Homology spheres and property R
Min Hoon Kim JungHwan Park






School of Mathematics, Korea Institute for Advanced Study, Seoul, Republic of Korea
kminhoon@kias.re.kr

School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia, United States
junghwan.park@math.gatech.edu
 Mathematics Subject Classification
subjclassname@1991=subjclassname@2000=
  
  57M27, 
  
  57M25






 We present infinitely many homology spheres which contain two distinct knots whose 0-surgeries are S^1 × S^2. This resolves a question posed by Kirby and Melvin in 1978.



Homology spheres and property R
    JungHwan Park
    December 30, 2023
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§ INTRODUCTION

A knot K in S^3 is said to satisfy property R, if surgery on K cannot give S^1× S^2. A celebrated result of Gabai <cit.> states that every non-trivial knot in S^3 satisfies property R. Now, we replace S^3 with a homology sphere. A homology sphere M contains a knot whose 0-surgery is S^1× S^2 if and only if M is the boundary of a contractible 4-manifold with a 0, 1, and 2-handle. The natural question for such M, asked by Kirby and Melvin in <cit.> (see also <cit.>), is if there is only one knot in M that can produce S^1× S^2 up to equivalence. In this article, we answer this question in the negative. Recall that two knots in a homology sphere M are equivalent if there is an orientation-preserving homeomorphism of M that takes one knot to the other. 

 There exist infinitely many homology spheres which contain two distinct knots whose 0-surgeries are S^1 × S^2.


The proof of the theorem proceeds in two steps. First, we construct two component links with unknotted components and linking number 1. The homology spheres obtained by performing the 0-surgeries on the links have a property that the 0-surgery of the meridian of either component of the link is S^1 × S^2. The second step is to show that the meridians are inequivalent knots in infinitely many such homology spheres. This is achieved by using Thurston's hyperbolic Dehn surgery theorem <cit.> and the uniqueness of the JSJ decomposition of a 3-manifold  <cit.>.



  §.§.§ Acknowledgments

This project started when the first named author was visiting the Georgia Institute of Technology and he thanks the Georgia Institute of Technology for its generous hospitality and support. We would also like to thank Stefan Friedl, Kouki Sato, and Jennifer Hom for helpful conversations. Kyle Hayden, Thomas E. Mark, and Lisa Piccirillo informed us that they have an independent proof of Theorem <ref> in their paper on exotic Mazur manifolds <cit.> which appeared on the arXiv after the first version of this article.





§ PROOF OF THEOREM <REF>









For each integer n, we consider the homology sphere M_n and the knots K and K' in M_n described in Figure <ref>. We show in Proposition <ref> that the 0-surgeries on M_n along K and K' are S^1 × S^2. We complete the proof by observing that there is a strictly increasing sequence {n_i}_i=1^∞ of integers such that the homology spheres M_n_i are mutually distinct and the knots K and K' are not equivalent.

 Let K and K' be the knots in M_n described in Figure <ref>, then the 0-surgeries on K and K' are S^1 × S^2.

 Consider the surgery diagram of M_n given in Figure <ref>. Each component of the surgery link is unknotted and K and K' are the meridians of the components. Since K is the meridian of the left component, doing 0-surgery on K removes the left component from the diagram. The result is S^1× S^2 as the right component is unknotted. The argument is symmetric for K'.







It remains to find a sequence {n_i}_i=1^∞ so that the knots K and K' are distinct and M_n_i are mutually distinct. For this purpose, we use Thurston's hyperbolic Dehn filling theorem <cit.> which we recall for completeness.

Let M be a cusped hyperbolic 3-manifold with ℓ cusps. Let ∂M = T_1 ∪ T_2 ∪⋯∪ T_ℓ and fix generators μ_i, λ_i of fundamental group π_1(T_i), for each i. Let (s_1, s_2, …, s_ℓ) be a sequence where s_i is either the symbol ∞ or a rational number p_i/q_i such that p_i and q_i are coprime integers. Let M(s_1, s_2, …, s_ℓ) be the 3-manifold obtained from M by filling in T_i with a solid torus using the slope p_i/q_i if s_i=p_i/q_i and by not filling in if s_i=∞. 

 Let M be a cusped hyperbolic 3-manifold with ℓ cusps. Then M(s_1, s_2, …, s_ℓ) is hyperbolic for all but finitely many sequences (s_1,s_2,…,s_ℓ). Moreover, 

    (M(s_1, s_2, …, s_ℓ))↗(M)
 as all p^2_i+q^2_i approach infinity for all nontrivial framings p_i/q_i.



 
To decompose M_n K and M_n K' along an embedded torus, we consider the 3-manifolds N and N' with tori boundary given by the surgery diagrams of Figure <ref>. In Figure <ref>, the surgery links are in S^1× D^2 so that they represent 3-manifolds with tori boundary.






Let Σ⊂ M_n be an embedded torus in Figure <ref>. For each n≥ 1, M_n decomposes into

    M_n=N_n∪_Σ N'

 where N_n is the 3-manifold obtained from the left diagram of Figure <ref> by performing -1/n-surgery on γ. As depicted in Figure <ref>, we remark that the knots K and K' lie in N_n and N', respectively.

For sufficiently large n, the knot exteriors M_n∖ K and M_n∖ K' have JSJ decompositions (N_n∖ K )∪_Σ N' and N_n ∪_Σ (N'∖ K'), respectively. Moreover, each of the JSJ pieces and N∖γ are hyperbolic and (N∖γ)=7.32772⋯ and (N')=3.66386⋯.


 By using Snappy <cit.> within Sage, we have checked that N∖γ, N∖ (γ∪ K), N', and N'∖ K' are hyperbolic and (N∖γ)=7.32772⋯ and (N')=3.66386⋯:
 


By Theorem <ref>, N_n∖ K and N_n are also hyperbolic for sufficiently large n. The proof is complete by noting that gluing two hyperbolic pieces gives a JSJ decomposition.

We are now ready to prove Theorem <ref>. 

 
Let K and K' be the knots in the homology sphere M_n as described above.
By Proposition <ref>, 0-surgeries on K and K' are S^1× S^2. Suppose they are equivalent, then by the uniqueness of JSJ decomposition and Proposition <ref>, N_n is homeomorphic to N', for large enough n. This is not possible since by Proposition <ref> 
    (N_n)↗(N∖γ)=7.32772⋯ and (N')=3.66386⋯.


 
Lastly, we show that there exists a sequence  {n_i}_i=1^∞ where M_n_i and M_n_j are homeomorphic if and only if n_i = n_j. Since N∖γ is hyperbolic by Proposition <ref>, for large enough n, N_n is hyperbolic by Theorem <ref> and M_n has a JSJ decomposition N_n ∪ N'. Moreover, by the uniqueness of JSJ decomposition and by Theorem <ref>, there exists a sequence {n_i}_i=1^∞ where M_n_i and M_n_j are homeomorphic if and only if n_i = n_j. The proof is completed, by choosing the homology spheres to be M_n_i and the knots to be K and K'.


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