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Networking Seifert Surgeries on Knots

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Networking Seifert Surgeries on Knots Networking Seifert surgeries on knots Kimihiko Motegi joint work with Arnaud Deruelle and Katura Miyazaki Dehn surgery K: a knot in the 3-sphere S3. l N(K) m remove N(K) g glue m p g =pm +q l q p 3 K(q): a 3-manifold obtained from S p by q-Dehn surgery on K In the following, we consider nontrivial surgery p =6 1 q 0 f Dehn surgery g = f knots in S3 g £ Q 1 Non-hyperbolic surgeries Let K be a hyperbolic knot in S3, i.e. S3 ¡ K admits a complete hyperbolic metric of ¯nite volume. Q = H[R[T[S[C H= fr j K(r) is hyperbolicg, R= fr j K(r) is reducibleg, T = fr j K(r) is toroidalg, S= fr j K(r) is Seifert g and C= fr j K(r) is a counterexample to the Ge- ometrization conjectureg. 2 Thurston's hyperbolic Dehn surgery theory shows: jR [ T [ S [ Cj < 1. Solution of the Geometrization conjecture implies C= ;. [Perelman] The cabling conjecture (Gonz¶alez-Acu~na-Short) asserts R= ;. Therefore it is conjectured: Q = H [ T [ S for any hyperbolic knot. 3 \Most" knots are hyperbolic, and for \most" hyperbolic knots we have Q = H. Theorem 1 (Miyazaki-M) An arbitrary knot in S3 can be deformed into a hyperbolic knot with Q = H by a single crossing change. Furthermore, there are in¯nitely many such crossing changes for the given knot. 4 Surgeries on torus knots S3 has in¯nitely many Seifert ¯brations and a torus knot is a regular ¯ber in some Seifert ¯bration of S3. Wq,p torusknotTp,q Vp,q 3 S=Vp,qq,pU W 3 The exterior S ¡ intN(Tp;q) is also Seifert ¯bered. Thus Dehn surgeries on a torus knot naturally create Seifert ¯ber spaces except for the surgery along the ¯ber slope. Moser (1971) conjectured that only surgeries on torus knots can produce Seifert ¯ber spaces. 5 Seifert surgeries on hyperbolic knots P(-2,3,7)pretzelknot (17)=Seifert (18)=Seifert ()lens (19)=Seifert ()lens [Fintushel-Stern] 6 figure-eightknot (-1)=Seifert (-2)=Seifert (-3)=Seifert [Thurston] 7 ¶ Question 1 ³ How can we prove (-1) isaSeifertfiberspace? ´ ² Kirby-Rolfsen's surgery calculus ² Montesinos trick ² double-primitive, primitive/Seifert-¯bered con- structions [Berge], [Dean] ² deformation theory of geometric structures may answer the above question. 8 (-1) isaSeifertfiberspace? Here is another question: ¶ Question 2 ³ Why ´ 9 How about the following question in our prac- tical life? ¶ Question 1 ³ How can I show that I have a cold? ´ ¶ Question 2 ³ Why do I have a cold? ´ 10 WhydoIhaveacold? 11 Ihaveacold 12 Ihaveacold Ihaveacold 13 Ihaveacold Ihaveacold Ihaveacold 14 Ihaveacold Ihaveacold Ihaveacold Ihaveacold 15 Ihaveacold Ihaveacold Ihaveacold Ihaveacold Ihaveacold Ihaveacold 16 Often we can understand the reason why we haveIhaveacold cold by considering our \network". Ihaveacold Ihaveacold Ihaveacold Ihaveacold Ihaveacold So it might be possible to understand .... 17 WhyamIaSeifertsurgery? 18 WhyamIaSeifertsurgery? 19 IamaSeifertsurgery 20 IamaSeifertsurgery IamaSeifertsurgery 21 IamaSeifertsurgery IamaSeifertsurgery IamaSeifertsurgery 22 IamaSeifertsurgery IamaSeifertsurgery IamaSeifertsurgery IamaSeifertsurgery 23 IamaSeifertsurgery IamaSeifertsurgery IamaSeifertsurgery IamaSeifertsurgery IamaSeifertsurgery IamaSeifertsurgery 24 Proposal Let us look all Seifert surgeries as a network and try to make a global picture of them. (K(4,1),-127)-11,13 (K(4,9),1)-11,13 c 1 = (K(-11,13,4),-127) c (P(-2,3,7),18) T-11,13 (O,3) c2 --1 (T,-7)-3,2 c 4 (K(-11,13,1),-139) (O,2) (O,1) (T,-22)-11,2 (P(-3,3,5),1) (T,-143)-11,13 c’1 (K(-2,0,0),56) (O,0) (K(2,0,0),40) (T,-6)-3,2 (O,-1) (C,(T),)-2n(n-1)+1,2 -n,n-1 -4n(n-1) (T,15) (O,-2) 3,5 (D(2,7,3),23) (O,-3) c (T,6)3,2 (K,31)2 1 c n (K,16n-1)n -1 Kn 25 Seifert Surgery Network A pair (K; m) is a Seifert surgery if K(m) is a Seifert ¯ber space. (K; m)=(K0; m0) if K and K0 are isotopic in S3 and m = m0. vertices = Seifert surgeries (K; m)'s -1 -1 -1 ( , ) ( , ) ( , ) figure-eight unknot trefoil edges = ? 26 \seiferters" for Seifert surgeries s q K=Tp,q sm s p c (= s¹; sp; sq) enjoys the following property. ² c is a trivial knot in S3. ² c is a Seifert ¯ber in Tp;q(m) for any integer m =6 pq. We call s¹; sp and sq basic seiferters. 27 c c also enjoys the following property. ² c is a trivial knot in S3. ² c is a Seifert ¯ber in K(¡1);K(¡2) and K(¡3). We say that c is a seiferter for Seifert surg- eries (K; ¡1); (K; ¡2); (K; ¡3). 