Not all boundary slopes are strongly detected by the character variety

Chesebro, Eric; Tillmann, Stephan

doi:10.4310/cag.2007.v15.n4.a2

Cited by 8 publications

(10 citation statements)

References 10 publications

(37 reference statements)

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“…. p n /q n ) with n ≥ 3 as also presented in [1], [9], [11], and [8]. See those for details, but briefly [6] associates candidate surfaces to admissible edgepath systems in a graph D in the uv-plane whose vertices (u, v) correspond to projective curve systems [a, b, c] on the 4-punctured sphere carried by the train track in Fig.…”

Section: Preliminariesmentioning

confidence: 99%

Denominators and differences of boundary slopes for (1,1)-knots

Callahan

2015

Bol. Soc. Mat. Mex.

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Abstract. We show that every nonzero integer occurs in the denominator of a boundary slope for infinitely many (1,1)-knots and that infinitely many (1,1)-knots have boundary slopes of arbitrarily small difference. Specifically, we prove that for any integers m, n > 1 with n odd the exterior of the Montesinos knot K(−1/2, m/(2m±1), 1/n) in S 3 contains an essential surface with boundary slope r = 2(n − 1) 2 /n if m is even and 2(n + 1) 2 /n if m is odd. If n ≥ 4m + 1, we prove that K(−1/2, m/(2m + 1), 1/n) also has a boundary slope whose difference with r is (8m − 2)/(n 2 − 4mn + n), which decreases to 0 as n increases. All of these knots are (1,1)-knots.

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Section: Preliminariesmentioning

confidence: 99%

Denominators and differences of boundary slopes for (1,1)-knots

Callahan

2015

Bol. Soc. Mat. Mex.

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“…, y 3 , xy 2 z, z C[x] is free with rank six. Now suppose that h is an element of the intersection (5). Then xh is an element of both 1, .…”

Section: Ranks and Examplesmentioning

confidence: 99%

“…It is natural to ask if, whenever N contains a closed essential surface, there is a closed essential surface in N detected by X(N ). The author is not certain if the answer is known to be no, however there are compelling reasons to believe that the answer is no, see for example [5] or Remark 5.1 in [11].…”

Section: Algebraic Geometrymentioning

confidence: 99%

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Closed surfaces and character varieties

Chesebro

2013

Algebr. Geom. Topol.

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The powerful character variety techniques of Culler and Shalen can be used to find essential surfaces in knot manifolds. We show that module structures on the coordinate ring of the character variety can be used to identify detected boundary slopes as well as when closed surfaces are detected. This approach also yields new number theoretic invariants for the character varieties of knot manifolds.Comment: 28 pages, 1 figur

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“…Schanuel and Zhang [11] gave the rst examples of non-ber (and non semi-ber) boundary slopes that are not strongly detected, although they are weakly detected. Chesebro and Tillmann [2] give an innite family of hyperbolic knots, each of which has at least one boundary slope of an incompressible surface (non-ber and non semi-ber) that is not strongly detected. Theorem 1.7.…”

Section: Introductionmentioning

confidence: 99%

Detection of incompressible surfaces in hyperbolic punctured torus bundles

Segerman

2010

Geom Dedicata

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Culler and Shalen, and later Yoshida, give ways to construct incompressible surfaces in 3-manifolds from ideal points of the character and deformation varieties, respectively. We work in the case of hyperbolic punctured torus bundles, for which the incompressible surfaces were classied by Floyd and Hatcher. We convert non ber incompressible surfaces from their form to the form output by Yoshida's construction, and run his construction backwards to give (for non semi-bers, which we identify) the data needed to construct ideal points of the deformation variety corresponding to those surfaces via Yoshida's construction. We use a result of Tillmann to show that the same incompressible surfaces can be obtained from an ideal point of the character variety via the Culler-Shalen construction. In particular this shows that all boundary slopes of non ber and non semi-ber incompressible surfaces in hyperbolic punctured torus bundles are strongly detected.

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Not all boundary slopes are strongly detected by the character variety

Abstract: It has been an open question whether all boundary slopes of hyperbolic knots are strongly detected by the character variety. The main result of this paper produces an infinite family of hyperbolic knots each of which has at least one strict boundary slope that is not strongly detected.

Cited by 8 publications

References 10 publications

Denominators and differences of boundary slopes for (1,1)-knots

Denominators and differences of boundary slopes for (1,1)-knots

Closed surfaces and character varieties

Detection of incompressible surfaces in hyperbolic punctured torus bundles

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