot s an n-sphere embedded nto the (n + -sphere In ths paper we assume that Introducton all knots are smooth A non-trval example of a -knot was frst ntroduced by Artn [] whch was constructed from a -knot n S 3 by rotatng t n S 4 along a trval axs It s called a spun knot One of the mportant propertes of spun knots s that ts knot group s somorphc to the knot group of the -knot of rotaton Afterwards Zeeman generalzed Artn s constructon by addng a twst durng the rotaton called a twst spun knot [4] Such a twstng s realzed by an S -acton on S 4 havng two fxed ponts and one famly of exceptonal orbts A twst spun knot s then obtaned as the unon of the fxed ponts and the premage of a (possbly knotted arc n the orbt space connectng the mage of the two fxed ponts Suppose that S 4 has an effectve locally smooth S -acton Let E m be the set of exceptonal orbts of Z m -type where m s an nteger greater than and F be the fxed pont set Let Em and F denote the mage of the orbt map of E m and F respectvely In the study of S - acton on S 4 Montgomery and Yang [8] showed that effectve locally smooth S -actons are classfed nto the followng four types: ( D 3 (S 3 (3S 3 m and (4 (S 3 K m n whch are called orbt data The 3-ball D 3 and the 3-sphere S 3 n these notatons represent the orbt spaces Type ( has no exceptonal orbt and F s the whole boundary of D 3 Type ( has no exceptonal orbt and F conssts of two ponts Type (3 has one type of exceptonal orbts E m whose mage Em consttutes an arc n the orbt space S 3 The mage F s the endponts of the arc Em Type (4 has two dfferent types of exceptonal orbts E m and E n where m n are larger than and relatvely prme The mages Em and En are arcs wth the endponts F such that Em En F consttutes a -knot n S 3 Conversely Fntushel [5] and Pao [9] showed that there s a unque weak equvalence class of effectve locally smooth S -actons on S 4 for each orbt data Here the weak equvalence of S -manfolds M and 00 Mathematcs Subject Classfcaton Prmary 57Q45; Secondary 57M60 57M7

3 680 M Fukuda M s defned by a homeomorphsm H : M M satsfyng H(θx = a(θh(x forθ S and x M where a s an automorphsm of S The sets E m F and E n F n type (4 are dffeomorphc to -spheres and the mage of all exceptonal orbts and the fxed pont set s a -knot K = E m E n F n S 3 An (m n-branched twst spn of K s defned to be the -knot E n F n the case of (4 whch corresponds to the case m n > and defned smlarly n type ( and (3 See Secton for precse defnton In ths paper by fxng the orentaton of S 4 we generalze the defnton of (m n-branched twst spns for (m n Z N where m and n are coprme Note that f n = then t s of type (3 and corresponds to the m-twst spun knot and f m = 0andn = then t s of type ( and corresponds to the spun knot An n-knot K s sad to be equvalent to another n-knot K denoted by K K f there exsts a smooth sotopy H t : S n+ S n+ such that H 0 = d and H (K = K It s known by Hllman and Plotnck that f an -knot K s torus or hyperbolc then K mn s not reflexve for m n and m 3 [7] Here a -knot s sad to be reflexve f ts Gluck reconstructon s equvalent to ether ts mrror mage or ts orentaton reverse Snce the trval -knot s reflexve K mn s non-trval f K s torus or hyperbolc and m n and m 3 Though ther result detects non-trvalty of these branched twst spns n the best of my knowledge there s no result whch dstngushes non-equvalent non-trval branched twst spns Inthspaperwegveasuffcent condton to dstngush non-trval branched twst spns by usng knot determnants Our man theorem s the followng: Theorem Let K m n K m n be branched twst spns constructed from -knots K and K n S 3 respectvely ( If m m are even and ΔK ( ΔK ( then K m n / K m n ( If m s even m s odd and Δ K ( then K m n / K m n To prove ths theorem we frst gve a presentaton of the fundamental group of the complement of a branched twst spn whch generalzes a presentaton of Plotnck and obtan the frst elementary deal from ths presentaton by usng Fox calculus Then we obtan certan equatons of elementary deals and the knot determnants appear when we substtute for the varable t of the Alexander polynomals Ths paper s organzed as follows: In Secton we gve the defnton of a branched twst spn and the presentaton of ts fundamental group In Secton 3 we gve concrete generators of the frst elementary deal and prove Theorem Branched twst spns Branched twst spns For a pont x M anorbtg(x s called G/H-type f the sotropy group G x = gx = x g G of x s H G If the G-acton s locally smooth each orbt of G/H-type has a lnear slce S such that H acts orthogonally on S See [3] for the termnologes of transformaton groups Suppose that S 4 hasaneffectve locally smooth S -acton We consder the S -actons of type (3 and type (4 n Secton Let E m and E n be the sets of exceptonal orbts of Z m -type and Z n -type respectvely and F be the fxed pont set Here when n = the S -actonsof type (3 Let E m E n and F denote the mage of the orbt map of E m E n and F respectvely

