Seifert surgery on knots via Reidemeister torsion and Casson-Walker-Lescop invariant II

# Seifert surgery on knots via Reidemeister torsion and Casson-Walker-Lescop invariant II

Teruhisa Kadokami, Noriko Maruyama and Tsuyoshi Sakai
June 3, 2015
###### Abstract

For a knot with in a homology -sphere, let be the result of -surgery on . We show that an appropriate assumption on the Reidemeister torsion of the universal abelian covering of implies , if is a Seifert fibered space.

00footnotetext: 2010 Mathematics Subject Classification: 11R04, 11R27, 57M25, 57M27. Keywords: Reidemeister torsion, Casson-Walker-Lescop invariant, Seifert fibered space.

## 1 Introduction

The first auther [Kd1] studied the Reidemister torsion of Seifert fibered homology lens spaces, and showed the following:

###### Theorem 1.1

([Kd1, Theorem 1.4]) Let be a knot in a homology -sphere such that the Alexander polynomial of is . The only surgeries on that may produce a Seifert fibered space with base and with have coefficients and , and produce Seifert fibered space with three singular fibers. Moreover (1) if the coefficient is , then the set of multiplicities is where , and (2) if the coefficient is , then the set of multiplicities is where .

It is conjectured that Seifert surgeries on non-trivial knots are integral (except some cases). We [KMS] have studied the -Seifert surgery, one of the remaining cases of the above theorem, by applying the Reidemister torsion and the Casson-Walker-Lescop invariant, and have given sufficient conditions to determine the integrality of ([KMS, Theorems 2.1, 2.3]).

In this paper, we give another condition for the integrality of (Theorem 2.1). Like as in [KMS], the condition is also suggested by computations for the figure eight knot ([KMS, Example 2.2]).

We note two differences of this paper from [KMS]; one is that the surgery coefficient appears in the condition instead of the Casson-Walker-Lescop invariant, and another is that we need more delicate estimation for the Dedekind sum to prove the result.

(1) Let be a homology -sphere, and let be a knot in . Then denotes the Alexander polynomial of , and denotes the result of -surgery on .

(2) The first author [Kd2] introduced the norm of polynomials and homology lens spaces: Let be a primitive -th root of unity. For an element of , denotes the norm of associated to the algebraic extension over . Let be a Laurent polynomial over . We define by

 |f(t)|d=|Nd(f(ζd))|=∣∣ ∣∣∏i∈(Z/dZ)×f(ζid)∣∣ ∣∣.

Let be a homology lens space with . Then there exists a knot in a homology -sphere such that ([BL, Lemma 2.1]). We define by

 |X|d=|ΔK(t)|d,

where is a divisor of . Then is a topological invariant of (Refer to [Kd2] for details).

(3) Let be a closed oriented -manifold. Then denotes the Lescop invariant of ([Le]). Note that .

## 2 Result

Let be a knot in a homology -sphere . Let be the result of -surgery on : . Let be the universal abelian covering of (i.e. the covering associated to ). Since , is the -fold unbranched covering.

In [KMS], we have defined by the following formula, if is defined:

 |K|(q,d):=|X|d.

Assume that the Alexander polynomial of is . Then, as noted in [KMS], and is defined.

We then have the following.

###### Theorem 2.1

Let be a knot in a homology 3-sphere . We assume the following.

(2.1) ,

(2.2) ,

(2.3) ,

(2.4) .

Then is not a Seifert fibered space.

###### Remark 2.2

Let be the figure eight knot in . Note that . Then by [KMS, Example 2.2]. Hence (2.4) holds if .

###### Remark 2.3

Theorem 2.1 seems to suggest studying the asymptotic behavior of as a function of .

## 3 An inequality for the Dedekind sum

To prove Theorem 2.1, we need the following inequality for the Dedekind sum ([RG]):

###### Proposition 3.1

([Ma, Lemma 3]) For an even integer and for an odd integer such that and , we have

 |s(q,p)|

where .

By this proposition, we immediately have the following.

###### Lemma 3.2

For an even integer and for an integer such that and , we have

 |s(q∗,p)|

Proof. By assumptions, there exists such that and . Hence by Proposition 3.1, we have

 |s(q∗,p)|=|s(q,p)|<(p−1)(p−5)24p

###### Remark 3.3

The estimation given in Proposition 3.1 has a natural application ([Ma]).

