On minimality of two-bridge knots

Nagasato, Fumikazu; Suzuki, Masaaki; Tran, Anh T.

doi:10.1142/s0129167x17500203

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2017

2024

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Cited by 7 publications

(2 citation statements)

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“…A knot is called minimal if its knot group admits epimorphisms onto the knot groups of only the trivial knot and itself. Many types of minimal knots are already shown in [10], [17], [4], [11], [13], [15], and [14]. By using the main theorem of this paper, we obtain several types of minimal knots.…”

Section: Introductionmentioning

confidence: 76%

Genera of two-bridge knots and epimorphisms of their knot groups

Suzuki

Tran

2018

Topology and its Applications

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Section: Introductionmentioning

confidence: 76%

Genera of two-bridge knots and epimorphisms of their knot groups

Suzuki

Tran

2018

Topology and its Applications

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“…Note that Claim 6.1 was also proved in [12,Proposition 3.1]. Another way in which non-canonical components of characters of irreducible representations can arise in the character variety is when the knot has a certain nice symmetry described by Ohtsuki.…”

Section: Two Examplesmentioning

confidence: 82%

Detecting essential surfaces as intersections in the character variety

Chu

2017

Algebr. Geom. Topol.

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Symplectic quandles and parabolic representations of 2-bridge knots and links

Kim

2020

Int. J. Math.

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In this paper, we study the parabolic representations of 2-bridge links by finiding arc coloring vectors on the Conway diagram. The method we use is to convert the system of conjugation quandle equations to that of symplectic quandle equations. In this approach, we have an integer coefficient monic polynomial [Formula: see text] for each 2-bridge link [Formula: see text], and each zero of this polynomial gives a set of arc coloring vectors on the diagram of [Formula: see text] satisfying the system of symplectic quandle equations, which gives an explicit formula for a parabolic representation of [Formula: see text]. We then explain how these arc coloring vectors give us the closed form formulas of the complex volume and the cusp shape of the representation. As other applications of this method, we show some interesting arithmetic properties of the Riley polynomial and of the trace field, and also describe a necessary and sufficient condition for the existence of epimorphisms between 2-bridge link groups in terms of divisibility of the corresponding Riley polynomials.

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On minimality of two-bridge knots

Abstract: Abstract. A knot is called minimal if its knot group admits epimorphisms onto the knot groups of only the trivial knot and itself. In this paper, we determine which two-bridge knot b(p, q) is minimal where q ≤ 6 or p ≤ 100.

Cited by 7 publications

References 18 publications

Genera of two-bridge knots and epimorphisms of their knot groups

Genera of two-bridge knots and epimorphisms of their knot groups

Detecting essential surfaces as intersections in the character variety

Symplectic quandles and parabolic representations of 2-bridge knots and links

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