Seifert surgery on knots via Reidemeister torsion and Casson-Walker-Lescop invariant II
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.
The first auther [Kd1] studied the Reidemister torsion of Seifert fibered homology lens spaces, and showed the following:
([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]).
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
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
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 .
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:
Assume that the Alexander polynomial of is . Then, as noted in [KMS], and is defined.
We then have the following.
Let be a knot in a homology 3-sphere . We assume the following.
Then is not a Seifert fibered space.
Let be the figure eight knot in . Note that . Then by [KMS, Example 2.2]. Hence (2.4) holds if .
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]):
([Ma, Lemma 3]) For an even integer and for an odd integer such that and , we have
By this proposition, we immediately have the following.
For an even integer and for an integer such that and , we have
Proof. By assumptions, there exists such that and . Hence by Proposition 3.1, we have
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),
By (2.1), (2.2) and [Le, 1.5 T2], we have . Hence , and hence
We now consider defined as follows:
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
As in [KMS], we show that implies a contradiction: Suppose that . Since , we have and . Hence
Since , , and as in [KMS], we have
This contradicts .
We next show that implies a contradiction: Suppose that . By (4.1), . Since , . Hence . Since , . Hence
and hence we have the following equation.
Since and are odd (see Figure 1), must be even. Since , we have . We then have
In fact, since is odd, . Hence by (4.5),
Now suppose that . Then . This is impossible since . Next suppose that . Then . This is also impossible since . Thus holds.
Substituing in (4.4),
where (since ). By and Lemma 3.2,
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
Remaining part of the proof is similar to that in the case .
This completes the proof of Theorem 2.1.
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- [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).
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Department of Mathematics, East China Normal University,
Dongchuan-lu 500, Shanghai, 200241, China
Musashino Art University,
Ogawa 1-736, Kodaira, Tokyo 187-8505, Japan
Department of Mathematics, Nihon University,
3-25-40, Sakurajosui, Setagaya-ku, Tokyo 156-8550, Japan