arXiv:math/0011119v1 [math.GT] 17 Nov 2000 ( ∇ hr ( where dniyn hi onaisi uhawythat way a such in boundaries their identifying hoe 1. Theorem lary. L where rd.Wietecifigein nterpofwsasuyo h Alex the of study a polynomia was Alexander proof the study their in we links spaces, in ric’ lens ingredient in knots chief of the polynomial While motivated classifications is of Brody. method history Our for homeomorphisms. implemen [6] orientation-preserving was to reader This the [5]. refer We 2], which [3]. Problem Mean homeomorphisms Brody [4, [7]. to by see 1952 up Hauptvermutung; in classification the Moise to topolog require by aproach Hauptvermutung the an the to outlined of had generalized proof was the classification with gory This [9]. homeomorphisms lk ik ymti link. symmetric link, [ n sdfie sflos[] 2.Let [2]. [1], follows as defined is x i ( L K n [ and ] ∈ 1 = ,q p, eoepoigTerm1 eoti h lsicto fln space lens of classification the obtain we 1, Theorem proving Before 1991 e ∆ Let Let nti ae,w ieacniepofo lsicto fln space lens of classification a of proof concise a give we paper, this In orientation-pres to up spaces lens classified Reidemeister 1935, In o noine 3-manifold oriented an For e od n phrases. and words Key ( ( p,q Z t − Then . ) , ) c T 1 ) The 2). ([ ahmtc ujc Classification. Subject Mathematics ,q p, / htrpeet eeao of generator a represents that 1 · Abstract. td fteAeadrplnma f‘ymti’lnsin links ‘symmetric’ of polynomial Alexander the of ingred study chief a The homeomorphisms. orientation-preserving y YMTYO IK N LSIIAINOF CLASSIFICATION AND LINKS OF SYMMETRY K 2 K y of ] and − ( ] and 1 = ) steitreto ubrof number intersection the is , t S ethe be ) [ t K 1 H 3 T / Let . )= ]) esspace lens 2 2 1 ,where ), ( esldtr,adlet and tori, solid be M J ρ egv ocs ro facasfiaino essae pto up spaces lens of classification a of proof concise a give We ZFH RYYK N KR YASUHARA AKIRA AND PRZYTYCKI H. OZEF ´ q/p ; : owynraie lxne polynomial Alexander Conway-normalized Z q q epciey ups that Suppose respectively. ) S lk − or essae ikn om lxne oyoil ylccov cyclic Polynomial, Alexander form, linking space, lens 3 M ∇ L −→ pr = K : ( lk ,q p, H ( 1. = q/p M z M 1 L steCna polynomial. Conway the is ) ( sa3mnfl hti bandfrom obtained is that 3-manifold a is ) ([ M ESSPACES LENS ( x x ,q p, ihfiiefis oooygop the group, homology first finite with in ] ; and , Z [ Q ) y ) m )= ]) / ethe be × rmr 72;Scnay57M25. Secondary 57M27; Primary Z y i H c . 1 H and e1cce in 1-cycles be 1 and 1 c ( ( L n · M l y ( p i ,q p, fl ylccvrand cover cyclic -fold y ; m etemrda n ogtd of longitude and meridian the be ∈ Z . 2 ) ); Q = nx −→ / Z Z ) pl If . , onsa2-chain a bounds 1 Q M + / eti u ro is proof our in ient Z qm htrpeetelements represent that ∆ S 3 1 ρ . − and of 1 ( K K ) yta fFox- of that by l ( .. ∆ i.e., , 2 fln spaces. lens of t f‘symmet- of l siscorol- its as s 1 = ) T K = ikn form linking 1 erving hl,Fox while, c ol not would ql and r periodic er, ntin knot a clcate- ical e later ted o some for 1 pto up s + K then , ander T ( rm t 2 = ) PL by T 1 i , 2 JOZEF´ H. PRZYTYCKI AND AKIRA YASUHARA

Corollary 2. Two lens spaces L(p, q) and L(p, q′) are equivalent up to orientation- preserving homeomorphisms if and only if q ≡ q′ (mod p) or qq′ ≡ 1 (mod p). Proof. The ‘if’ part of the corollary is well-known and easy to see. It is enough to show the ‘only if’ part. Since pl1 +qm1 bounds a meridian disk D2 of T2, pl1 bounds a 2-chain qD1 ∪ D2, where qD1 is a disjoint union of q copies of a meridian disk D1 −1 of T1. So we have lkL(p,q)([l1], [l1]) = (qD1 ∪ D2) · l1/p = q/p. Note that ρ (l1) is −1 the trivial knot, and hence ∆ρ (l1)(t) = 1. Suppose that there is an orientation- ′ preserving homeomorphism f : L(p, q) −→ L(p, q ). Then lkL(p,q′)([f(l1)], [f(l1)]) = −1 −1 lkL(p,q)([l1], [l1]) and ∆ρ (f(l1))(t) = ∆ρ (l1)(t). Therefore, by Theorem 1, we have q/p ≡ q′/p or ≡ q′/p (mod 1). This implies that q ≡ q′ (mod p) or qq′ ≡ 1 (mod p).