4 Branched Twst Spns and Knot Determnants 68 Then E m F and E n F are dffeomorphc to -spheres and the mage of all exceptonal orbts and the fxed pont set consttutes an -knot K = Em En F Now we gve the defnton of (m n-branched twst spns for (m n Z N To do ths we need to fx orentatons of S 4 and the S -acton on S 4 and observe the drecton of twstng n neghborhoods of the exceptonal orbts Let (m n be a par of ntegers n Z N such that m and n are coprme Here we further assume that m 0 Frst we decompose the orbt space S 3 nto fve peces The set F conssts of two ponts say x and x and let D3 be a small compact ball n S 3 centered at x for = Choose a compact tubular neghborhood N(K ofk suffcently small such that N(K \ nt(d 3 D3 hastwo connected components N m and N n wth N m Em and N n En Set Em c and En c to be the connected components of N(K \ nt(d 3 where Ec N m and E c D3 respectvely Note that N m and N n are dffeomorphc to Em c D and En c Let X be the closure of S 3 \((E c m D (E c n D D 3 D3 of K Then we have a decomposton S 3 = X (Em c D (En c m n N n D respectvely whch s the knot complement D D 3 D3 Fg The complement X Let p : S 4 S 3 be the orbt map Each pont of X s the mage of a free orbt Thus p X S s a prncpal S -bundle The premage p (X sdffeomorphc to X S and p X S : X S X s the frst projecton snce H (X; Z = H (X X; Z = 0 (cf [ Chapter ] sdffeomor- Note that the acton on s the cone of the acton of Let B 4 be a lnear slce at p (x whch s a closed 4-ball By [5] p (D 3 phc to B 4 and the acton on p (D 3 ss -equvalent to that on B 4 B 4 s the cone of the acton of B 4 andsotheactononp (D 3 p ( D 3 Choosng a pont z m n Em c letd z be a -dsk n S 3 centered at z m m Em c and transversal to Em c The premage p (D z s a sold torus V m whose core s the exceptonal orbt of m Z m -type In the same way choosng a pont z n En c letd z be a -dsk n S 3 centered n at z n En c and transversal to En c The premage p (D z s a sold torus V n whose core n s the exceptonal orbt of Z n -type Note that snce V m V n = p ( D 3 s a 3-sphere p (y (y D z m s a curve on V m rotatng up to orentaton m tmes along the merdan and n tmes along the longtude of V m where n s determned module m due to the selfhomeomorphsms of V m Now we fx the orentatons of V m and E c m as follows: Frst fx the orentaton of S 4 and those of orbts such that they concde wth the drecton of the S -acton These orentatons determne the orentaton of V m E c m Let(φ θ be a preferred merdan-longtude par of X

