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This questions is actually exercise 10D4 from Rolfsen's Knots and Links. In example 8D7 Rolfsen computes a presentation matrix for $\Sigma _n$ the n-fold cyclic cover of $S^3$ branched over the trefoil. It is then clear that the $6n\pm 1$th cyclic covers are homology spheres. I am trying to distinguish these homology spheres, and then show that even their homotopy types differ.

Using standard surgery maneuvers, I have shown that all of these homology spheres result from a $-1/m$ surgery along the trefoil, where $0 \not= m\in \mathbb{Z}$. Call the resulting manifold of such a surgery $X_m$. Following the methods of the section, I have computed that $\pi _1 (X_m) = \langle x,z\ |\ x^{3m+1} = (z^3x^{-3})^m,\ (zx)^2 = z^3 \rangle$. I am fairly confident in this presentation, as setting $m=-1$ results in the binary icosahedral group, the fundamental group of the Poincare homology sphere, as it should. The last step would be to show that these groups are actually different for different values of $m$, I don't know what to do.. Any ideas?

Also, are there any theorems that state anything about different surgeries on a nontrivial knot giving different manifolds?

lev
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1 Answers1

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All of these branched covers have distinct fundamental groups. This is proved in Theorem 9.7 of Cannon, Floyd, Lambert, Parry, and Purcell, "Bitwist manifolds and two-bridge knots."

http://arxiv.org/pdf/1306.4564v1.pdf

As you can see there, the proof is self-contained and fairly straightforward. The trick is to look at the group modulo its center. The quotient is a certain triangle group $\Delta(2,3,k)$, where $k = 6n \pm 1$ in your notation. The largest order of a torsion element in $\Delta(2,3,k)$ is $k$, which shows the groups are all distinct.

As for your question about different surgeries on a knot: this is a hard question in general. There are tools coming from hyperbolic geometry and Heegaard Floer theory.

Joe
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