A note on lens space surgeries: orders of fundamental groups versus Seifert genera
aa r X i v : . [ m a t h . G T ] J a n A NOTE ON LENS SPACE SURGERIES: ORDERS OFFUNDAMENTAL GROUPS VERSUS SEIFERT GENERA
TOSHIO SAITO
Abstract.
Let K be a non-trivial knot in the 3-sphere with a lens spacesurgery and L ( p, q ) a lens space obtained by a Dehn surgery on K . We studya relationship between the order | p | of the fundamental group of L ( p, q ) andthe Seifert genus g of K . Considering certain infinite families of knots withlens space surgeries, the following estimation is suggested as a conjecture:2 g + 2 √ g + 15 + 35 ≤ | p | ≤ g + 3 except for ( g, p ) = (5 , Backgrounds A Dehn surgery on a knot K is an operation of removing a regular neighborhoodof K and filling a solid torus along the resulting boundary. In particular, a Dehnsurgery yielding a lens space is called a lens space surgery . Gordon and Luecke [6]showed that a non-trivial surgery on a non-trivial knot in the 3-sphere S cannotyield S . Gabai [4] proved that S × S never comes from a Dehn surgery ona non-trivial knot in S . Moser [11] completely classified all the Dehn surgerieson the torus knots in S . Also, Bleiler and Litherland [2], Wang [17] and Wu[18] independently characterized the lens space surgeries on satellite knots in S .The Cyclic Surgery Theorem , obtained by Culler, Gordon, Luecke and Shalen [3],implies that if a non-trivial, non-torus knot in S admits a lens space surgery, thenthe surgery must be longitudinal. In early 1990s, Berge [1] introduced a conceptof doubly primitive knots and proved that any doubly primitive knot yields a lensspace by a Dehn surgery along a surface slope (see [1] or [15] for details). In thisarticle, such a surgery will be called Berge’s surgery . He also gave examples ofdoubly primitive knots divided into 12 infinite families. We remark that one ofthe 12 families, which is of type I, is for the torus knots and another, which is oftype II, is for the satellite knots with lens space surgeries. Every doubly primitiveknot is expected to belong to one of the 12 families. Though it is still open thatwhich hyperbolic knots in S admit lens space surgeries, Gordon [9, Problem 1.78]conjectured that such a knot would be doubly primitive.Let K be a non-trivial knot in S with a lens space surgery and L ( p, q ) a lensspace obtained by a Dehn surgery on K . The purpose of this article is to studya relationship between the order | p | of the fundamental group of L ( p, q ) and theSeifert genus g of K . To the author’s knowledge, Goda and Teragaito [5] firstmentioned their relationship: if K is hyperbolic, then | p | ≤ g −
7. They alsoproposed the following in the same paper:
Conjecture 1.1 (Goda-Teragaito [5, Conjecuture]) . Let K be a hyperbolic knot in S with a lens space surgery and L ( p, q ) a lens space obtained by a Dehn surgery The author was supported by JSPS Postdoctoral Fellowships for Research Abroad. | p | g
100 200 300 400 500 | p | = 4 g − | p | = 2 g + 8 Figure 1. on K . Then K is fibered and g + 8 ≤ | p | ≤ g − , where g is the Seifert genus of K . We remark that Ni [12] proved that such a knot is always fibered. We nowfocus on the inequality which is the latter part of Conjecture 1.1. There are someestimates of the order | p | by the Seifert genus g . Rasmussen [14] proved | p | ≤ g + 3without assuming hyperbolicity. Kronheimer, Mrowka, Ozsv´ath and Szab´o [10]showed 2 g − ≤ | p | . The both results are remarkably close to the inequality inConjecture 1.1. In particular, if we do not assume hyperbolicity, then Rasmussen’sinequality is sharp as he mentioned in [14]. In fact, the (4 k + 3)-surgery on thetorus knot T (2 , k + 1) satisfies the equality. If one considers sharpness in a sensethat there are infinite families satisfying equalities, the inequality in Conjecture1.1 seems not to be sharp. Figure 1, if that helps, is a scatter diagram of theorder | p | and the Seifert genus g for Berge’s surgery on all the hyperbolic knots inBerge’s examples with | p | ≤ Lower bound for non-hyperbolic knots
Recall that the Dehn surgeries on the torus knots are completely classified byMoser [11]. Hence we begin with calculating a sharp lower bound of | p | for lensspace surgeries on non-trivial torus knots. For simplicity, we assume r > s > T ( r, s ). Proposition 2.1.
