C. M. Gordon and J. Luecke, Knots are determined by their complements, J. Amer. Math. Soc. 2 (1989) 371---415; Bull. Amer. Math. Soc. 20 (1989) 83---87; MR 90a:57006a---b.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Since the volumes of the complements of these knots Kn , for high values of n, are less than, but converge to, the volume of the complement of K p;q (whose volumes, in turn, converge to that of the three component link K [ in nitely many of the knot complements are distinct. Therefore [GL] in nitely many of the knots are distinct. Hyperbolicity can be veri ed by showing that the topological hypotheses of Thurston s Geometrization Theorem [Th3] hold, that is, that L is not a split, satellite, or torus link. The basic outline of the proof of this follows section 3 of [Br1] We let ....

C. Gordon and J. Luecke, Knots are determined by their complements, J. Amer. Math. Soc. 2 (1989), 371-415.

....absolute grading on the homology theories, and also which hold for other surgery coecients, see [26] Corollary 1. 9 should be compared with the result of Gordon and Luecke which states that no non trivial surgery on a non trivial knot in the three sphere can give back the three sphere, see [12] [13], see also [5] 1.2. Second application: bounding the number of gradient trajectories. We give another application, to Morse theory over homology three spheres. Consider the following question. Let Y be a integral homology three sphere. Equip Y with a self indexing Morse function f : Y R ....

C. McA. Gordon and J. Luecke. Knots are determined by their complements. Bull. Amer. Math. Soc. (N.S.), 20(1):83-87, 1989.

....an absolute grading on the homology theories, and also which hold for other surgery coecients, see [26] Corollary 1. 9 should be compared with the result of Gordon and Luecke which states that no non trivial surgery on a non trivial knot in the three sphere can give back the three sphere, see [12], 13] see also [5] 1.2. Second application: bounding the number of gradient trajectories. We give another application, to Morse theory over homology three spheres. Consider the following question. Let Y be a integral homology three sphere. Equip Y with a self indexing Morse function f : Y ....

C. McA. Gordon and J. Luecke. Knots are determined by their complements. J. Amer. Math. Soc., 2(2):371-415, 1989.

....is asserted by Thurston in [85] without proof. Although the statement is purely topological we present here a proof which uses hyperbolic geometry in what seems like an essential way. We have considered applying the boundary slope techniques of GordonLitherland and Gordon Luecke et al. 35] [36], 28] but without success. In fact, consider the following more general situation. Let M be the interior of a compact 3 manifold M , such that M admits a complete hyperbolic metric #. Suppose that F is a collection of torus components of #M , and let M i be a sequence of Dehn fillings of M , ....

C. McA. Gordon and J. Luecke, Knots are determined by their complements, Bull. Amer. Math. Soc. 20 (1989), 83--87.

.... to solve tangle equations of the form N(U P ) K 1 and N(U R) K 2 [2,4,9,16] In many biological applications the knots links involved are 4 plats, and in many of these cases, it is possible to prove that U is ambient isotopic to a sum of rational tangles, and P and R are rational tangles [11,7 9,4]. In this situation it is possible to list all solutions for U and R as a function of P . When U , P , and R are known, orientation can also determined. In section 2 a brief introduction to tangles is given. The system of unoriented tangle equations N(U P ) K 1 and N(U R) K 2 is solved in ....

C. Gordon and J. Luecke. Knots are determined by their complements. J. Amer. Math. Soc., 2:371-415, 1989.

....the Cabling Conjecture are true for all arborescent knots. These conjectures are still among the most interesting problems in Dehn surgery theory. The property P conjecture asserts that surgery on a nontrivial knot will not produce a homotopy sphere. Gordon and Luecke s Knot Complement Theorem [GL] implies that this would follow from the Poincare conjecture. The conjecture has also been proved for satellite knots [Ga1] and symmetric knots [CGLS] Recently Delman and Roberts [DR] proved it for alternating knots. The Cabling Conjecture asserts that all surgeries on hyperbolic knots are ....

....toroidal. Problem 4.3. Determine if all the surgeries in the above theorems are atoroidal. The Strong Property P Conjecture of Gabai [Ga3] asserts that all knots in S 3 have Strong Property P, i.e, K(#) does not contain fake 3 ball for all K and #. By Gordon and Luecke s Knot Complement Theorem [GL], this implies the Property P Conjecture. Since all closed 3 manifolds are obtained by surgery on links, the Poincare Conjecture is equivalent to the following link version of Strong Property P Conjecture. Conjecture 4.4. All links L in S 3 have strong property P, that is, surgeries on L ....

