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.



Negative surgeries on negative knots
$(-7)$-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.].


