Parameterizations of 1-Bridge Torus Knots

Choi, Doo Ho; Ko, Ki Hyoung

doi:10.1142/s0218216503002445

Cited by 21 publications

(20 citation statements)

References 18 publications

(32 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A representation of this type will be called "standard". Note that a similar result, using a rank three free subgroup of MCG 2 (T ), has been obtained in [6,Theorem 3].…”

Section: Standard Decompositionsupporting

confidence: 63%

(1, 1)-knots via the mapping class group of the twice punctured torus

2004

Advg

View full text Add to dashboard Cite

We develop an algebraic representation for (1, 1)-knots using the mapping class group of the twice punctured torus M CG 2 (T ). We prove that every (1, 1)-knot in a lens space L(p, q) can be represented by the composition of an element of a certain rank two free subgroup of M CG 2 (T ) with a standard element only depending on the ambient space. As notable examples, we obtain a representation of this type for all torus knots and for all two-bridge knots. Moreover, we give explicit cyclic presentations for the fundamental groups of the cyclic branched coverings of torus knots of type (k, ck + 2).

show abstract

“…A representation of this type will be called "standard". Note that a similar result, using a rank three free subgroup of MCG 2 (T ), has been obtained in [6,Theorem 3].…”

Section: Standard Decompositionsupporting

confidence: 63%

(1, 1)-knots via the mapping class group of the twice punctured torus

2004

Advg

View full text Add to dashboard Cite

show abstract

“…A different parametrization of (1, 1)-knots, involving four parameters for the knot and two additional parameters for the ambient space, can be found in [8].…”

Section: Proposition 3 ([6]) the Two-bridge Knot Having Conway Parammentioning

confidence: 99%

Representations of (1,1)-knots

Cattabriga

Mulazzani

2005

Fund. Math.

View full text Add to dashboard Cite

Abstract. We present two different representations of (1, 1)-knots and study some connections between them. The first representation is algebraic: every (1, 1)-knot is represented by an element of the pure mapping class group of the twice punctured torus PMCG 2 (T ). Moreover, there is a surjective map from the kernel of the natural homomorphism Ω : PMCG 2 (T ) → MCG(T ) ∼ = SL(2, Z), which is a free group of rank two, to the class of all (1, 1)-knots in a fixed lens space. The second representation is parametric: every (1, 1)-knot can be represented by a 4-tuple (a, b, c, r) of integer parameters such that a, b, c ≥ 0 and r ∈ Z 2a+b+c . The strict connection of this representation with the class of Dunwoody manifolds is illustrated. The above representations are explicitly obtained in some interesting cases, including two-bridge knots and torus knots.

show abstract

“…A priori working with the class of one-bridge knots does not simplify matters much. Indeed, there are clearly infinitely many one-bridge knots in L(p, q); in particular, it contains torus knots -those knots which can be isotoped to lie in the Heegaard torus -as a proper subset (see [6] for a classification scheme). However, amongst the one-bridge knots in L(p, q) is a particularly simple finite subfamily, which we call simple (or grid-number one) knots.…”

Section: Definition 11 (One-bridge) a Knot (L(p Q) K ) Is Called mentioning

confidence: 99%

On Floer homology and the Berge conjecture on knots admitting lens space surgeries

Hedden

2011

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. We complete the first step in a two-part program proposed by Baker, Grigsby, and the author to prove that Berge's construction of knots in the three-sphere which admit lens space surgeries is complete. The first step, which we prove here, is to show that a knot in a lens space with a threesphere surgery has simple (in the sense of rank) knot Floer homology. The second (conjectured) step involves showing that, for a fixed lens space, the only knots with simple Floer homology belong to a simple finite family. Using results of Baker, we provide evidence for the conjectural part of the program by showing that it holds for a certain family of knots. Coupled with work of Ni, these knots provide the first infinite family of non-trivial knots which are characterized by their knot Floer homology. As another application, we provide a Floer homology proof of a theorem of Berge.

show abstract

Parameterizations of 1-Bridge Torus Knots

Cited by 21 publications

References 18 publications

(1, 1)-knots via the mapping class group of the twice punctured torus

(1, 1)-knots via the mapping class group of the twice punctured torus

Representations of (1,1)-knots

On Floer homology and the Berge conjecture on knots admitting lens space surgeries

Contact Info

Product

Resources

About