C. Gordon and J. Luecke, Knots are determined by their complements, Jour. Amer. Math. Soc. 2 (1989), 371--415.

....invariants and skein theories fall into this category. The second approach could be called three manifold knot theory. Here the emphasis is on the complement of the knot, which is a three manifold with torus boundary. Nothing is lost in this approach since, by the result of Gordon and Luecke [GL89] knots are determined by their complements. By focusing on the knot complements, the geometry and topology of three manifolds come into play. Techniques involving Dehn surgery, surfaces, and hyperbolic geometry are often used. Until the developments of Thurston, there had been few useful ....

C. McA. Gordon and J. Luecke, Knots are determined by their complements, J. Amer. Math. Soc. 2 (1989), 371-415.

.... generally, the study of the fundamental group of a knot complement and the knot complement alone was the main topic of research in knot theory for the next fifty years, culminating in 1988 in the proof of Tietze [120] conjecture (that a knot is determined by its complement) by Gordon and Luecke [46]. We can refer to the survey articles by Ralph Hartzler Fox (1913 1973) 42] and Gordon [45] or books [12, 20, 64, 98, 100] in this respect. However Alexander observed also that if three oriented links, L ; L Gamma and L 0 ; have diagrams which are identical except near one crossing where they ....

C.McA. Gordon, J. Luecke, Knots are Determined by their Complements, Bulletin of the A.M.S, 20 (1989), 83-87.

....Theorem 3. In section 1.5 we state the philosophical idea that underlines the present paper. 1.5. A philosophical comment. To justify the title exceptional in the title of the next section, we mention that for generic framings, Dehn surgery on a knot gives rise to irreducible 3 manifolds, [GoLu], however, for special knots and special framings, exceptional decompositions occur. It is a general philosophy in arithmetic, in algebraic geometry, in singularity theory, and in many other areas that exceptional phenomena are very important in the study of the generic situations. From this ....

C.Gordon, J.Luecke, Knots are determined by their complement, Journal A.M.S. 22 (1989) 371-415.

....the Cabling Conjecture are true for all arborescent knots. These conjectures are still among the most interesting problems in Dehn surgery theory. The property P conjecture asserts that surgery on a nontrivial knot will not produce a homotopy sphere. Gordon and Luecke s Knot Complement Theorem [GL] implies that this would follow from the Poincare conjecture. The conjecture has also been proved for satellite knots [Ga1] and symmetric knots [CGLS] Recently Delman and Roberts [DR] proved it for alternating knots. The Cabling Conjecture asserts that all surgeries on hyperbolic knots are ....

....Problem 4.3. Determine if all the surgeries in the above theorems are atoroidal. The Strong Property P Conjecture of Gabai [Ga3] asserts that all knots in S 3 have Strong Property P, i.e, K(fl) does not contain fake 3 ball for all K and fl. By Gordon and Luecke s Knot Complement Theorem [GL], this implies the Property P Conjecture. Since all closed 3 manifolds are obtained by surgery on links, the Poincare Conjecture is equivalent to the following link version of Strong Property P Conjecture. Conjecture 4.4. All links L in S 3 have strong property P, that is, surgeries on L ....

C. Gordon and J. Luecke, Knots are determined by their complements, Jour. Amer. Math. Soc. 2 (1989), 371--415.

....can be assumed that no circle component of S # T bounds a disc in T . But it does not hold for S in general. We further assume that the number of loop components of S # T is minimal up to an isotopy of S. 4 The arc components of S #T define graphs G S in # S and G T in # T as follows [4, 16], where # S is the closed surface obtained by capping #S o# by a disc. Let G S be the graph in # S obtained by taking as the (fat) vertex the disc # S IntS and as edges the arc components of S # T in # S. Similarly, G T is the graph in # T whose vertices are the discs # T ....

C. McA. Gordon and J. Luecke. Knots are determined by their complements. J. Amer. Math. Soc. 2 (1989), 371--415.

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C. M. Gordon and J. Luecke, Knots are determined by their complements, J. Amer. Math. Soc. 2 (1989) 371---415; Bull. Amer. Math. Soc. 20 (1989) 83---87; MR 90a:57006a---b.

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C. McA. Gordon and J. Luecke, Knots are determined by their complements, J. Amer. Math. Soc. 2 (1989), 371-415.

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C. McA. Gordon and J. Luecke. Knots are determined by their complements. Bull. Amer. Math. Soc. (N.S.), 20(1):83-87, 1989.

No context found.

C. McA. Gordon and J. Luecke. Knots are determined by their complements. J. Amer. Math. Soc., 2(2):371-415, 1989.

No context found.

C. Gordon and J. Luecke, Knots are determined by their complements, J. Amer. Math. Soc. 2 (1989), 371-415.

No context found.

C. Gordon and J. Luecke, Knots are determined by their complements, Jour. Amer. Math. Soc. 2 (1989), 371--415.

No context found.

C. Gordon and J. Luecke, Knots are determined by their complements, Jour. Amer. Math. Soc. 2 (1989), 371--415.

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