Some numerical computations on Reidemeister torsion for homology 3-spheres obtained by Dehn surgeries along the figure-eight knot

Kitano, Teruaki

doi:10.48550/arxiv.1603.03728

Cited by 3 publications

(10 citation statements)

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“…We check the integrality for several examples below, and naturally we conjecture it is always true. This is a curiosity, and the integral property of torsion has been already reported in the mathematical literature [85]. One crucial difference is that they consider torsions in the fundamental representation, while we consider here the adjoint representation.…”

Section: Consistency Check: Integrality Of Twisted Indicesmentioning

confidence: 93%

Precision microstate counting for the entropy of wrapped M5-branes

Gang

Kim

Zayas

2020

J. High Energ. Phys.

101

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We study the large N expansion of twisted partition functions of 3d N = 2 superconformal field theories arising from N M5-branes wrapped on a hyperbolic 3manifold, M 3 . Via the 3d-3d correspondence, the partition functions of these 3d N = 2 superconformal field theories are related to simple topological invariants on the 3-manifold. The partition functions can be expressed using only classical and one-loop perturbative invariants of P SL(N, C) Chern-Simons theory around irreducible flat connections on M 3 . Using mathematical results on the asymptotics of the invariants, we compute the twisted partition functions in the large N limit including perturbative corrections to all orders in 1/N . Surprisingly, the perturbative expansion terminates at finite order. The leading part of the partition function is of order N 3 and agrees with the Bekenstein-Hawking entropy of the dual black holes. The subleading part, in particular the log N -terms in the field theory partition function is found to precisely match the one-loop quantum corrections in the dual eleven dimensional supergravity. The field theory results of other terms in 1/N provide a stringent prediction for higher order corrections in the holographic dual, which is M-theory.

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Section: Consistency Check: Integrality Of Twisted Indicesmentioning

confidence: 93%

Precision microstate counting for the entropy of wrapped M5-branes

Gang

Kim

Zayas

2020

J. High Energ. Phys.

101

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“…Therefore, one of the SL(2, R)-representations gives a Perron number. Such a phenomenon is numerically seen in [11] as well. Also, the Reidemeister torsions of the Brieskorn homology 3-sphere Σ(p, q, r) are real numbers (see Appendix A).…”

Section: 1mentioning

confidence: 67%

“…It is worth mentioning that while the Brieskorn homology 3-spheres are not hyperbolic, S 3 p/q (4 1 ) is hyperbolic unless p/q is an integer with |p/q| ≤ 4 or p/q = ∞. In this paper, we rigorously prove the observation in [11].…”

mentioning

confidence: 73%

“…As a consequence, for the homology 3-sphere obtained by Dehn surgery along a (right-handed) torus knot T p,q , its Reidemeister torsion τ ρ (S 3 −1/n (T p,q )) is also an algebraic integer since Σ(p, q, pqn + 1) = S 3 −1/n (T p,q ) for n > 0. Also, such a phenomenon was numerically observed for the figure-eight knot 4 1 in [11]. It is worth mentioning that while the Brieskorn homology 3-spheres are not hyperbolic, S 3 p/q (4 1 ) is hyperbolic unless p/q is an integer with |p/q| ≤ 4 or p/q = ∞.…”

mentioning

confidence: 74%

“…Reidemeister torsion is sometimes defined to be 3 i=0 det[b i bi−1 /c i ] (−1) i . Our τ ρ (M ) is the same as Reidemeister torsion in [8,11,14], but the inverse of Reidemeister torsion in [9,10]. This difference is crucial for the results in this paper.…”

Section: Preliminariesmentioning

confidence: 90%

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An algebraic property of Reidemeister torsion

Kitano¹,

Nozaki²

2022

Preprint

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For a 3-manifold M and an acyclic SL(2, C)-representation ρ of its fundamental group, the SL(2, C)-Reidemeister torsion τρ(M ) ∈ C × is defined. If there are only finitely many conjugacy classes of irreducible representations, then the Reidemeister torsions are known to be algebraic numbers. Furthermore, we prove that the Reidemeister torsions are not only algebraic numbers but also algebraic integers for most Seifert fibered spaces and infinitely many hyperbolic 3-manifolds. Also, for a knot exterior E(K), we discuss the behavior of τρ(E(K)) when the restriction of ρ to the boundary torus is fixed. Contents 9 4. Continuous variation of Reidemeister torsion 16 Appendix A. Reidemeister torsion of Seifert fibered spaces 18 Appendix B. Computational results 21 References 28

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An algebraic property of Reidemeister torsion

Kitano

Nozaki

2022

Trans London Maths Soc

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For a 3‐manifold M$M$ and an acyclic SLfalse(2,double-struckCfalse)$\mathit {SL}(2,\mathbb {C})$‐representation ρ$\rho$ of its fundamental group, the SLfalse(2,double-struckCfalse)$\mathit {SL}(2,\mathbb {C})$‐Reidemeister torsion τρ(M)∈C×$\tau _\rho (M) \in \mathbb {C}^\times$ is defined. If there are only finitely many conjugacy classes of irreducible representations, then the Reidemeister torsions are known to be algebraic numbers. Furthermore, we prove that the Reidemeister torsions are not only algebraic numbers but also algebraic integers for most Seifert fibered spaces and infinitely many hyperbolic 3‐manifolds. Also, for a knot exterior Efalse(Kfalse)$E(K)$, we discuss the behavior of τρ(Efalse(Kfalse))$\tau _\rho (E(K))$ when the restriction of ρ$\rho$ to the boundary torus is fixed.

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Some numerical computations on Reidemeister torsion for homology 3-spheres obtained by Dehn surgeries along the figure-eight knot

Abstract: We show some computations on representations of the fundamental group in SL(2; C) and Reidemeister torsion for a homology 3-sphere obtained by Dehn surgery along the figure-eight knot.

Cited by 3 publications

References 7 publications

Precision microstate counting for the entropy of wrapped M5-branes

Precision microstate counting for the entropy of wrapped M5-branes

An algebraic property of Reidemeister torsion

An algebraic property of Reidemeister torsion

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