Suppose a non-trivial torus knot K T in S yields the lens space L ( p, q ) by a Dehn surgery. Then g + √ g + 1 ≤ | p | , where g is the Seifert genusof K T . Moreover, the equality holds if and only if K T is the torus knot T ( j + 1 , j )( j ≥ and the surgery coefficient is j + j − . NOTE ON LENS SPACE SURGERIES 3 b strands a strands b strands a strands1-full twist(i) (ii) Figure 2.
Proof.
If the torus knot T ( r, s ) yields the lens space L ( p, q ) by m/n -surgery, then | p | = nrs ± | p | = rs −
1. Since the Seifert genus of T ( r, s ) is ( r − s − , it is enough toshow: ( r − s −
1) + p r − s −
1) + 1 ≤ rs − p r − s −
1) + 1 ≤ r + s −
2. Since( r + s − − (4( r − s −
1) + 1) = ( r − s ) − | r − s | = 1and hence r − s = 1. (cid:3) Remark 2.2.
We can similarly show the following: if a satellite knot K S in S yields the lens space L ( p, q ) by a Dehn surgery, then g + √ g + 1 ≤ | p | , where g is the Seifert genus of K S , and the equality holds if and only if K S is the (2 j ( j +1) + 1 , -cable of the torus knot T ( j + 1 , j ) . We, however, omit the proof sincethis lower bound of | p | is higher than that in Proposition 2.1, which means it hasan insignificant effect on a lower bound of | p | for lens space surgeries on hyperbolicknots.3. Knots on a genus one fiber surface of the figure-eight knot
For a pair ( a, b ) of coprime positive integers, let k − ( a, b ) be a knot on a genusone fiber surface of the figure-eight knot as illustrated in Figure 2 (i). This is knownto be of type VIII in Berge’s families of doubly primitive knots. It is shown in [1]that ( a + ab − b )-surgery, which we call Berge’s surgery, on k − ( a, b ) yields a lensspace. For simplify, we assume a > b >
0. We first notice:
Observation 3.1.
For each integer j ≥ , k − ( j + 1 , is the torus knot T ( j + 1 , j ) . This implies that the torus knots satisfying the equality in Proposition 2.1 arealso of type VIII. Since the other knots of type VIII should be hyperbolic, it wouldmake sense to find a lower bound of | p | for lens space surgeries on knots of type TOSHIO SAITO
VIII. Once one finds an expected lower bound, it would not be so difficult to proveit.
Theorem 3.2.
Suppose a knot K = k − ( a, b ) yields the lens space L ( p, q ) by Berge’ssurgery , i.e., | p | = a + ab − b . Then g + 2 √ g + 15 + 35 ≤ | p | , where g is theSeifert genus of K and the equality holds if and only if K = k − (2 j + 1 , j ) ( j ≥ .Proof. We notice that k − ( a, b ) can be set in a closed positive braid position asillustrated in Figure 2 (ii). Hence Seifert’s algorithm detects its fiber surface andtherefore we see that the Seifert genus of k − ( a, b ) is equal to a + ab − b − a +12 . Wenow only have to show:( a + ab − b − a + 1) + 2 p a + ab − b − a + 1) + 15 + 35 ≤ a + ab − b ,or equivalently p a + ab − b − a + 1) + 1 ≤ a −
4. Since(5 a − − (20( a + ab − b − a + 1) + 1) = 5( a − b ) − | a − b | = 1and hence a − b = 1. (cid:3) Theorem 3.3.
For each integer j ≥ , k − (2 j + 1 , j ) is hyperbolic. To prove the theorem above, we consider the dual knot in the lens space L ( p, q )obtained by Berge’s surgery. Here, the dual knot is a core loop of the filling solidtorus. It is shown by Berge [1] that the dual knot of a doubly primitive knot in L ( p, q ) is a 1-bridge braid and is represented by K ( L ( p, q ); u ) (see also [15] fordetails). We now prepare the following notations. Definition 3.4.
Let ( p, q ) be a pair of coprime integers with p, q >
0. Let { φ i } ≤ i ≤ p be the finite sequence with φ i ≡ iq (mod p ) and 0 ≤ φ i < p . For an integer u with0 < u < p , Ψ p,q ( u ) denotes the integer i with φ i = u , and Φ p,q ( u ) denotes thenumber of elements of the following set: { φ i | ≤ i < Ψ p,q ( u ) , φ i < u } .Also, e Φ p,q ( u ) denotes the following: e Φ p,q ( u ) = min { Φ p,q ( u ) , Φ p,q ( u ) − Ψ p,q ( u ) + p − u, Ψ p,q ( u ) − Φ p,q ( u ) − , u − Φ p,q ( u ) − } .It has been proven in [16, Corollary 4.6] that e Φ p,q ( u ) is an invariant for K ′ = K ( L ( p, q ); u ) if K ′ yields S by a longitudinal surgery. In such a case, e Φ p,q ( u ) ishereafter denoted by Φ( K ′ ). Lemma 3.5 ([16, Theorem 1.3]) . Suppose K ′ = K ( L ( p, q ); u ) yields S by a lon-gitudinal surgery. Then Φ( K ′ ) ≥ if and only if K ′ is hyperbolic. Proof of Theorem 3.3.