28 c c also enjoys the following property. ² c is a trivial knot in S3. ² c is a Seifert ¯ber in K(¡6);K(¡7) and K(¡8). We say that c is a seiferter for Seifert surg- eries (K; ¡6); (K; ¡7); (K; ¡8). 29 ¶ seiferter ³ A knot c in S3 ¡ K is called a seiferter for a Seifert surgery (K; m) if c: ² is unknotted in S3, ² becomes a Seifert ¯ber in K(m). ´ K c seiferter Theorem 2 (Miyazaki-M) If a Seifert surgery (K; r) has a seiferter, then r is an integer ex- cept when K is a torus knot or a cable of a torus knot. \Existence of seiferters" implies the conjecture \Seifert surgery is integral". 30 \Inheritance" property 3 K =T S 0 -3,2 Kp 2p+2 p-twisting c along c c trefoil Dehnsurgery Dehnsurgery ()-1 2p+2 ()-1 c surgeryalong c thefiber c Seifertfiberspace Seifertfiberspace 31 p’ q r surgery along c c Seifert p q r (lensspace) c c 0 q r Seifert lens#lens \an exceptional ¯ber of index 0" | degenerate case. 32 edges , seiferters Two Seifert surgeries (K; m) and (K0; m0) are connected by an edge if we have a seiferter c such that (K0; m0) is obtained from (K; m) by a single (meaning §1) twisting along the seiferter c. K-2 K-1 K -1 -1 -1 ( ) ( , ) ( ) C , C C , figure-eight unknot trefoil 33 edges , seiferters Two Seifert surgeries (K; m) and (K0; m0) are connected by an edge if we have a seiferter c such that (K0; m0) is obtained from (K; m) by a single (meaning §1) twisting along the seiferter c. K-2 K-1 K -1 -1 -1 ( ) ( , ) ( ) C , C C , figure-eight unknot-1-twist trefoil 34 edges , seiferters Two Seifert surgeries (K; m) and (K0; m0) are connected by an edge if we have a seiferter c such that (K0; m0) is obtained from (K; m) by a single (meaning §1) twisting along the seiferter c. K-2 K-1 K -1 -1 -1 ( ) ( , ) ( ) C , C C , figure-eight -1-twist unknot-1-twist trefoil 35 edges , seiferters Two Seifert surgeries (K; m) and (K0; m0) are connected by an edge if we have a seiferter c such that (K0; m0) is obtained from (K; m) by a single (meaning §1) twisting along the seiferter c. K-2 K-1 K -1 -1 -1 ( ) ( , ) ( ) C , C C , figure-eight -1-twist unknot-1-twist trefoil Then we obtain a 1-dimensional complex which we call Seifert Surgery Network. 36 Seiferters, S-families and S-lattices Each Seifert surgery (K; m) with a seiferter c provides us with in¯nitely many Seifert surg- eries (Kp; mp) p 2 Z, which form a subcom- plex consisting of Seifert surgeries{ S-family: K-2 K-1 K Sc(K; m). -1 -1 -1 ( ) ( , ) ( ) C , C C , figure-eight -1-twist unknot-1-twist trefoil 37 Combinatorial structure of the network Theorem 3 Sc(K; m) forms a 1-dimensional lattice: if (Kp; mp) = (Kq; mq), then p = q. We call Sc(K; m) an S-lattice. S-lattice (K,m)-3-3 (K,m)-2-2 (K,m)-1-1 (K,m) (K,m)11 (K,m)22 (K,m)33 Thiscannothappen 38 Theorem 4 Any two S-lattices intersect in at most ¯nitely many vertices. S-lattices In fact it is expected at most once. 39 Theorem 5 The Seifert Surgery Network is not a tree. In fact, it has in¯nitely many cycles consisting of Seifert surgeries on hyperbolic knots. Proposition 6 (O; m) has in¯nitely many seifer- ters, and hence there are in¯nitely many edges adjacent to (O; m). (O,m) 40 Theorem 7 (1) The subnetwork T consist- ing of Seifert surgeries on torus knots and edges corresponding to basic seiferters is con- nected. (2) The subnetwork G consisting of Seifert surgeries on graph knots and edges corre- sponding to basic seiferters for companion torus knots is connected. (T,2202)71,31 (T,2204)71,31 (T,37)9,4 (T,280)9,31 T isconnectedbybasicseiferters (O,1) (O,5)=(T,5)1,4 41 Connectivity problem Is the Seifert Surgery Network connected? More precisely, for every Seifert surgery (K; m), does there exist a path from (K; m) to a Seifert surgery on a torus knot? (K(4,1),-127)-11,13 (K(4,9),1)-11,13 c 1 = (K(-11,13,4),-127) c (P(-2,3,7),18) T-11,13 (O,3) c2 --1 (T,-7)-3,2 c 4 (K(-11,13,1),-139) (O,2) (O,1) (T,-22)-11,2 (P(-3,3,5),1) (T,-143)-11,13 c’1 (K(-2,0,0),56) (O,0) (K(2,0,0),40) (T,-6)-3,2 (O,-1) (C,(T),)-2n(n-1)+1,2 -n,n-1 -4n(n-1) (T,15) (O,-2) 3,5 (D(2,7,3),23) (O,-3) c (T,6)3,2 (K,31)2 1 c n (K,16n-1)n :Seifertsurgeryon torusknot -1 Kn portionofthenetwork 42 To obtain Seifert surgeries on hyperbolic knots from those on torus knots, we need non-basic seiferters.
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