Exceptional cosmetic surgeries on \(S^{3}\) (Q272877)

Let \(K\) be a knot in \(S^3\), \(S^3_K\) the exterior of \(K\) and \(S^3_K(r)\) the result of a Dehn surgery on \(K\) with slope \(r\). Let \(r, s \in \mathbb{Q}\cup\{\infty \}\) be two distinct slopes on \(\partial N(K)\) such that \(S^3_K(r)\) and \(S^3_K(s)\) are homeomorphic, then the surgeries are called cosmetic; moreover if there is a homeomorphism between \(S^3_K(r)\) and \(S^3_K(s)\) that preserves orientation, then the surgeries are said to be truly cosmetic.NEWLINENEWLINEWhen int\((S^3_K)\) admits a complete finite volume Riemannian metric, the knot \(K\) is called hyperbolic. For such knots it is known that for all but finitely many slopes \(\alpha\) on \(\partial N(K)\), \(S^3_K(\alpha)\) is hyperbolic. A slope for which \(S^3_K(\alpha)\) is not hyperbolic is called an exceptional slope.NEWLINENEWLINEThe article under review focuses on cosmetic surgeries which are also exceptional surgeries. It is proved that for a hyperbolic knot \(K\) which admits two distinct exceptional slopes \(r,s\) on \(\partial N(K)\) so that \(S^3_K(r)\) is homeomorphic to \(S^3_K(s)\) as oriented manifolds, then the surgery must be toroidal and non-Seifert fibered, and moreover \(\{r,s \}=\{+1, -1\}\). As consequences of this statement the author deduces that there are no exceptional truly cosmetic surgeries on non-trivial algebraic knots, on alternating hyperbolic knots and on arborescent knots in \(S^3\).NEWLINENEWLINEThe author also gives the following results related to Heegaard Floer theory. Given a hyperbolic knot \(K\subset S^3\) which admits an exceptional truly cosmetic surgery, then the Heegaard Floer correction term vanishes for any \(\frac{1}{n}\) surgery on \(K\). If \(Y\) is a 3-manifold obtained as an exceptional truly cosmetic surgery on a hyperbolic knot \(K\subset S^3\), the following inequality holds: NEWLINE\[NEWLINE| t_0(K)| + 2 \sum_{i=1}^{n} | t_i(K) | \leq rank(HF_{red}(Y)) NEWLINE\]NEWLINE where \(t_i(K)\) is the torsion invariant of the Alexander polynomial \(\Delta(K)\), \(n\) is the degree of \(\Delta(K)\) and \(rank(HF_{red}(Y))\) is the rank of the reduced Heegaard Floer homology of \(Y\).