Set K = k − (2 j + 1 , j ) and K ∗ the dual knot of K in thelens space obtained by Berge’s surgery. It follows from the formula given in [16,Theorem 6.2 (2)] that K ∗ is represented by K ( L (5 j + 5 j + 1 , j − j − K ( L (5 j + 5 j + 1 , j + 9); 5 j + 8) which is easier to deal with. Toavoid useless complications, we set p = 5 j + 5 j + 1, q = 5 j + 9 and u = 5 j + 8.Since L ( p, q ) comes from a longitudinal surgery on K , we see that K ∗ yields S by NOTE ON LENS SPACE SURGERIES 5 | p | g
100 200 300 400 500 | p | = 3 g + 3 | p | = 2 g + √ g +15 + Figure 3. a longitudinal surgery. Hence calculating Φ( K ∗ ) is sufficient to prove the theorem.Since (5 j + 2) q ≡ u (mod p ), we have Ψ p,q ( u ) = 5 j + 2. We now consider thefollowing finite sequence to prove Φ p,q ( u ) = 4: { φ i } ≤ i ≤ p with φ i ≡ iq (mod p )and 0 ≤ φ i ≤ p . It is easy to see that φ j +1 < u , φ j +1 < u , φ j +1 < u and φ j +1 < u . It also follows from easy calculations that φ i > u for any integer i which satisfies 1 ≤ i < Ψ p,q ( u ) and i = j + 1 , j + 1 , j + 1 , j + 1. This impliesΦ p,q ( u ) = 4 and hence we have Φ( K ∗ ) = 4 if j ≥
2. Thus we see that K ∗ ishyperbolic by Lemma 3.5. Since the exterior of K in S is homeomorphic to thatof K ∗ in the lens space, we have the desired result. (cid:3) Prospective estimation
By consideration in the previous section together with computer experiments,we propose the following:
Conjecture 4.1.
Suppose that a hyperbolic knot in S yields L ( p, q ) by a Dehnsurgery. Then g + 2 √ g + 15 + 35 ≤ | p | ≤ g + 3 except for ( g, p ) = (5 , . One might think it is over-optimistic, but this estimation is correct for Berge’ssurgery on all the hyperbolic knots in Berge’s examples with | p | ≤ K in Berge’s examples isparametrized by a triplet of integers ( p, q, u ). Actually, we have formulas to obtainsuch a parametrization from a doubly primitive position of K [16, Section 6]. Wecan also determine whether K is hyperbolic by Lemma 3.5. We can calculate theAlexander polynomial of K from ( p, q, u ) [8]. Since any doubly primitive knot isfibered [13], the degree of the Alexander polynomial of K is equal to twice theSeifert genus of K . Therefore we can determine the Seifert genus g of K . TOSHIO SAITO j + 1 strands(3 j + 1 , j − K T (i) (ii) Figure 4.
The exception in Conjecture 4.1 is due to 19-surgery on the pretzel knot of type( − , , sharp ina sense that there are infinite families satisfying equalities. The sharpness of thelower bound is assured by Theorems 3.2 and 3.3. For the upper bound case, wehave the following. Proposition 4.2.
Let K j ( j ≥ be the knot which is obtained from K by twisting ( j − times along T as illustrated in Figure 4 (i). Then K j is a hyperbolic knotand satisfies the upper equality in Conjecture 4.1.Proof. We first notice that each K j is known to be a doubly primitive knot of typeV in Berge’s examples. Hence we see that 9 j -surgery on K j yields L (9 j, j − K j has a form of the closure of a positive braid illustrated in Figure 4 (ii), wealso see that the Seifert genus of K j is equal to 3 j −
1. Therefore K j satisfies theupper equality in Conjecture 4.1. Finally, it follows from [16, Theorem 6.1] that thedual knot K ∗ j of K j is represented by K ( L (9 j, j − j + 1). Using the same wayin the proof of Theorem 3.3, we have Φ( K ∗ j ) ≥ K j is hyperbolic. (cid:3) Remark 4.3.
The author heard from Ni that Greene [7] had obtained a lowerbound of | p | which is fairly close to that in Conjecture 4.1. By subsequent pri-vate correspondence with Greene, the author knew that he verified that the lowerinequality in Conjecture 4.1 holds for all the knots in Berge’s examples. Acknowledgements
The author would like to thank Dr. J. Greene for many useful conversations. Hewould also like to thank the referee for a careful reading.
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