5 68 M Fukuda From the decomposton of the orbt space S 3 we can see that φ s regarded as a coordnate of the second factor of V m Em c We assgn the orentaton of V m so that the orentaton of V m Em c concdes wth the gven orentaton of S 4 Fnally we choose the merdan and longtude par (Θ H ofv m D S such that H becomes the merdan of V n n the decomposton V m V n = p ( D 3 and the orbts of the S -acton are n the drecton εnθ+ m H wth n > 0 where ε = fm 0andε = fm < 0 Fg The premage of y Defnton (Branched twst spn For each par (m n Z N wth m 0 such that m and n are coprme let K mn be the -knot E n F If(m n = (0 then let K 0 be the spun knot of K The -knot K mn s called an (m n-branched twst spn of K Note that the branched twst spn K m constructed from (S 3 K m s an m-twst spun knot of K Fg3 The mage of E m E n F n S 3 Let K be an n-knot n S n+ The fundamental group of the knot complement S n+ \ntn(k of an n-knot K s called the knot group of K where N(K s a tubular neghborhood of K Lemma Let K be an -knot and K mn be the (m n-branched twst spn of K wth (m n Z N where m and n are coprme Let x x s r r t be a presentaton of the knot group of K such that x s a merdan Then the knot group of K mn has the presentaton ( π (S 4 \ ntn(k mn x x s h r r t x hx h x m hβ where β s an nteger such that nβ ε (mod m Recall that ε = f m 0 and ε = f m < 0

6 Branched Twst Spns and Knot Determnants 683 Remark 3 The presentaton of roll spun knots s smlar to the presentaton n Lemma For twst-roll spun knots the condtons of dstngushng two twst roll-spns and nontrvalty s studed by Teragato [ 3] Proof The knot complement of K mn s gven by X S f V m Em c where the map f : D Em c S V m Em c s the attachng map specfed by the decomposton explaned n ths secton Let h s the coordnate of the second factor of X S whose drecton concdes wth the S -acton By the above dscusson of the orentatons the nduced map f : H ( D Em c S H (V m Em c must satsfy ( α εn ( f ([θ] f ([h] = ([Θ] [H] β m where α and β are ntegers satsfyng mα + nβ = ε The nverse of f satsfes the relaton ( f ( m εn ([Θ] f ([H] = ([θ] [h] β α hence f ([Θ] = m [θ] + β[h] and f ([H] = εn[θ] + α[h] hold Snce f ([Θ] s null-homologous n V m E c m π (S 4 \ ntn(k mn x x s h r r t x hx h x m hβ holds by Van Kampen s theorem Remark 4 Snce D 4 s the unon of V m and V n the merdan μ of K mn s that of V n named H n the above proof From the relaton f ([H] = εn[θ] + α[h] we have μ = θ εn h α 3 Proof of man theorem Let K be an n-knot Assume that a presentaton x x l r r k of π (S n+ \ 3 Proof of man theorem ntn(k s gven Let a : π (X H (X Z t be the quotent map Here t s the nfnte cyclc group Ths map nduces a map a : Zπ (X Z[t t ] naturally The matrx A defned by A = a r x j s called the Alexander matrx of K Note that H (X Z holds for all n-knots and a takes a merdan of K to the generator t of Z In the case of a -knot t s very common to use a Wrtnger presentaton for descrbng the knot group of K Then the quotent map of the abelanzaton maps each generator to the generator t of H (X; Z Two Alexander matrces A and A s sad to be equvalent denoted by A A fa s obtaned from A by the followng operatons: (Permutng rows or permutng columns ( Adjonng to a row or a column a lnear combnaton of other rows or columns respectvely

