$3$-manifold that is a surgery on a knot

Question

By the Lickorish-Wallace theorem, every oriented closed $3$-manifold can be obtained by a surgery on a link in $S^3$. In the statement of this result, links are required: not every such manifold can be obtained by a surgery on a knot in $S^3$. One way to see this is that $H_1(S^3_{p/q}(K))$ is cyclic for any knot.

1. What are some other necessary or sufficient conditions on a $3$-manifold to be a surgery on a knot in $S^3$?

The same question can be asked if we restrict ourselves to $\mathbb{Z}HS^3$. I am aware of some strengthenings of the Lickorish-Wallace theorem from the work of Hilden and Matveev, but they still require links.

2. What is some $\mathbb{Z}HS^3$ that cannot be obtained by a surgery on a knot in $S^3$? Is there a characterization of $\mathbb{Z}HS^3$s that can be obtained by a surgery on a knot in $S^3$? If not, what are some necessary or sufficient conditions?

I know that these questions are equivalent to asking which surgery diagrams can be reduced to knots by the Kirby moves, so I do not expect a complete classification exists.

3. Are there any partial results for branched double covers of knots?

Answer 1

This is an extensively studied question and is far from being understood in general. Here are some other conditions beyond the fact that $H_1(M)$ is cyclic.

• the fundamental group should have weight 1, that is it is trivialized by adding a single relation, since killing the meridian of the knot kills the fundamental group (and hence that of any Dehn surgery)

• If $H_1(M)=\mathbb{Z}$, then $M$ must be irreducible by Property R. Other obstructions are known.

• If $M$ is reducible, then the number of connected summands is at most 3. The cabling conjecture would imply that the manifold must be a connected sum of a surgery on a knot and a lens space and in principle would reduce to the prime case.

• If $M$ is a $\mathbb{Z}$-homology 3-sphere, then it must have a non-cyclic $SU(2)$ representation. Examples of rational homology spheres with weight 1 fundamental group and no non-cyclic $SU(2)$ representation are known and which are not dehn surgery on a knot.

• If $M$ is a rational homology sphere then it must bound a definite 4-manifold. Examples of rational homology 3-spheres which bound no definite 4-manifold are known.

• If $M$ has finite fundamental group (so is spherical), then it is known precisely which such manifolds are obtained by Dehn surgery on a knot.

Aside from these general considerations, there are families of manifolds that are known not to be surgery on a knot. There are new examples using Furuta’s $\frac{10}{8}$ theorem. There are also announced examples of arbitrarily large surgery number.

Answer 2

About your third question:

Are there any partial results for branched double covers of knots?

The Montesinos trick tells you that if $K$ has unknotting number 1, then its double cover is $\pm\frac{\det K}2$-surgery along a knot $J$. So, for homology spheres, if $\det K = 1$, then $\Sigma_2(K)$ is a $\pm\frac12$-surgery along some knot $J$.