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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.

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    $\begingroup$ what is negative knots? Also can you please elaborate your example, it is not clear to me. $\endgroup$ Aug 4, 2020 at 14:49
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    $\begingroup$ @Anubhav A negative knot is a knot that has a knot diagram with only negative crossings. $\endgroup$
    – Henry
    Aug 4, 2020 at 16:51
  • $\begingroup$ To your convention, the right-handed trefoil is negative knot, right? $\endgroup$ Aug 9, 2020 at 13:41
  • $\begingroup$ @Oguz Right-handed trefoil is positive. $\endgroup$
    – Henry
    Aug 9, 2020 at 15:23

2 Answers 2

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$(-7)$-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.].

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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 $S^3$.

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    $\begingroup$ Moser's notation is somehow non-standard. When she says $(p,q)$-surgery, she means gluing the meridian $\mu$ to $q\mu+p\lambda$, I think. Most people define this as the $(q,p)$-surgery. I am also not sure if the orientation of the lens spaces is really the one induced via the group action on $S^3$ or if she is working with the opposite one. $\endgroup$
    – Marc Kegel
    Aug 10, 2020 at 20:17
  • $\begingroup$ @MarcKegel thanks for your attention. You are right. To the common convention, what L. Moser did is $(q,p)$-surgery on the torus knot in $S^3$. I will edit the post thinking about the orientations. $\endgroup$ Aug 11, 2020 at 10:12

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