Celebratio Mathematica — Gordon — Working with Cameron

Working with Cameron: from exceptional surgeries to the L-space conjecture

Works connected to John Edwin Luecke

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M. Culler, C. M. Gordon, J. Luecke, and P. B. Shalen: “Dehn surgery on knots,” Ann. of Math. (2) 125 : 2 (1987), pp. 237–300. MR 881270 Zbl 0633.57006 article

John Edwin Luecke	Related
Marc Edward Culler	Related

C. M. Gordon and J. Luecke: “Only integral Dehn surgeries can yield reducible manifolds,” Math. Proc. Cambridge Philos. Soc. 102 : 1 (July 1987), pp. 97–101. MR 886439 Zbl 0655.57500 article

Let \( K \) be a knot in \( S^3 \), and let \( K(a/b) \) denote the closed oriented 3-manifold obtained by \( a/b \)-Dehn surgery on \( K \). (We parametrize the Dehn surgeries on \( K \) by \( \mathbb{Q}\cap \{1/0\} \) as in [Rolfsen 1976]; in particular, \( K(1/0) = S^3 \).) If \( K \) is a \( (p,q) \)-cable knot, then \( K(pq) \) is always reducible (see Section 3), and it is conjectured by González-Acuña and Short in [1986] that these are the only examples where Dehn surgery on a knot in \( S^3 \) yields a reducible manifold. One of the results in this direction proved in [Rolfsen 1976] is that if \( \pi_1(K(a/b)) \) is a non-trivial free product then \( |b|\leq 5 \). We show that this may be sharpened to the assertion in the title.

C. M. Gordon and J. Luecke: “Reducible manifolds and Dehn surgery,” Topology 35 : 2 (April 1996), pp. 385–409. MR 1380506 Zbl 0859.57016 article