Twisted Alexander polynomials of 2-bridge knots for parabolic representations

Morifuji, Takayuki; Tran, Anh T.

doi:10.2140/pjm.2014.269.433

Cited by 21 publications

(11 citation statements)

References 29 publications

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“…By [MT,Section 2], the assignment (2.1) gives a non-abelian representation ρ : π 1 (K) → SL 2 (C) if and only if (M, y) ∈ C * × C satisfies the equation…”

Section: Knot Groupsmentioning

confidence: 99%

On left-orderable fundamental groups and Dehn surgeries on knots

Tran¹

2015

J. Math. Soc. Japan

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“…By [MT,Section 2], the assignment (2.1) gives a non-abelian representation ρ : π 1 (K) → SL 2 (C) if and only if (M, y) ∈ C * × C satisfies the equation…”

Section: Knot Groupsmentioning

confidence: 99%

On left-orderable fundamental groups and Dehn surgeries on knots

Tran¹

2015

J. Math. Soc. Japan

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“…From the proof of Theorem 2, one can easily see that the twisted Alexander polynomial associated to any parabolic SL 2 (C)-representation of the knot group of the m-twist knot K m , m > 0, determines the fiberedness and genus of K m . This gives another proof of [MT, Theorem 1.1] and hence of a conjecture of Dunfield, Friedl and Jackson [DFJ] for K m .…”

mentioning

confidence: 73%

ON THE TWISTED ALEXANDER POLYNOMIAL FOR REPRESENTATIONS INTO SL₂(ℂ)

Tran

2013

J. Knot Theory Ramifications

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We study the twisted Alexander polynomial ∆ K,ρ of a knot K associated to a non-abelian representation ρ of the knot group into SL 2 (C). It is known for every knot K that if K is fibered, then for every non-abelian representation, ∆ K,ρ is monic and has degree 4g(K) − 2 where g(K) is the genus of K. Kim and Morifuji recently proved the converse for 2-bridge knots. In fact they proved a stronger result: if a 2-bridge knot K is non-fibered, then all but finitely many non-abelian representations on some component have ∆ K,ρ non-monic and degree 4g(K)−2. In this paper, we consider two special families of non-fibered 2-bridge knots including twist knots. For these families, we calculate the number of non-abelian representations where ∆ K,ρ is monic and calculate the number of non-abelian representations where the degree of ∆ K,ρ is less than 4g(K) − 2.2010 Mathematics Subject Classification. 57M27.

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“…For a hyperbolic knot K, the holonomy representationρ 0 : Recently, the third author confirmed the conjecture for twist knots [32], and the third author and Tran did for a certain wider class of 2-bridge knots [33]. These are the first infinite families of knots where Conjecture 2.3 is verified.…”

Section: 3mentioning

confidence: 83%

Twisted Alexander Polynomials on Curves in Character Varieties of Knot Groups

2013

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For a fibered knot in the 3-sphere the twisted Alexander polynomial associated to an SL(2, C)-character is known to be monic. It is conjectured that for a nonfibered knot there is a curve component of the SL(2, C)-character variety containing only finitely many characters whose twisted Alexander polynomials are monic, i.e. finiteness of such characters detects fiberedness of knots. In this paper we discuss the existence of a certain curve component which relates to the conjecture when knots have nonmonic Alexander polynomials. We also discuss the similar problem of detecting the knot genus.2010 Mathematics Subject Classification. Primary 57M27, Secondary 57M05, 57M25.

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Twisted Alexander polynomials of 2-bridge knots for parabolic representations

Cited by 21 publications

References 29 publications

On left-orderable fundamental groups and Dehn surgeries on knots

On left-orderable fundamental groups and Dehn surgeries on knots

ON THE TWISTED ALEXANDER POLYNOMIAL FOR REPRESENTATIONS INTO SL₂(ℂ)

Twisted Alexander Polynomials on Curves in Character Varieties of Knot Groups

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Twisted Alexander polynomials of 2-bridge knots for parabolic representations

Cited by 21 publications

References 29 publications

On left-orderable fundamental groups and Dehn surgeries on knots

On left-orderable fundamental groups and Dehn surgeries on knots

ON THE TWISTED ALEXANDER POLYNOMIAL FOR REPRESENTATIONS INTO SL2(ℂ)

Twisted Alexander Polynomials on Curves in Character Varieties of Knot Groups

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ON THE TWISTED ALEXANDER POLYNOMIAL FOR REPRESENTATIONS INTO SL₂(ℂ)