デーン手術と４次元多様体(Lens spaces obtained by Dehn surgeries
on Akbulut–Yasui links )山田 裕一 （電気通信大学）
微分トポロジー１５〜微分トポロジーの過去・現在・未来〜
March 25, 2015

Abstract2014 年の秋, Mazur link の (4, 5) surgery が lens space であることに気がつきました．そこで，Akubult–Yasui Cork の図式を与える２成分 link にも対象を広げて類似の lens spacesurgery を探し，いくつかの族を見つけましたので報告します．Reidemeister torsion との密接な関係の下で，上記の surgery に関わる Alexander 多項式の合同式（円分体での等式）について得られたことも話します．

Theorem 1.1 There exist some lens space surgeries, and lens ]lens surgeries along themirror image of ) the Mazur link M L, see [Maz].
(M L; 2, 6) = L(11, 3)(M L; 2, 7) = L(13, 5)(M L; 3, 4) = L(11, 2)(M L; 3, 5) = L(2, 1)]L(7, 2)(M L; 3, 6) = L(17, 10)(M L; 4, 4) = L(3, 2)]L(5, 2)(M L; 4, 5) = L(19, 8)

m+1m

Definition 1.2 (Akbulut–Yasui link AYm ) We call the two-bridge link C(2m, 1, 2m + 2)
Akbulut–Yasui link AYm , which was used for describing “corks” in [AY]. Note that AY1 =M L.

AYm

ex. AY2

Theorem 1.3 There exist some lens space surgeries, and lens ]lens surgeries along Akbulut–
Yasui link AYm with m ≥ 2. For m = 1, see the previous theorem.
(AYm ; 2m, 2m + 4)
= L(4m2 + 8m − 1, 2m2 + 3m − 2)
(AYm ; 2m + 1, 2m + 2) = −L(4m2 + 6m + 1, 2m + 3)
(AYm ; 2m + 2, 2m + 3) = −L(4m2 + 10m + 5, 4m + 3)
(AYm ; 2m + 1, 2m + 3) = L(2, 1)] − L(2m2 + 4m + 1, 2m + 1)
(AYm ; 2m + 2, 2m + 2) =−L(2m + 1, m)] − L(2m + 3, 2)
1

M L = AY1

AYm(2, 6)(3, 5)](4, 4)](2m, 2m + 4)
(2m + 1, 2m + 2) (2m + 1, 2m + 3)](2m + 2, 2m + 2)]] means a lens]lens surgery.

(2, 7)(3, 6)(4, 5)(3, 4)(2m + 2, 2m + 3)

Conjecture 1.4 This is the complete list of lens space surgeries on AYm .
A closed relationship between Reidemeister–Turaev torsion and Alexander polynomial anda surgery formula are known, see [KY]. Related to the above lens space surgeries, we checked
the following formulas.
Lemma 1.5 The Alexander polynomial ∆AYm (x, y) of Akbulut–Yasui link AYm satisfies following congruences. They are all in the form: For (AYm ; p, q) = L(P, Q) with P = pq − 1,.
∆AYm (tq , t)(ti − 1)(tj − 1) = (t − 1)(tq − 1) mod (tP − 1)with ij = ±Q or ±Q in Z/P Z. Here Q is defined by QQ ≡ 1 mod P .
ex. 1 For (AYm ; 2m, 2m + 4) = L(4m2 + 8m − 1, 2m2 + 3m − 2),
22∆AYm (t2m , t) · (t2m +3m − 1)(t2m +5m − 1) · t4m ≡ (t − 1)(t2m − 1)
2mod (t4m +8m−1 − 1)
ex. 3 For (M L; 2, 7) = L(13, 5),∆M L (t7 , t) · (t2 − 1) · (t3 − 1) · t ≡ −(t − 1)(t7 − 1) mod (t13 − 1)
ex. 6 For (AY2 ; 6, 7) = L(41, 26),
∆AY2 (t7 , t) · (t6 − 1) · (t33 − 1) · t32 ≡ (t − 1)(t7 − 1) mod (t41 − 1)
Acknowledgement. The author was supported by JSPS KAKENHI (Grant-in-Aid forScientific Research) (C) Grant Number 24540070.
事後の補足：講演の翌日，蒲谷氏（京大）から以下のご指摘をいただきました．感謝いたします．Theorem 1.1 と 1.3 のデーン手術は Martelli–Petronio–Roukema [MPR] で考察された多量のリストに含まれている．
（ただし，上のデーン手術の族をそのリストと照合して，得られる非双曲多様体を具体的に調べることは面倒です．一旦 断念しました．
）

References[AY] S Akbulut, K Yasui, Corks, plugs and exotic structures, J. Gökova Geom. Topol. GGT 2 (2008),
40–82.
[Maz] B Mazur, A note on some contractible 4-manifolds. Ann. of Math. (2) 73 (1961) 221–228.
[KY] T Kadokami, Y Yamada, Lens space surgeries along certain 2-component links related with Park’s
rational blow down, and Reidemeister-Turaev torsion, J. Aust. Math. Soc. 96 (2014), no. 1, 78–126.
[MPR] B Martelli, C Petronio, F Roukema, Exceptional Dehn surgery on the minimally twisted fivechain link Comm. Anal. Geom. 22 (2014), no. 4, 689–735.

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