Tunnel number one, genus-one fibered knots

Baker, Kenneth L.; Johnson, Jesse; Klodginski, Elisabeth A.

doi:10.4310/cag.2009.v17.n1.a1

Cited by 9 publications

(11 citation statements)

References 16 publications

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“…We can choose arcs I x and I y in one fiber for the complement being a handlebody, for example. In [1], tunnel number one once punctured torus bundles are completely determined. They said that the tunnel number of a once punctured torus bundle is one if and only if there is a simple free loop on a fiber such that this loop and the image of this loop under the monodromy intersect once.…”

Section: The Case Where Tra φ =mentioning

confidence: 99%

Generalized torsions in once punctured torus bundles

Sekino¹

2021

Preprint

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A generalized torsion in a group, an non-trivial element such that some products of its conjugates is the identity. This is an obstruction for a group being bi-orderable. Though it is known that there is a non bi-orderable group without generalized torsions, it is conjectured that 3-manifold groups without generalized torsions are bi-orderable. In this paper, we find generalized torsions in the fundamental groups of once punctured torus bundles which are not bi-orderable. Our result contains a generalized torsion in a tunnel number two hyperbolic once punctured torus bundle.

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Section: The Case Where Tra φ =mentioning

confidence: 99%

Generalized torsions in once punctured torus bundles

Sekino¹

2021

Preprint

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“…The r -Hopf band is the fibered link in L(r, 1) whose fiber is an annulus and the monodromy is r -Dehn twists along the core curve; see [8].…”

Section: Lemma 14mentioning

confidence: 99%

A cabling conjecture for knots in lens spaces

Baker

2014

Bol. Soc. Mat. Mex.

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Closed 3-string braids admit many bandings to two-bridge links. By way of the Montesinos Trick, this allows us to construct infinite families of knots in the connected sum of lens spaces L(r, 1)#L(s, 1) that admit a surgery to a lens space for all pairs of integers (r, s) except (0, 0). These knots are typically hyperbolic. We also demonstrate that the previously known two families of examples of hyperbolic knots in non-prime manifolds with lens space surgeries of Eudave-Muñoz-Wu and Kang all fit this construction. As such, we propose a generalization of the cabling conjecture of González-Acuña-Short for knots in lens spaces. The main results, context, and conjecturesBanding the braid closure L = β of a closed 3-string braid β along a "level" framed arc a produces a two-bridge link L = β that is a plait closure of β and a dual framed Dedicated to the 70th birthday of Professor Fico González-Acuña.

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“…where c and b are curves on the once-punctured torus fiber T transversally intersecting once and τ a is a right-handed Dehn twist along the curve a. This monodromy is given as its conjugate τ c τ b τ c τ n b in [2]. The manifold M n is hyperbolic if and only if |n| > 2, contains an essential torus if and only if |n| = 2, and is a Seifert fiber space if and only if |n| < 1 as we observe in Lemma 2.8.…”

Section: Once-punctured Torus Bundles With Tunnel Number Onementioning

confidence: 99%

“…Substitution for x 2 can be obtained by considering φ ′ 2 −z 2 φ 1 and φ ′ 3 +(f n+1 (y)+ 1) 2 x 2 . These equations are linear in φ ′ 2 and φ ′ 3 , so the polynomials 2 . generate R as well.…”

Section: Computation Of the Psl 2 (C) Character Varietymentioning

confidence: 99%

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Character Varieties of Once-Punctured Torus Bundles With Tunnel Number One

Baker

Petersen

2013

Int. J. Math.

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Abstract. We determine the PSL 2 (C) and SL 2 (C) character varieties of the once-punctured torus bundles with tunnel number one, i.e. the once-punctured torus bundles that arise from filling one boundary component of the Whitehead link exterior. In particular, we determine 'natural' models for these algebraic sets, identify them up to birational equivalence with smooth models, and compute the genera of the canonical components. This enables us to compare dilatations of the monodromies of these bundles with these genera. We also determine the minimal polynomials for the trace fields of these manifolds. Additionally we study the action of the symmetries of these manifolds upon their character varieties, identify the characters of their lens space fillings, and compute the twisted Alexander polynomials for their representations to SL 2 (C).

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Tunnel number one, genus-one fibered knots

Abstract: We determine the genus-one fibered knots in lens spaces that have tunnel number one. We also show that every tunnel number one, once-punctured torus bundle is the result of Dehn filling a component of the Whitehead link in the 3-sphere.

Cited by 9 publications

References 16 publications

Generalized torsions in once punctured torus bundles

Generalized torsions in once punctured torus bundles

A cabling conjecture for knots in lens spaces

Character Varieties of Once-Punctured Torus Bundles With Tunnel Number One

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