Knots are determined by their complements

Gordon, C. McA.; Luecke, John

doi:10.1090/s0273-0979-1989-15706-6

Cited by 85 publications

(94 citation statements)

References 4 publications

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“…But if O is the complement of a non-trivial knot, then there is a unique way to Dehn fill (S 3 \ O ) in order to retrieve S 3 . In other words, if S 3 is obtained by Dehn surgery in the complement of a knot, and the knot is non-trivial, then this Dehn filling is unique up to ambient isotopy [15]. From Claim 4·1 we know that R ≈ (1) and P ≈ (0), therefore the Dehn fillings determined by P and R are not equivalent (the gluing diffeomorphisms are not ambient isotopic).…”

Section: Theorem 4·2 (Inversely Repeated Sites) Suppose That the Tanmentioning

confidence: 99%

Tangle analysis of Gin site-specific recombination

Vázquez¹,

Sumners²

2004

Math. Proc. Camb. Phil. Soc.

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We use the tangle model to study the action of the site-specific recombinase Gin, an enzyme that can introduce topological changes on circular DNA molecules. Gin and its bound DNA are modelled as a 2-string tangle which undergoes changes during recombination, thereby changing the topology of the DNA substrate. We show that the tangles involved in the analysis are all rational tangles. This technique allows us to prove that, under the model's assumptions, there is a unique topological description of the enzymatic action. The Gin system is one of the few to date where tangle analysis can be carried out systematically and rigorously, yielding a single, biologically reasonable solution. Preview and motivationSite-specific recombination alters the genome of an organism by moving, inserting or inverting DNA segments. When acting on circular DNA molecules, site-specific recombinases can change the topology and geometry of the molecules [25,31]. The tangle model provides mathematical tools to analyze such changes [26,27,28]. The detailed tangle formalism and an application of the model to the Tn3 resolvase system were first presented in [10].Here we analyze the enzymatic action of Gin, a site-specific recombinase encoded by bacteriophage Mu (a virus that infects bacteria). Gin inverts a DNA segment within the phage genome to extend the phage's range of infection. The phage genome contains two recombination sites that Gin recognizes, cleaves and rearranges to complete one round of recombination (Figure 1). Gin recombination is processive, i.e. it can carry out more than one recombination round while binding only once to its substrate. We study the case where, prior to recombination, the phage genome is a single circular DNA molecule, effectively the unknot in S 3 . Recombination can then be analyzed by using tangles to model an enzyme such as Gin together with the DNA bound to the enzyme (see: [29]). Tangles, defined formally below, are

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Section: Theorem 4·2 (Inversely Repeated Sites) Suppose That the Tanmentioning

confidence: 99%

Tangle analysis of Gin site-specific recombination

Vázquez¹,

Sumners²

2004

Math. Proc. Camb. Phil. Soc.

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“…If one cuts M 2 (K) along the core torus T one recovers the knot complement, from which the knot itself can be recovered by Gordon and Luecke [2]. Thus the theorem follows from the following lemma.…”

Section: The 2-generalized Knot Groupmentioning

confidence: 93%

“…Also, the peripheral subgroup of the knot group is recovered as the edge group for this edge. Finally, the knot is determined by knot group plus peripheral subgroup by Gordon and Luecke [2]. For n = 3 (and 2) one can use essentially the same argument, but there is an extra V 0 -vertex corresponding to the peripheral Z × Z of the knot group, and the vertex for π 1 (T ) is a V 1 -vertex.…”

Section: The N-generalized Knot Groupmentioning

confidence: 99%

The 2-Generalized Knot Group Determines the Knot

Nelson

Neumann

2008

Commun. Contemp. Math.

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“…We use the standard techniques of intersection graphs developed by Scharlemann [16] and by Gordon and Luecke [1], [6], [7]. In §2 below, we recall the construction of the intersection graphs in the particular context of this problem.…”

Section: Corollary 2 Under the Hypotheses Of Theorem 1 We Havementioning

confidence: 99%

“…Since x; y are consecutive edges of G i at each vertex of C , the edges of G Q \ between x and y at v belong to G 3 i and correspond to intersection points of k 0 with P 3 i . Since P 3 i is separating, it follows that x y is odd, and hence from the parity rule (see for example [6], p. 386) that all vertices of C have the same orientation.…”

mentioning

confidence: 99%

Can Dehn surgery yield three connected summands?

Howie¹

2010

Groups Geom. Dyn.

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Abstract.A consequence of the Cabling Conjecture of Gonzalez-Acuña and Short is that Dehn surgery on a knot in S 3 cannot produce a manifold with more than two connected summands. In the event that some Dehn surgery produces a manifold with three or more connected summands, then the surgery parameter is bounded in terms of the bridge number by a result of Sayari. Here this bound is sharpened, providing further evidence in favour of the Cabling Conjecture.Mathematics Subject Classification (2010). 57M25.

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Knots are determined by their complements

Abstract: Two (smooth or PL) knots K, K' in S 3 are equivalent if there exists a homeomorphism h: S 3 -• S 3 such that h(K) = K'. This implies that their complements S 3 -K and S 3 -K' are homeomorphic. Here we announce the converse implication.

Cited by 85 publications

References 4 publications

Tangle analysis of Gin site-specific recombination

Tangle analysis of Gin site-specific recombination

The 2-Generalized Knot Group Determines the Knot

Can Dehn surgery yield three connected summands?

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