##
**The lens space realization problem.**
*(English)*
Zbl 1276.57009

A knot \(K\) in \(S^3\) is a lens space knot if it admits a lens space surgery. There is a question asked by Moser: What are all the ways to construct lens spaces \(L(p, q)\) by Dehn surgeries along knots in \(S^3\)? The answer to this question remains open.

One of the best results was due to Berge, who enumerated many lens space knots, which are doubly primitive knots. The doubly primitive knots that Berge enumerated are called Berge knots. Berge further conjectured that his construction yields all the knots that allow integer lens space surgeries. On the other hand, Culler-Gordon-Luecke-Shalen proved that if a lens space knot is not a torus knot, then the surgery coefficient is an integer. So, to answer Moser’s question we just need to prove Berge’s conjecture affirmatively.

There are three questions that are closely related to this conjecture.

Q1: How to realize the lens spaces that are obtained from integer surgery along a knot in \(S^3\)?

Q2: Do the Berge knots include all the doubly primitive knots?

Q3: Can we bound the knot genus \(g(K)\) from above tightly in terms of the surgery slope \(p\)?

The following results in the paper under review answer these three questions.

{Theorem 1.2.} Suppose that negative integer surgery along a knot \(K\subset L(p,q)\) produces \(S^3\). Then \(K\) lies in the same homology class as a Berge knot \(B \subset L(p, q)\).

{Theorem 1.3.} Suppose that \(K \subset S^3\), \(p\) is a positive integer, and the surgered manifold by \(p\)-surgery along \(K\), \(K_p\), is a lens space. Then there exists a Berge knot \(B\subset S^3\) such that \(B_p \cong K_p\) and \(\widehat{HFK}(B)\cong\widehat{HFK}(K)\). Furthermore, every doubly primitive knot in \(S^3\) is a Berge knot.

{Theorem 1.4.} Suppose that \(K \subset S^3\), \(p\) is a positive integer, and \(K_p\) is a lens space. Then \(2g(K)-1\leq p-2 \sqrt{(4p+1)/5}\), unless \(K\) is the right-hand trefoil and \(p = 5\). Moreover, this bound is attained by the type VIII Berge knots specified by the pairs \((p, k) = (5n^2 +5n+1, 5n^2-1)\).

One of the best results was due to Berge, who enumerated many lens space knots, which are doubly primitive knots. The doubly primitive knots that Berge enumerated are called Berge knots. Berge further conjectured that his construction yields all the knots that allow integer lens space surgeries. On the other hand, Culler-Gordon-Luecke-Shalen proved that if a lens space knot is not a torus knot, then the surgery coefficient is an integer. So, to answer Moser’s question we just need to prove Berge’s conjecture affirmatively.

There are three questions that are closely related to this conjecture.

Q1: How to realize the lens spaces that are obtained from integer surgery along a knot in \(S^3\)?

Q2: Do the Berge knots include all the doubly primitive knots?

Q3: Can we bound the knot genus \(g(K)\) from above tightly in terms of the surgery slope \(p\)?

The following results in the paper under review answer these three questions.

{Theorem 1.2.} Suppose that negative integer surgery along a knot \(K\subset L(p,q)\) produces \(S^3\). Then \(K\) lies in the same homology class as a Berge knot \(B \subset L(p, q)\).

{Theorem 1.3.} Suppose that \(K \subset S^3\), \(p\) is a positive integer, and the surgered manifold by \(p\)-surgery along \(K\), \(K_p\), is a lens space. Then there exists a Berge knot \(B\subset S^3\) such that \(B_p \cong K_p\) and \(\widehat{HFK}(B)\cong\widehat{HFK}(K)\). Furthermore, every doubly primitive knot in \(S^3\) is a Berge knot.

{Theorem 1.4.} Suppose that \(K \subset S^3\), \(p\) is a positive integer, and \(K_p\) is a lens space. Then \(2g(K)-1\leq p-2 \sqrt{(4p+1)/5}\), unless \(K\) is the right-hand trefoil and \(p = 5\). Moreover, this bound is attained by the type VIII Berge knots specified by the pairs \((p, k) = (5n^2 +5n+1, 5n^2-1)\).

Reviewer: Yu Zhang (Harbin)

### MSC:

57M25 | Knots and links in the \(3\)-sphere (MSC2010) |

57N10 | Topology of general \(3\)-manifolds (MSC2010) |

### Software:

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\textit{J. E. Greene}, Ann. Math. (2) 177, No. 2, 449--511 (2013; Zbl 1276.57009)

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