7 684 M Fukuda ( ( A A 0 (3 A (4 A 0 0 For the Alexander matrx A M(p q Z[t t ] of K over Z[t t ] and non-negatve nteger kthek-th elementary deal E k (AofK s defned as follows: E k (A s the deal generated by determnants of all (q k (q k-submatrces of A f 0 < q k p E k (A = 0fq k > p E k (A = Z[t t ]fq k 0 For all k wehavee k E k+ If K K a presentaton of the knot group of K s obtaned from that of K by Tetze transformatons Snce Tetze transformatons preserve the equvalence class of Alexander matrces we denote E k (AbyE k (K Now we study the elementary deals of the knot group of K mn Hereafter we fx a Wrtnger presentaton of the knot group K as (3 x x l r r l Then (3 π (S 4 \ ntn(k mn x x l h r r l x hx h x m hβ holds by applyng Lemma Let r l+ be x hx h for each ( l and let r l+ be x m hβ Then usng the nduced map a and Fox calculus for ths presentaton the Alexander matrx A of K mn s wrtten as ( ( r A = a x j Lemma 3 The k-th elementary deal of K mn has the followng property: ( E 0 (K mn = 0 ( Let β be a postve nteger satsfyng nβ ε (mod m ThedealE (K mn s the deal generated by the followng elements: Δ K (t β ( t m ( t β t m β t β t m β t m G (t β ( t m ( t m ( t β t m β t β t m β t m ( t m l ( t m ( t β t m β t β t m β t m where l s the number of generators of the knot group of K and G (t are generators of E (K Especally E (K mn 0 Here the notaton PQ Q Q 3 Q 4 means PQ PQ PQ 3 PQ 4 Proof From Lemma we have the presentaton (3 of the knot group of K mn By Remark 4 the merdan of K mn s wrtten as x εn h α where mα+nβ = ε Snce the quatont map a sends x εn h α to the generator of H (X; Z dentfed wth t wehavea(x = = a(x l = t β a(h = t m Then the Alexander matrx A obtaned from (3 s gven by

8 (33 A = Branched Twst Spns and Knot Determnants 685 B 0 t m t β O O t m t β t m β t m β ( t m β t β 0 0 t m where B s the Alexander matrx of K obtaned from the Wrtnger presentaton (3 by replaced t wth t β Sncer s x x j x xk for each ( l entres n the -th row of A satsfy a x x j x xk t β (p = t β (p = j = x p (p = k 0 (p : others Therefore we have ( l x x p= a j x A = x k 0 x p = 0 Hence A s equvalent to B t m t β O 0 O t m 0 t m β t m β ( t m β t β 0 0 t m where B s an l (l matrx An (l + (l + submatrx of A should contan both the (l + -th row and the (l + -th row f t s determnant s not zero Then any (l + (l + submatrx has the form where ( B B 3 B t m 0 0 t β B 3 t m β t β 0 0 t m β ( t m β t m s an (l (l matrx and ts determnant has the factor

9 686 M Fukuda t m t β t m β t β t m β ( t m β t m whch s equal to zero Thus E 0 (K mn = 0 holds By checkng all the determnants of l l submatrces of A we can see that E (K mn s generated by the followng terms: Δ K (t β ( t m (t β t m β t β t m β ( t m β t m G (t β ( t m ( t m (t β t m β t β t m β ( t m β t m ( t m l ( t m (t β t m β t β t m β ( t m β t m Here Δ K (t s the Alexander polynomal of K whch s gven up to unt as the common factor of all determnants of (l (l submatrces of B andg (t are the generators of E (K Thus we have the asserton Proof of Theorem We prove the asserton by contraposton Suppose that K m n K m n LetG j (t be the generators of E (K From Lemma 3 for each = E (K m n s generated by Δ K (t β ( t m ( t β t m β t β G j (tβ ( t m ( t m l t m β t m ( t m ( t β t m β t β t m β t m ( t m ( t β t m β t β t m β t m Snce the deals E (K m n ande (K m n concde each generator of E (K m n s a lnear combnaton of generators of E (K m n over Z[t t ] Thus for nstance we have Δ K (t β ( t m =Δ K (t β P (t( t m + P (t( t β + P 3 (t t m β + j t β G j (tβ ( t m P j 5 (t( t m + P j 6 (t( tβ + P j 7 (t t m β t β + P j 8 (t t m β + ( t m l t m P 9 (t( t m + P 0 (t( t β + P (t t m β t β + P 4 (t t m β t m + P (t t m β t m where P k (t P j k (t Z[t t ] are Laurent polynomals Snce m and β are relatvely prme β s odd Substtutng for the above equaton s t wehave

