Cosmetic surgery and the \(SL(2,\mathbb{C})\) Casson invariant for two-bridge knots. (English) Zbl 1430.57002
Summary: We consider the cosmetic surgery problem for two-bridge knots in the 3-sphere. We first verify by using previously known results that all the two-bridge knots of at most \(9\) crossings admit no purely cosmetic surgery pairs except for the knot \(9_{27}\). Then we show that any two-bridge knot corresponding to the continued fraction \([0, 2x, 2, -2x, 2x, 2, -2x]\) for a positive integer \(x\) admits no cosmetic surgery pairs yielding homology 3-spheres, where \(9_{27}\) appears when \(x = 1\). Our advantage to prove this is using the \(SL(2,\mathbb{C})\) Casson invariant.
MSC:
| 57K10 | Knot theory |
| 57K30 | General topology of 3-manifolds |
| 57M50 | General geometric structures on low-dimensional manifolds |
References:
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| [22] | K. Ichihara and Z. Wu, A note on Jones polynomial and cosmetic surgery, to appear in Comm. Anal. Geom., arXiv:1606.03372. Kazuhiro Ichihara Department of Mathematics College of Humanities and Sciences Nihon University 3-25-40 Sakurajosui Setagaya-ku Tokyo 156-8550, Japan E-mail: ichihara@math.chs.nihon-u.ac.jp Toshio Saito Department of Mathematics Joetsu University of Education |
| [23] | Yamayashiki Joetsu Niigata 943-8512, Japan E-mail: toshio@juen.ac.jp |
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