Geometric types of twisted knots

Aït-Nouh, Mohamed; Matignon, Daniel; Motegi, Kimihiko

doi:10.5802/ambp.213

Cited by 8 publications

(17 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Further, it follows from Theorem 6.1(1) that it is not fibred for r = 1, but it is fibred for r > 1. These facts are also confirmed by evaluating the Alexander polynomial of B(482, 381), and using Proposition 4.1 (1).…”

Section: Proof By Theorem 23 For Anysupporting

confidence: 55%

“…In the final section, Section 14, we determine the genus one knots in our family of knots K(2α, β|r). In particular, we find satellite knots among them, and hence give a negative answer to the problem posed in [1].…”

Section: Fibredmentioning

confidence: 85%

“…In [10], we settled the problem for all double torus knots of type (1,1). A g1-b1 knot can be presented as a double torus knot of type either (1,2), (2,2) or (2,3). In this paper, we settle the problem for the g1-b1 knots presented as of type (1,2).…”

Section: Fibredmentioning

confidence: 99%

See 2 more Smart Citations

Fibered Torti-Rational Knots

Hirasawa

Murasugi

2010

J. Knot Theory Ramifications

View full text Add to dashboard Cite

A torti-rational knot, denoted by K(2α, β|r), is a knot obtained from the 2-bridge link B(2α, β) by applying Dehn twists an arbitrary number of times, r, along one component of B(2α, β). We determine the genus of K(2α, β|r) and solve a question of when K(2α, β|r) is fibered. In most cases, the Alexander polynomials determine the genus and fiberedness of these knots. We develop both algebraic and geometric techniques to describe the genus and fiberedness by means of continued fraction expansions of β/2α. Then, we explicitly construct minimal genus Seifert surfaces. As an application, we solve the same question for the satellite knots of tunnel number one.

show abstract

Section: Proof By Theorem 23 For Anysupporting

confidence: 55%

Section: Fibredmentioning

confidence: 85%

See 1 more Smart Citation

Fibered Torti-Rational Knots

Hirasawa

Murasugi

2010

J. Knot Theory Ramifications

View full text Add to dashboard Cite

show abstract

“…Let L be a generalized augmented link in S 3 with standard diagram. Suppose there is a disk E embedded in S 3 such that E has boundary some crossing circle C i , is disjoint from the other crossing circles, and E ∩ (∪ int(D k )) is empty. Then there exists such a disk F such that in addition, F intersects the projection plane in a single arc γ 1 , the intersections of F with knot strands all lie on γ 1 , and the number of intersections of F with knot strands is at most the number of intersections of E with the knot strands.…”

Section: Reducing Knot Diagramsmentioning

confidence: 99%

“…We can construct examples of atoroidal knots K whose geometric type (hyperbolic or Seifert fibered) does not agree with that of the corresponding reduced augmented link L. However, in all these examples, at least one generalized twist region contains fewer than 3 half-twists. In [3], Aït-Nouh, Matignon, and Motegi, working on a related question, show that when exactly one crossing circle is inserted into the diagram of an unknot, and then the unknot is twisted, inserting at least 4 half-twists, the geometric type of the resulting knot (Seifert fibered, toroidal, or hyperbolic) agrees with that of the unknot union the crossing circle. While these results do not apply to generalized augmented links, the result requiring only 4 half-twists is intriguing.…”

Section: Introductionmentioning

confidence: 99%

On multiply twisted knots that are Seifert fibered or toroidal

Purcell

2010

Communications in Analysis and Geometry

View full text Add to dashboard Cite

Abstract. We consider knots whose diagrams have a high amount of twisting of multiple strands. By encircling twists on multiple strands with unknotted curves, we obtain a link called a generalized augmented link. Dehn filling this link gives the original knot. We classify those generalized augmented links that are Seifert fibered, and give a torus decomposition for those that are toroidal. In particular, we find that each component of the torus decomposition is either "trivial", in some sense, or homeomorphic to the complement of a generalized augmented link. We show this structure persists under high Dehn filling, giving results on the torus decomposition of knots with generalized twist regions and a high amount of twisting. As an application, we give lower bounds on the Gromov norms of these knot complements and of generalized augmented links.

show abstract

Exceptional Dehn Surgeries on Some Infinite Series of Hyperbolic Knots and Links

Cavicchioli,

Spaggiari

2024

Mediterr. J. Math.

View full text Add to dashboard Cite

We study closed connected orientable 3-manifolds obtained by Dehn surgery along the oriented components of a link, introduced and considered by Motegi and Song (2005) and Ichihara et al. (2008). For such manifolds, we find a finite balanced group presentation of the fundamental group and describe exceptional surgeries. This allows us to construct an infinite family of tunnel number one strongly invertible hyperbolic knots with three parameters, which admit toroidal surgeries and Seifert fibered surgeries. Among the obtained results, we mention that for every integer $$n >5$$ n > 5 there are infinitely many hyperbolic knots in the 3–sphere, whose $$(n-2)$$ ( n - 2 ) and $$(n+1)$$ ( n + 1 ) -surgeries are toroidal, and $$(n-1)$$ ( n - 1 ) and n-surgeries are Seifert fibered.

show abstract

Geometric types of twisted knots

Cited by 8 publications

References 22 publications

Fibered Torti-Rational Knots

Fibered Torti-Rational Knots

On multiply twisted knots that are Seifert fibered or toroidal

Exceptional Dehn Surgeries on Some Infinite Series of Hyperbolic Knots and Links

Contact Info

Product

Resources

About