Celebratio Mathematica

M. Culler, C. M. Gordon, J. Luecke, and P. B. Shalen: “Dehn surgery on knots,” Bull. Am. Math. Soc. (N.S.) 13 : 1 (July 1985), pp. 43–45. MR 788388 Zbl 0571.57008 article

John Edwin Luecke	Related
Peter B. Shalen	Related
Marc Edward Culler	Related

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.

M. Culler, C. M. Gordon, J. Luecke, and P. B. Shalen: “Correction: ‘Dehn surgery on knots’,” Ann. Math. (2) 127 : 3 (1988), pp. 663. Correction to an article published in Ann. Math. 125:2 (1987). MR 942524 Zbl 0645.57006 article

John Edwin Luecke	Related
Peter B. Shalen	Related
Marc Edward Culler	Related

C. M. Gordon and J. Luecke: “Knots are determined by their complements,” J. Am. Math. Soc. 2 : 2 (1989), pp. 371–415. A much shorter version of this article was published in Bull. Am. Math. Soc. 20:1 (1989). MR 965210 Zbl 0678.57005 article

C. M. Gordon and J. Luecke: “Knots are determined by their complements,” Bull. Am. Math. Soc. (N.S.) 20 : 1 (January 1989), pp. 83–87. A much longer version of this article was published in J. Am. Math. Soc. 2:2 (1989). MR 972070 Zbl 0672.57009 article

C. M. Gordon and J. Luecke: “Links with unlinking number one are prime,” Proc. Am. Math. Soc. 120 : 4 (1994), pp. 1271–1274. MR 1195721 Zbl 0818.57008 article

C. M. Gordon and J. Luecke: “Dehn surgeries on knots creating essential tori, I,” Comm. Anal. Geom. 3 : 3–4 (1995), pp. 597–644. MR 1371211 Zbl 0865.57015 article

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

C. M. Gordon and J. Luecke: “Toroidal and boundary-reducing Dehn fillings,” Topology Appl. 93 : 1 (April 1999), pp. 77–90. MR 1684214 Zbl 0926.57019 article

C. M. Gordon and J. Luecke: “Dehn surgeries on knots creating essential tori, II,” Comm. Anal. Geom. 8 : 4 (2000), pp. 671–725. MR 1792371 Zbl 0970.57010 article

C. M. Gordon and J. Luecke: “Non-integral toroidal Dehn surgeries,” Comm. Anal. Geom. 12 : 1–2 (2004), pp. 417–485. MR 2074884 Zbl 1062.57006 article

If we perform a non-trivial Dehn surgery on a hyperbolic knot in the 3- sphere, the result is usually a hyperbolic 3-manifold. However, there are exceptions: there are hyperbolic knots with surgeries that give lens spaces [Berge], small Seifert fiber spaces [Bleiler and Hodgson 1992; Dean 2003; Eudave-Muñoz 2002; Mattman et al. 2006], and toroidal manifolds, that is, manifolds containing (embedded) incompressible tori [Eudave-Muñoz 1997, 2002]. In particular, Eudave-Muñoz [1997] has explicitly described an infinite family of hyperbolic knots \( k(l,m,n,p) \), each of which has a specific half-integral toroidal surgery. (These are the only known examples of non-trivial, non-integral, non-hyperbolic surgeries on hyperbolic knots.) Here we show that these knots are the only hyperbolic knots with non-integral toroidal surgeries.

C. M. Gordon and J. Luecke: “Knots with unknotting number 1 and essential Conway spheres,” Algebr. Geom. Topol. 6 (2006), pp. 2051–2116. MR 2263059 Zbl 1129.57009 ArXiv math.GT/0601265 article

For a knot \( K \) in \( S^3 \), let \( \mathbf{T}(K) \) be the characteristic toric sub-orbifold of the orbifold \( (S^3,K) \) as defined by Bonahon–Siebenmann. If \( K \) has unknotting number one, we show that an unknotting arc for \( K \) can always be found which is disjoint from \( \mathbf{T}(K) \), unless either \( K \) is an EM-knot (of Eudave-Muñoz) or \( (S^3,K) \) contains an EM-tangle after cutting along \( \mathbf{T}(K) \). As a consequence, we describe exactly which large algebraic knots (ie, algebraic in the sense of Conway and containing an essential Conway sphere) have unknotting number one and give a practical procedure for deciding this (as well as determining an unknotting crossing). Among the knots up to 11 crossings in Conway’s table which are obviously large algebraic by virtue of their description in the Conway notation, we determine which have unknotting number one. Combined with the work of Ozsváth–Szabó, this determines the knots with 10 or fewer crossings that have unknotting number one. We show that an alternating, large algebraic knot with unknotting number one can always be unknotted in an alternating diagram.

As part of the above work, we determine the hyperbolic knots in a solid torus which admit a non-integral, toroidal Dehn surgery. Finally, we show that having unknotting number one is invariant under mutation.

K. L. Baker, C. Gordon, and J. Luecke: “Obtaining genus 2 Heegaard splittings from Dehn surgery,” Algebr. Geom. Topol. 13 : 5 (2013), pp. 2471–2634. MR 3116298 Zbl 1294.57013 article

John Edwin Luecke	Related
Kenneth Lee Baker	Related

K. L. Baker, C. Gordon, and J. Luecke: “Bridge number, Heegaard genus and non-integral Dehn surgery,” Trans. Am. Math. Soc. 367 : 8 (2015), pp. 5753–5830. MR 3347189 Zbl 1329.57017 ArXiv 1202.0263 article

John Edwin Luecke	Related
Kenneth Lee Baker	Related

K. L. Baker, C. Gordon, and J. Luecke: “Bridge number and integral Dehn surgery,” Algebr. Geom. Topol. 16 : 1 (2016), pp. 1–40. MR 3470696 Zbl 1339.57005 ArXiv 1303.7018 article

In a 3-manifold \( M \), let \( K \) be a knot and \( \hat{R} \) be an annulus which meets \( K \) transversely. We define the notion of the pair \( (R,\hat{K}) \) being caught by a surface \( Q \) in the exterior of the link \( K\cup\partial\hat{R} \). For a caught pair \( (\hat{R},K) \), we consider the knot \( K^n \) gotten by twisting \( K \) \( n \) times along \( \hat{R} \) and give a lower bound on the bridge number of \( K^n \) with respect to Heegaard splittings of \( M \)–as a function of \( n \), the genus of the splitting, and the catching surface \( Q \). As a result, the bridge number of \( K^n \) tends to infinity with \( n \). In application, we look at a family of knots \( \{K^n\} \) found by Teragaito that live in a small Seifert fiber space \( M \) and where each \( K^n \) admits a Dehn surgery giving \( S^3 \). We show that the bridge number of \( K^n \) with respect to any genus 2 Heegaard splitting of \( M \) tends to infinity with \( n \). This contrasts with other work of the authors as well as with the conjectured picture for knots in lens spaces that admit Dehn surgeries giving \( S^3 \).

John Edwin Luecke	Related
Kenneth Lee Baker	Related

year	title	people

			clear

Cameron McAllan Gordon

Complete Bibliography

Works connected to John Edwin Luecke

Filter the Bibliography List