Negative surgeries on negative knots

Question

This question is two-fold.

The first question is rather specific: what are some small examples of negative surgeries on negative knots that give rise to the same 3-manifold? I know one class of examples coming from Borromean rings. By performing $-1/m$ and $-1/n$ surgery on two components of the Borromean rings, we get the double twist knot $K_{m,n}$ which is negative. Now, $-1/l$ surgery on $K_{m,n}$ is just $-1/l$, $-1/m$ and $-1/n$ surgery on the Borromean rings, and by the symmetry of the Borromean rings, it is the same as $-1/m$ surgery on $K_{l,n}$ and $-1/n$ surgery on $K_{l,m}$. I would like to know some other simple examples (preferably with knots with small number of crossings).

The second question is a bit vague: what is known about the class of 3-manifolds obtained as negative surgeries on negative knots? I am curious to know if there are some theorems saying this class of 3-manifolds are "nice" in some way. Any kind of input would be greatly appreciated.

Answer 1

$(-7)$-surgery on the left-handed trefoil yields the lens space $L(7,2)$ which is defined to be the $(-7/2)$-surgery along the unknot.

Similarly one can get more examples along negative torus knots producing lens spaces. Moser classified all surgeries along torus knots in [L. Moser, Elementary surgery along a torus knot, Pacific J. Math. 38 (1971), 737–745.].

Answer 2

In general, to find explicit examples for the first part of your question is a hard problem, sometimes impossible. Actually, it is related to the notion of cosmetic surgeries, see Ni and Wu's paper, and further articles.

You may predict conjectures or obtain obstructions due to Thurston's theorem: all but finitely many surgeries on a hyperbolic knot result in hyperbolic manifolds.

On the other hand, as Kegel said, L. Moser completely classified surgeries along torus knots as follows:

Theorem: Let $K$ be an $(r,s)$ torus knot in $S^3$ and let $Y$ be the $3$-manifold obtained by performing a $(p,q)$-surgery along $K$. Set $\sigma =rsp−q$.

(a). If $|\sigma|>1$, then $Y$ is the Seifert manifold $\Sigma(\alpha_1, \alpha_2, \alpha_3)$ over $S^2$ with three exceptional fibers of multiplicities $\alpha_1=s, \alpha_2=r$ and $\alpha_3=|\sigma|$.

(b). If $\sigma =±1$, then $Y$ is the lens space $L(|q|,ps^2)$.

(c). If $\sigma =0$, then $Y$ is the connected sum of lens spaces $L(r,s) \#L(s,r)$.

EDIT: Considering mirror symmetry of knots and following the common convention on surgeries, we have for $n \geq 1$,

1. $\Sigma(r,s,rsn-1)$ is obtained by $(-1,n)$-surgery along the left-handed $(r,s)$ torus knot.
2. $\Sigma(r,s,rsn+1)$ is obtained by $(-1,n)$-surgery along the right-handed $(r,s)$ torus knot.

Note that these are only integral homology spheres obtained by surgery on a torus knot in $S^3$.