0:Lens spa es whi h are unobtainableby surgery on knotsToshio SAITO(Osaka University)joint work with Kazuhiro I hihara(Osaka Sangyo University)

1:Lens spa esLens spa e L(p; q) : the 3-manifold obtained byp=q-surgery on the trivial knot in S3.[(p; q)- urve

2:ProblemDe ide the lens spa es obtainable bysurgery on non-trivial knots in S3.

3:Known results� (Moser)M : obtained by p=q-surgery on a (r; s)-torus knot.M is a lens spa e ) M = L(jpj; qs2).� (Bleiler-Litherland, Wang, Wu)M : obtained by surgery on a sattelite knot K.M is a lens spa e ) 8><>:K : the (2pq � 1; 2) ableon a (p; q) torus knot;M = L(4pq � 1; 4q2):

4:� (Hirasawa-Shimokawa)L(2p; 1) is unobtainable by surgery on stronglyinvertible knots.� (Goda-Teragaito)Lens spa es are unobtainable by surgery on genusone hyperboli knots.� (Kronheimer-Mrowka-Ozsv�ath-Szab�o)L(p; 1) is unobtainable by p-surgery on knots.

5:Gordon's onje tureM : obtained by non-trivial surgeryon a non-trivial knot.M is a lens spa e ) M is obtained by Berge'ssurgery on a doubly primitive knot.

6:Doubly primitive knots(V1; V2;S) : a genus two Heegaard splitting of S3.K : a simple loop on S.K is a doubly primitive knot if K representsa free generator of both �1(V1) and �1(V2).

7:`Conje ture' (I hihara-Teragaito)Lens spa es ontaining Klein bottles areunobtainable by surgery on non-torus knots.

7:`Conje ture' (I hihara-Teragaito)Lens spa es ontaining Klein bottles areunobtainable by surgery on non-torus knots.TheoremLens spa es ontaining Klein bottles areunobtainable by Berge's surgery onnon-torus doubly primitive knots.

8:Dual knotsK: a knot in a 3-manifold N .N 0 := E(K;N) [ V , where V is a �lling solid torus.K�: a ore loop of V .Then K� is the dual knot of K.

8:Dual knotsK: a knot in a 3-manifold N .N 0 := E(K;N) [ V , where V is a �lling solid torus.K�: a ore loop of V .Then K� is the dual knot of K.Theorem (Berge)K: a doubly primitive knot.Berge's surgeryS3 L(p; q)[ [K K�

8:Dual knotsK: a knot in a 3-manifold N .N 0 := E(K;N) [ V , where V is a �lling solid torus.K�: a ore loop of V .Then K� is the dual knot of K.Theorem (Berge)K: a doubly primitive knot.Berge's surgeryS3 L(p; q)[ [K K� = K(L(p; q);u)

9:K(L(p; q);u)�D2�D1

9:K(L(p; q);u)�D2�D1P0 Pp�1P1 Pu

10:tui : a simple ar in Di joining P0 to Pu.

10:tui : a simple ar in Di joining P0 to Pu.�D2tu1 �D1P0 Pp�1P1 Pu

10:tui : a simple ar in Di joining P0 to Pu.�D2tu1 �D1P0 Pp�1P1 Pu

tu2

10:tui : a simple ar in Di joining P0 to Pu.�D2tu1 �D1P0 Pp�1P1 Pu

tu2K(L(p; q);u) := tu1 [ tu2

11:Basi sequen esFor positive oprime integers p and q,fqj (mod p)gpj=1 is alled a basi sequen e.�D2 �D1P0

12:ObservationL(p; 2) is unobtainable by Berge's surgeryon non-torus doubly primitive knots.

12:ObservationL(p; 2) is unobtainable by Berge's surgeryon non-torus doubly primitive knots.K: a doubly primitive knot.Berge's surgeryS3 L(p; 2)[ [K K� = K(L(p; 2);u)

12:ObservationL(p; 2) is unobtainable by Berge's surgeryon non-torus doubly primitive knots.K: a doubly primitive knot.Berge's surgeryS3 L(p; 2)[ [K K� = K(L(p; 2);u)f2j (mod p)gpj=1 : 2; 4; : : : ; p� 1; 1; 3; : : : ; p� 2; 0

13:Case 1. u � 0 (mod 2).f2j (mod p)gpj=1 : 2; : : : ; u; : : : ; p� 1; 1; : : : ; p� 2; 0�D2tu1 �D1P0 Pp�1P2 Pu

tu2

13:Case 1. u � 0 (mod 2).f2j (mod p)gpj=1 : 2; : : : ; u; : : : ; p� 1; 1; : : : ; p� 2; 0�D2�D1P0 Pp�1P2 Pu

14:Case 2. u � 1 (mod 2).f2j (mod p)gpj=1 : 2; : : : ; p� 1; 1; : : : ; u; : : : ; p� 2; 0�D2tu1 �D1P0 Pp�1P1 Pu

tu2

14:Case 2. u � 1 (mod 2).f2j (mod p)gpj=1 : 2; : : : ; p� 1; 1; : : : ; u; : : : ; p� 2; 0�D2�D1P0 Pp�1P1 Pu

15:K(L(p; q);u) with S3-surgeryThere is an algorithm to determine whethera given lens spa e is obtainableby Berge's surgery on a doubly primitive knot.

15:K(L(p; q);u) with S3-surgeryThere is an algorithm to determine whethera given lens spa e is obtainableby Berge's surgery on a doubly primitive knot.Step 1 Consider a 2-bridge link S(p; q).

uth

16:Step 1 Consider a 2-bridge link S(p; q).Step 2 Atta h a band to S(p; q) as follows.

uth

16:Step 1 Consider a 2-bridge link S(p; q).Step 2 Atta h a band to S(p; q) as follows.=: K0

16:Step 1 Consider a 2-bridge link S(p; q).Step 2 Atta h a band to S(p; q) as follows.=: K1

16:Step 1 Consider a 2-bridge link S(p; q).Step 2 Atta h a band to S(p; q) as follows.Step 3 Che k whether K0 or K1 is a trivial knot.

16:Step 1 Consider a 2-bridge link S(p; q).Step 2 Atta h a band to S(p; q) as follows.Step 3 Che k whether K0 or K1 is a trivial knot.Con lusionK(L(p; q);u) admits integral S3-surgery, K0 or K1 is a trivial knot.