On the volume and Chern–Simons invariant for 2-bridge knot orbifolds

Ham, Ji-Young; Lee, Joongul; Mednykh, Alexander; Rasskazov, Aleksei

doi:10.1142/s0218216517500821

Cited by 8 publications

(9 citation statements)

References 43 publications

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“…Let v = x + M 2 + M −2 . Theorem 2.3 can be found in [8,32]. We include the proof for readers' convenience.…”

Section: The Rileymentioning

confidence: 99%

“…Theorem 2.3. [8,32] ρ is a representation of π 1 (X 2m 2n ) if and only if x is a root of the following Riley-Mednykh polynomial,…”

Section: The Rileymentioning

confidence: 99%

“…Chern-Simons invariants of hyperbolic knot orbifolds are computed explicitly for a few infinite families in [9,10,11] using the Schläfli formula. In [8], Chern-Simons invariants of hyperbolic alternating knot orbifolds are computed explicitly using ideal triangulations by extending Neumann's Methods. This paper is written to compare the Chern-Simons invariants computed in [8] with those of this paper.…”

Section: Introductionmentioning

confidence: 99%

“…In [8], Chern-Simons invariants of hyperbolic alternating knot orbifolds are computed explicitly using ideal triangulations by extending Neumann's Methods. This paper is written to compare the Chern-Simons invariants computed in [8] with those of this paper. We wish to provide an efficient algorithm to compute the Chern-Simons invarinats of hyperbolic orbifolds of knots and links with SnapPy in a near future.…”

Section: Introductionmentioning

confidence: 99%

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Explicit formulae for Chern-Simons invariants of the hyperbolic $J(2n,-2m)$ knot orbifolds

Ham¹,

Lee²

2017

Preprint

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We calculate the Chern-Simons invariants of the hyperbolic J(2n, −2m) knot orbifolds using the Schläfli formula for the generalized Chern-Simons function on the family of cone-manifold structures of J(2n, −2m) knot. We present the concrete and explicit formula of them. We apply the general instructions of Hilden, Lozano, and Montesinos-Amilibia and extend the Ham and Lee's methods to a bi-infinite family. We dealt with even slopes just as easily as odd ones. As an application, we calculate the Chern-Simons invariants of cyclic coverings of the hyperbolic J(2n, −2m) knot orbifolds. For the fundamental group of J(2n, −2m) knot, we take and tailor Hoste and Shanahan's. As a byproduct, we give an affirmative answer for their question whether their presentation is actually derived from Schubert's canonical 2-bridge diagram or not.

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“…Let v = x + M 2 + M −2 . Theorem 2.3 can be found in [8,32]. We include the proof for readers' convenience.…”

Section: The Rileymentioning

confidence: 99%

“…Theorem 2.3. [8,32] ρ is a representation of π 1 (X 2m 2n ) if and only if x is a root of the following Riley-Mednykh polynomial,…”

Section: The Rileymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Explicit formulae for Chern-Simons invariants of the hyperbolic $J(2n,-2m)$ knot orbifolds

Ham¹,

Lee²

2017

Preprint

Self Cite

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“…Recently, the volume of double twist knot cone-manifolds has been computed in [Tr3]. We should remark that a formula for the volume of the cone-manifold of C(2n, 4) has just been given in [HLMR1]. However, with an appropriate change of variables, this formula has already obtained in [Tr3], since C(2n, 4) is the double twist knot J(2n, −4).…”

mentioning

confidence: 99%

The A-polynomial 2-tuple of twisted Whitehead links

Tran

2018

Int. J. Math.

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We compute the A-polynomial 2-tuple of twisted Whitehead links. As applications, we determine canonical components of twisted Whitehead links and give a formula for the volume of twisted Whitehead link cone-manifolds. IntroductionThe A-polynomial of a knot in the 3-sphere S 3 was introduced by Cooper, Culler, Gillet, Long and Shalen [CCGLS] in the 90's. It describes the SL 2 (C)-character variety of the knot complement as viewed from the boundary torus. The A-polynomial carries a lot of information about the topology of the knot. For example, it distinguishes the unknot from other knots [BoZ, DG] and the sides of the Newton polygon of the A-polynomial give rise to incompressible surfaces in the knot complement [CCGLS]. The A-polynomial was generalized to links by Zhang [Zh] about ten years later. For an m-component link in S 3 , Zhang defined a polynomial m-tuple link invariant called the Apolynomial m-tuple. The A-polynomial 1-tuple of a knot is nothing but the A-polynomial defined in [CCGLS]. The A-polynomial m-tuple also caries important information about the topology of the link. For example, it can be used to construct concrete examples of hyperbolic link manifolds with non-integral traces [Zh].Finding an explicit formula for the A-polynomial is a challenging problem. So far, the A-polynomial has been computed for a few classes of knots including two-bridge knots C(2n, p) (with 1 ≤ p ≤ 5) in Conway's notation [HS, Pe, Ma, HL], (−2, 3, 2n + 1)-pretzel knots [TY, GM]. It should be noted that C(2n, p) is the double twist knot J(2n, −p) in the notation of [HS]. Moreover, C(2n, 1) is the torus knot T (2, 2n + 1) and C(2n, 2) is known as a twist knot. A cabling formula for the A-polynomial of a cable knot in S 3 has recently been given in [NZ]. Using this formula, Ni and Zhang [NZ] has computed the A-polynomial of an iterated torus knot explicitly.In this paper we will compute the A-polynomial 2-tuple for a family of 2-component links called twisted Whitehead links. As applications, we will determine canonical components of twisted Whitehead links and give a formula for the volume of twisted Whitehead link cone-manifolds. For k ≥ 0, the k-twisted Whitehead link W k is the 2-component link depicted in Figure 1. Note that W 0 is the torus link T (2, 4) and W 1 is the Whitehead link. Moreover, W k is the two-bridge link C(2, k, 2) in Conway's notation and is b(4k +4, 2k +1) in Schubert's notation. These links are all hyperbolic except for W 0 .The A-polynomial 2-tuple of the twisted Whitehead link W k is a polynomial 2-tuple [A 1 (M, L), A 2 (M, L)] given as follows.

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New formulas for the character varieties of two-bridge links

Cavicchioli

Spaggiari

2021

J. Geom.

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On the volume and Chern–Simons invariant for 2-bridge knot orbifolds

Cited by 8 publications

References 43 publications

Explicit formulae for Chern-Simons invariants of the hyperbolic $J(2n,-2m)$ knot orbifolds

Explicit formulae for Chern-Simons invariants of the hyperbolic $J(2n,-2m)$ knot orbifolds

The A-polynomial 2-tuple of twisted Whitehead links

New formulas for the character varieties of two-bridge links

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