Chirally cosmetic surgeries and Casson invariants

Abstract

We study chirally cosmetic surgeries, that is, a pair of Dehn surgeries on a knot producing homeomorphic 3-manifolds with opposite orientations. Several constraints on knots and surgery slopes to admit such surgeries are given. Our main ingredients are the original and the $SL(2,\mathbb{C})$ version of Casson invariants. As applications, we give a complete classification of chirally cosmetic surgeries on alternating knots of genus one.

20 pages, 2 figures. Ver. 2. a gap in the proof of Theorem 2.1 was fixed. Due to this, results in Section 2 and 6 were modified or removed. On the other hand, by further improvements, the final application can be extended to alternating knots of genus one

References in corpus (3)

• Positive knots, closed braids and the Jones polynomial
• Boundary slopes and the numbers of positive/negative crossings for Montesinos knots
• Cosmetic surgeries and non-orientable surfaces