10 Branched Twst Spns and Knot Determnants 687 Δ K ( (P ( + β P 4 ( =Δ K (( β ( ( m If m s even then P ( + β P 4 ( = 0 snce Δ K ( 0 for any -knot K Ifm s odd then Δ K ( Δ K ( = Δ K ( = P ( + β P 4 ( Z snce β can be chosen to be even and Δ K ( = for any -knot K The same arguments for other generators Δ K (t β E (K m n lead the followng table: The generators of E (K m n t m t β ( tβ t m β t β t m β t β t m β t m of t m β t m (m β β = (e o o P Z/ P Z/β (m β β = (o o e Z/ P Z/ m P The second column explans the case of the generator Δ K (t β ( t m whch we have seen above If (m β β = (e o o where e and o stands for even and odd respectvely then we have P ( + β P 4 ( = 0 whch s represented P n the table Note that we cannot get any nformaton of Δ K (t andδ K (t from the nformaton P If (m β β = (o o e then Δ K (( β Δ K ( Z/ whch s represented by Z/ The 3rd 4th and 5th columns are flled by the same way for the other generators Δ K (t β ( t β Δ K (t β t m β Δ t β K (t β t m β respectvely t m In the case where m s odd from ths table we have Δ K (( β Δ K ( Z/ Z/ m Wemay choose β to be even then Δ K ( = holds In the case where m s even snce β s odd by the same argument we have Δ K ( Δ K ( Z By applyng the same argument wth exchangng K and K wehave Δ K ( Δ K ( Z Thus Δ K ( = Δ K ( Acknowledgements The author s grateful to hs supervsor Masaharu Ishkawa for many helpful suggestons References [] E Artn: Zur Isotope zwedmensonalen Flachen m R 4 Abh Math Sem Unv Hambg 4 ( [] DE Blar: Remannan Geometry of Contact and Symplectc manfolds Progress n Mathematcs 03 Brkhäuser Boston Inc Boston 00 [3] GE Bredon: Introducton to compact transformaton groups Academc Press New York-London Pure and Appled Mathematcs Vol46 97 [4] RH Cromwell and RH Fox: Introducton to Knot Theory Graduate Texts n Math 57 Sprnger-Verlag 977 [5] R Fntushel: Locally smooth crcle actons on homotopy 4-spheres Duke Math J 43 (

11 688 M Fukuda [6] R Jacoby: One-parameter transformaton groups of the three-sphere Proc Amer Math Soc 7 ( [7] JA Hllman and SP Plotnck: Geometrcally fberd two-knots Math Ann 87 ( [8] D Montgomery and CT Yang: Groups on S n wth prncpal orbts of dmenson n-3 I II Illnos J Math 4 ( ; 5 (96 06 [9] PS Pao: Non-lnear crcle actons on the 4-sphere and twstng spun knots Topology 7 ( [0] S Plotnck: Fbered knots n S 4 -twstng spnnng rollng surgery and branchng Contemp Math vol 5 Amer Math Soc Provdence RI [] D Rolfsen: Knots and Lnks Math Lec Seres 7 Publsh or Persh Inc Berkeley 976 [] M Teragato: Roll-spun knots Math Proc Cambrdge Phlos Soc 3 ( [3] M Teragato: Twst-roll spun knots Proc Amer Math Soc ( [4] EC Zeeman: Twstng spun knotstransammathsoc5 ( Graduate School of Scence Tohoku Unversty Senda Japan e-mal: fukudamzukr5@dctohokuacjp