KNOTS ARE DETERMINED BY THEIR COMPLEMENTS

The Annals of Mathematics, 1987

Let M be a compact, connected, orientable, irreducible 3-manifold such that dM is a torus. An isotopy class c of unoriented simple closed curves in dM will be called a slope. A closed 3-manifold M(c) may be constructed by attaching a solid torus / to M so that c bounds a disk in J. If c and d are two slopes, we denote their (minimal) geometric intersection number by A(c, d). ) < 1. In particular, there are at most three slopes c such that TT 1 ( Af(c)) is cyclic. This result is sharp; Fintushel-Stern and Berge have given examples of hyperbolic knots in S 3 for which two Dehn surgeries give lens spaces. In the statements of the following corollaries we use rational numbers as in [R] to parametrize the nontrivial Dehn surgeries on a knot K in S 3 . We will denote by K(r) the result of r-surgery on K. COROLLARY 2. If K is a nontrivial knot and r e Q is not equal to 1 or -1 then K(r) is not simply-connected. Moreover, K(l) and K(-l) cannot both be simply-connected. COROLLARY 3. Up to unoriented equivalence, there are at most two knots whose complements are of a given topological type. COROLLARY 4. If K is a nontrivial amphicheiral knot and r e Q -{0}, then Tr^Kir)) is not cyclic. In particular, K has Property P. COROLLARY 5. Knots of Arf invariant 1 are determined up to unoriented equivalence by their complements. Whitten [W], using work of Johannson [Jol], shows that Corollary 1 implies the following result. COROLLARY 6. Prime knots with isomorphic groups have homeomorphic complements.

Proceedings of the American Mathematical Society, 1976

Modulo the conjecture that the complement of a prime knot in the 3-sphere is determined by its fundamental group, we show that at most finitely many mutually inequivalent knots can have homeomorphic complements.

2020

We show that the only way of changing the framing of a knot or a link by ambient isotopy in an oriented $3$-manifold is when the manifold has a properly embedded non-separating $S^2$. This change of framing is given by the Dirac trick, also known as the light bulb trick. The main tool we use is based on McCullough&#39;s work on the mapping class groups of $3$-manifolds. We also relate our results to the theory of skein modules.

Topology, 1985

Mathematical Proceedings of the Cambridge Philosophical Society, 1998

The cabling conjecture states that a non-trivial knot K in the 3-sphere is a cable knot or a torus knot if some Dehn surgery on K yields a reducible 3-manifold. We prove that symmetric knots satisfy this conjecture. (Gordon and Luecke also prove this independently ([GLu3]), by a method different from ours.)

Analele Universitatii "Ovidius" Constanta - Seria Matematica, 2012

We introduce the concept of regular diagrams for knots and links in lens spaces, proving that two diagrams represent equivalent links if and only if they are related by a finite sequence of seven Reidemester type moves. As a particular case, we obtain diagrams and moves for links in RP 3 previously introduced by Y.V. Drobothukina.

Journal of Differential Geometry

This paper studies knots that are transversal to the standard contact structure in R 3 , bringing techniques from topological knot theory to bear on their transversal classification. We say that a transversal knot type T K is transversally simple if it is determined by its topological knot type K and its Bennequin number. The main theorem asserts that any T K whose associated K satisfies a condition that we call exchange reducibility is transversally simple. As a first application, we prove that the unlink is transversally simple, extending the main theorem in [10]. As a second application we use a new theorem of Menasco [17] to extend a result of Etnyre [11] to prove that all iterated torus knots are transversally simple. We also give a formula for their maximum Bennequin number. We show that the concept of exchange reducibility is the simplest of the constraints that one can place on K in order to prove that any associated T K is transversally simple. We also give examples of pairs of transversal knots that we conjecture are not transversally simple.

2013

Abstract. We classify Dehn surgeries on (p, q, r) pretzel knots that result in a manifold of finite fundamental group. The only hyperbolic pretzel knots that admit non-trivial finite surgeries are (−2, 3, 7) and (−2, 3, 9). Agol and Lackenby’s 6-theorem reduces the argument to knots with small indices p, q, r. We treat these using the Culler-Shalen norm of the SL(2, C)-character variety. In particular, we introduce new techniques for demonstrating that boundary slopes are detected by the character variety. In [19] Mattman showed that if a hyperbolic (p, q, r) pretzel knot K admits a non-trivial finite Dehn surgery of slope s (i.e., a Dehn surgery that results in a manifold of finite fundamental group) then either • K = (−2, 3, 7) and s = 17, 18, or 19, • K = (−2, 3, 9) and s = 22 or 23, or • K = (−2, p, q) where p and q are odd and 5 ≤ p ≤ q. In the current paper we complete the classification by proving Theorem 1. Let K be a (−2, p, q) pretzel knot with p, q odd and 5 ≤ p ≤ q. Then...

Topology, 1963

2006

We give two geometric methods of constructing plane curves giving cable knots of torus knots via A'Campo's divide knot theory, related to both singularity theory and knot theory. We point out a relationship between "area" of the plane curves and the coefficients of finite Dehn surgery, which is Dehn surgery yielding three-dimensional manifolds with finite fundamental group.