## 4 Proof of Theorem 2.1

Suppose that is a Seifert fibered space. Then, as shown in [KMS], we may assume that

: has a framed link presentation as in Figure 1,

where and .

Also as shown in [KMS], . Hence by (2.4),

 (αβ)2>4q2 (4.1)

By (2.1), (2.2) and [Le, 1.5 T2], we have . Hence , and hence

 |λ(M)|<αβ2 (4.2)

We now consider defined as follows:

 e:=q12α+q22β+q35.

According to the sign of , we treat two cases separetely: We first consider the case . Then the order of is . Since , , and . Hence by and [Le, Proposition 6.1.1], we have

 λ(M)=(−45)αβ+5β24α+5α24β+1120αβ−14−T (4.3)

where .

By (4.2), we have

 −αβ2<λ(M).

Hence by (4.3),

 −αβ2<(−45)αβ+5β24α+5α24β+1120αβ−14+|T|.

Consequently

 310αβ<−14+524αβ+524(αβ)+1120αβ+|T| (4.4)

As in [KMS], we show that implies a contradiction: Suppose that . Since , we have and . Hence

 35β<−14+524⋅2β+524+1120⋅2⋅3+|T|.

Since , , and as in [KMS], we have

 |T|≤|s(q1,2α)|+|s(q2,2β)|+|s(q3,5)|≤β3+15.

Hence

 35β<−14+548β+524+1120⋅6+(β3+15).

Thus

 (35−548−13)β<−14+524+1120⋅6+15.

Therefore

 39240β<1240(38+13)<39240.

We next show that implies a contradiction: Suppose that . By (4.1), . Since , . Hence . Since , . Hence

 q12+q22β+q35=110β

and hence we have the following equation.

 (5β)q1+5q2+(2β)q3=1 (4.5)

Since and are odd (see Figure 1), must be even. Since , we have . We then have

 (♯):q2≢±1(mod 2β).

In fact, since is odd, . Hence by (4.5),

 β+5q2≡1(mod 2β).

Now suppose that . Then . This is impossible since . Next suppose that . Then . This is also impossible since . Thus holds.

Substituing in (4.4),

 310β<−14+524β+524β+1120β+|T|

where (since ). By and Lemma 3.2,

 |s(q2,2β)|<2β24=β12.

Hence

 |T|≤|s(q2,2β)|+|s(q3,5)|<β12+15.

Since ,

 310β<−14+524β+524⋅8+1120⋅8+(β12+15).

Thus

 (310−524−112)β<−14+524⋅8+1120⋅8+15

and hence . This is a contradiction, and ends the proof in the case .

We finally consider the case . Then . By and [Le, Proposition 6.1.1], we have

 λ(M)=−{(−45)αβ+5β24α+5α24β+1120αβ−14+T}.

Remaining part of the proof is similar to that in the case .

This completes the proof of Theorem 2.1.

## References

• [BL] S. Boyer and D. Lines, Surgery formulae for Casson’s invariant and extensions to homology lens spaces, J. Reine Angew. Math., 45 (1990), 181–220.
• [Kd1] T. Kadokami, Reidemeister torsion of Seifert fibered homology lens spaces and Dehn surgery, Algebr. Geom. Topol., 7 (2007), 1509–1529.
• [Kd2] T. Kadokami, Reidemeister torsion and lens surgeries on knots in homology -spheres II, Top. Appl., 155, no.15 (2008), 1699–1707.
• [KMS] T. Kadokami, N.Maruyama and T. Sakai, Seifert surgery on knots via Reidemeister torsion and Casson-Walker-Lescop invariant, Top. Appl., 188 (2015), 64–73.
• [Le] C. Lescop, Global surgery formula for the Casson-Walker invariant, Ann. of Math. Studies, Princeton Univ. Press., 140 (1996).
• [Ma] N. Maruyama, On a distribution of rational homology 3-spheres captured by the CWL invariant, Preprint (2009).
• [RG] H. Rademacher and E. Grosswald, Dedekind sums, The Carus Mathematical Monograph, 16 (1972).

Department of Mathematics, East China Normal University,

Dongchuan-lu 500, Shanghai, 200241, China

Noriko Maruyama

Musashino Art University,

Ogawa 1-736, Kodaira, Tokyo 187-8505, Japan

maruyama@musabi.ac.jp

Tsuyoshi Sakai

Department of Mathematics, Nihon University,

3-25-40, Sakurajosui, Setagaya-ku, Tokyo 156-8550, Japan

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