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

Baker, Kenneth L.; Petersen, Kathleen L.

doi:10.1142/s0129167x13500481

Cited by 7 publications

(4 citation statements)

References 31 publications

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“…give the structure of an algebraic set in C 3 when the fundamental group π 1 (M n ) are generated by two generators α and β (for details see [BP,Proposition 5.20]). This means that the variables x, y and z play a role as a coordinate (x, y, z) of the character variety X(π 1 (M n )).…”

Section: Preliminariesmentioning

confidence: 99%

“…There are 3 cases on the parametrization of X(π 1 (M n )) (for details we refer to [BP,Propositions 5.23 & 5.32 and Remark 5.33]):…”

Section: Preliminariesmentioning

confidence: 99%

“…It is also shown in [BP,Theorem 5.1] that the character variety X(π 1 (M n )) consists of the unique component C when n ≡ 2 (mod 4) and X(π 1 (M n )) consists of two components C and L, where L is parametrized by the case (3) when n ≡ 2 (mod 4). We observe the sum of inverse of the adjoint Reidemeister torsion on the components C and L for the conjecture by Gang-Kim-Yoon.…”

Section: Preliminariesmentioning

confidence: 99%

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Adjoint Reidemeister torsions of once-punctured torus bundles

Tran¹,

Yamaguchi²

2021

Preprint

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Gang, Kim and Yoon have recently proposed a conjecture on a vanishing identity of adjoint Reidemeister torsions of hyperbolic 3-manifolds with torus boundary, from the viewpoint of wrapped M5-branes. In this paper, we provide infinitely many new supporting examples and an infinite family of counterexamples to this conjecture. These families come from hyperbolic once-punctured torus bundles with tunnel number one. We also propose a modified conjecture to exclude our counterexamples and show that it holds true for all hyperbolic once-punctured torus bundles with tunnel number one.

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Section: Preliminariesmentioning

confidence: 99%

“…There are 3 cases on the parametrization of X(π 1 (M n )) (for details we refer to [BP,Propositions 5.23 & 5.32 and Remark 5.33]):…”

Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

See 1 more Smart Citation

Adjoint Reidemeister torsions of once-punctured torus bundles

Tran¹,

Yamaguchi²

2021

Preprint

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show abstract

“…As in the proof of [BP,Lemma 6.8], we will use the following theorem of Ljunggren [Lj]. Consider a polynomial of the form R(t) = t k 1 + ε 1 t k 2 + ε 2 t k 3 + ε 3 where ε j = ±1 for j = 1, 2, 3.…”

Section: Since ρ(Abmentioning

confidence: 99%

On nonabelian representations of twist knots

Dean

Tran

2016

Involve

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We study representations of the knot groups of twist knots into SL 2 (C). The set of nonabelian SL 2 (C) representations of a twist knot K is described as the zero set in C × C of a polynomial P K (x, y) = Q K (y) + x 2 R K (y) ∈ Z[x, y], where x is the trace of a meridian. We prove some properties of P K (x, y). In particular, we prove that P K (2, y) ∈ Z[y] is irreducible over Q. As a consequence, we obtain an alternative proof of a result of Hoste and Shanahan that the degree of the trace field is precisely two less than the minimal crossing number of a twist knot.

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Rank 1 character varieties of finitely presented groups

2017

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Let X(F,G) be the G-character variety of F where G is a rank 1 complex affine algebraic group and F is a finitely presentable discrete group. We describe an algorithm, which we implement in Mathematica, SageMath, and in Python, that takes a finite presentation for F and produces a finite presentation of the coordinate ring of X(F,G). We also provide a new description of the defining relations and local parameters of the coordinate ring when F is free. Although the theorems used to create the algorithm are not new, we hope that as a well-referenced exposition with a companion computer program it will be useful for computation and experimentation with these moduli spaces.Comment: 30 pages, Mathematica program at http://math.gmu.edu/~slawton3/trace-identities.nb, SageMath program at http://math.gmu.edu/~slawton3/Main.sagews, Python program at http://math.gmu.edu/~slawton3/charvars.py, accepted for publication at Geometriae Dedicat

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

Cited by 7 publications

References 31 publications

Adjoint Reidemeister torsions of once-punctured torus bundles

Adjoint Reidemeister torsions of once-punctured torus bundles

On nonabelian representations of twist knots

Rank 1 character varieties of finitely presented groups

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