2 2 Remark 3. Since lkL(p,q)([nl1], [nl1]) = n q/p, the set {n q/p|1 ≤ nAlexander polynomial of links with a certain kind of symmetry. Lemma 4. Let r be a prime and s a positive integer. Let L be an rs-periodic link in S3 and let L′ be a link obtained from L by changing a set of crossings that is the Zrs -orbit of a single crossing in the periodic diagram of L. For an integer q, let L(q) (resp. L′(q)) be a link obtained from L (resp. L′) by adding −q-full twists as illustrated in Figure 1; equivalently we perform a 1/q-surgery along the fixed point set of the Zrs -action. Then rs ∆L(q)(t) ≡ ∆L′(q)(t) mod (t − 1, r). Proof. In Figure 1(b), a tangle T and the full-twists part are commutative. There- fore by arguments similar to that in the proof of Lemma 2.3 in [8], we obtain −1/2 1/2 rs ∆L(q)(t) ≡ ∆L′(q)(t) mod ((t − t ) , r). s s s Since (t−1/2 − t1/2)r ≡ t−r /2 − tr /2 (mod r), we have the conclusion. A link in S3, the universal cover of a lens space, that covers a knot in the lens space has the above-mentioned symmetry. By Lemma 4, we obtain the following proposition. Proposition 5. Let ρ : S3 −→ L(p, q) be the p-fold cyclic cover and let K and K′ be knots in L(p, q). If K and K′ are homologous in L(p, q) and if p is divisible by the sth power rs of a prime integer r, then rs ∆ρ−1(K)(t) ≡ ∆ρ−1(K′)(t) mod (t − 1, r). ′ Proof. We may assume that both K and K are contained in T1. We note that 3 the p-fold cover S is obtained from T˜1 and T˜2 by identifying their boundaries in such a way thatm ˜2 = l˜1 + qm˜1, where T˜i is a solid torus that is the p-fold cover of Ti,andm ˜i and l˜i are the meridian and longitude of T˜i (i = 1, 2). If two knots in L(p, q) are homologous, then they are also homotopic. So they differ in a finite number of crossings. Thus it suffices to consider the case in which K′ is obtained from K by changing a single crossing c. Then ρ−1(K) and ρ−1(K′) are p-periodic SYMMETRY OF LINKS AND CLASSIFICATION OF LENS SPACES 3

Figure 1.

−1 ′ −1 −1 in T˜1, ρ (K ) is obtained from ρ (K) by changing the crossings ρ (c), and a s covering translation φ of T˜1 generates the Zp-action on T˜1. Set p = ur . Then −1 j−1 we note that ρ (c) = {c11, ..., c1u}∪···∪{crs1, ..., crsu}, where φ (ck1) = ckj u(k−1) s and φ (c1j ) = ckj for k = 1, ..., r , j = 1, ..., u. This implies that there is a −1 −1 ′ s sequence ρ (K)= L0, ..., Lu = ρ (K ) of r -periodic links in T˜1 such that Lj is obtained from Lj−1 by changing the crossings c1j , ..., crsj (j =1, ..., u). By Lemma 4, we have rs ∆Lj−1 (t) ≡ ∆Lj (t) mod (t − 1, r). This completes the proof.

Proof of Theorem 1. Let Kn be a knot in ∂T1 ⊂ L(p, q) that represents n[l1]+ [m1] ∈ H1(L(p, q); Z). We may assume that K is homologous to Kn for some n −1 3 ((n,p) = 1). Then ρ (Kn) is the (n,p − qn)-torus knot since S is obtained from T˜1 and T˜2 by identifying their boundaries such thatm ˜2 = l˜1 +qm˜1. It is well-known that n(p−qn) −g (1 − t)(1 − t ) ∆ −1 = t , ρ (Kn) (1 − tn)(1 − tp−qn) where g = (n(p − qn)+1 − (p − qn + n))/2. By Proposition 5, n(p−qn) − (1 − t)(1 − t ) s t g ≡ 1 mod (tr − 1, r), (1 − tn)(1 − tp−qn) where rs is a divisor of p and r is a prime integer. By elementary calculations, we obtain n2 ≡ 1 or ≡ q2 (mod rs). So we have n2 ≡ 1 or ≡ q2 (mod p). Since 2 lkL(p,q)([K], [K]) = n q/p, we have the required result.

References 1. J.W. Alexander, Note on two 3-dimensional manifolds with the same group, Trans. Amer. Math. Soc. 20 (1919), 339–342. 2. J.W. Alexander, New topological invariants expressible as tensors, Proc. Nat. Acad. Sci. 10 (1924), 99–101. 3. E.J. Brody, The topological classification of the lens spaces, Ann. of Math. 71 (1960), 163–184. 4. S. Eilenberg, On the problems of topology, Ann. of Math. 50 (1949), 247–260. 4 JOZEF´ H. PRZYTYCKI AND AKIRA YASUHARA

5. R.H. Fox, Recent development of knot theory at Princeton, Proceedings of the International Congress of Mathematics, Cambridge, Mass., 1950, vol. 2, pp. 453–457, Amer. Math. Soc., Providence, R.I., 1952. 6. C.McA. Gordon, 3-dimensional topology up to 1960, History of topology, pp. 449–489, North- Holland, Amsterdam, 1999. 7. E. Moise, Affine structures in 3-manifolds V. The triangulation theorem and Hauptvermu- tung, Ann. of Math. 56 (1952), 96–114. 8. J.H. Przytycki, On Murasugi’s and Traczyk’s criteria for periodic links, Math. Ann. 283 (1989), 465–478. 9. K. Reidemeister, Homotopieringe und Linsenr¨aume, Abh. Math. Sem. Univ. Hamburg 11 (1935), 102–109. 10. J.H.C. Whitehead, On incidence matrices, nuclei and homotopy types, Ann. of Math. 42 (1941), 1197–1239.

Department of Mathematics, The George Washington University, Washington, DC 20052, USA E-mail address: [email protected]

Department of Mathematics, Tokyo Gakugei University, Nukuikita 4-1-1, Koganei, Tokyo 184-8501, Japan Current address, October 1, 1999 to September 30, 2001: Department of Mathematics, The George Washington University, Washington, DC 20052, USA E-mail address: [email protected]