結び目に沿った矯飾的手術予想について

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結び目に沿った矯飾的手術予想について

市原一裕

(日本大学文理学部)

斎藤敏夫

(

上越教育大学

)

鄭仁大

(

近畿大学理工学部

)

日本数学会

2016

年度年会

2016.3.16 @

筑波大学

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Cosmetic surgery

Cosmetic Surgery Conjecture

[Bleiler (Kirby’s list Problem 1.81(A))]

Two surgeries on inequivalent slopes are never purely cosmetic.

i.e., if K(r ₁ ) ∼ = K(r ₂ ) for inequivalent slopes r ₁ , r ₂ , then the homeomorphism is orientation reversing.

• Two slopes are equivalent if there exists

a homeo. of E(K ) taking one slope to the other.

• Two surgeries on K along slopes r ₁ , r ₂ are purely cosmetic

if there exists an ori.-pres. homeo. between K(r ₁ ) & K (r ₂ ),

and chirally cosmetic if the homeo. is orientation-reversing.

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2-bridge knots with at most 9 crossings

Proposition [I.-Saito].

All the two-bridge knots of at most 9 crossings other than 9 ₂₇ admit no cosmetic surgery pairs.

Table: 2-bridge knots of at most 9 crossings with τ = 0

Name Schubert Form Alexander Polynomial ∆^′′_K(1)

4₁ S(5,2) t⁻¹−3 +t 2

6₁ S(9,7) 2t⁻¹−5 + 2t 4

6₃ S(13,5) t⁻²−3t⁻¹+ 5−3t+t² 2

7₇ S(21,8) t⁻²−5t⁻¹+ 9−5t+t² -2

8₁ S(13,11) 3t⁻¹−7 + 3t 6

8₃ S(17,4) 4t⁻¹−9 + 4t 8

8₈ S(25,9) 2t⁻²−6t⁻¹+ 9−6t+ 2t² 4

89 S(25,7) t⁻³−3t⁻²+ 5t⁻¹−7 + 5t−3t²+t³ 4

812 S(29,12) t⁻²−7t⁻¹+ 13−7t+t² -6

8₁₃ S(29,11) 2t⁻²−7t⁻¹+ 11−7t+ 2t² 2

9₁₄ S(37,14) 2t⁻²−9t⁻¹+ 15−9t+ 2t² -2

9₁₉ S(41,16) 2t⁻²−10t⁻¹+ 17−10t+ 2t² -4

−3− ⁻² ⁻¹− − ² ³

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A family including 9 27

Theorem [I.-Saito].

Let K _x be a 2-bridge knot C[2x, 2 − 2x, 2x, 2, − 2x] with x ≥ 1.

Then K x admits no cosmetic surgery pairs yielding homology 3-spheres.

i.e., any _n ¹ - and _m ¹ -surgeries are not purely cosmetic for K _x .

Remark:

For K x , ∆ ^′′ _K

_x

(1) = 0 and τ (K x ) = 0 hold. In particular, K 1 = 9 27 .

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New Example

• This knot K admits

a (chirally) cosmetic banding.

• The double branched cover M along K is hyperbolic.

• The knot K ¯ in M corresponding to the banding is hyperbolic.

• The knot K ¯ is not amphicheiral.

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結び目に沿った矯飾的手術予想について