nLab hyperbolic solid torus

Idea

The solid torus of dimension 3 admits various structures of a hyperbolic 3-manifold. Equipped with any such it is called a hyperbolic solid torus.

Properties

Knot complement

The solid torus is homeomorphic to the knot complement of the unknot in the 3-sphere.

Infinite volume

The volume of the hyperbolic solid torus is not finite. Therefore this hyperbolic 3-manifold knot complement does not count as a “knot complement with hyperbolic structure” in the sense of Thurston‘s classification of 3-manifolds? (see also this MO discussion).

As Euclidean BTZ black hole and thermal $AdS_3$

The Euclidean BTZ black hole as well as thermal AdS3 is a hyperbolic 3-manifold homeomorphic to (the interior of) the hyperbolic solid torus, hence to the knot complement of the unknot in the 3-sphere.

(e.g Gukov 03, appendix A, Kraus 06, around Fig. 1, BKR 07, 2.1)

Related concepts

• hyperbolic 3-manifold

  • hyperbolic 3-space

References

General

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As BTZ black hole and thermal $AdS_3$

• Kirill Krasnov, Holography and Riemann Surfaces, Adv. Theor. Math. Phys. 4 (2000) 929-979 (arXiv:hep-th/0005106)

• Kirill Krasnov, around Figure 6 of: Analytic Continuation for Asymptotically AdS 3D Gravity, Class. Quant. Grav. 19 (2002) 2399-2424 (arXiv:gr-qc/0111049)

• Sergei Gukov, Appendix A of: Three-Dimensional Quantum Gravity, Chern-Simons Theory, and the A-Polynomial, Commun. Math. Phys. 255: 577-627, 2005 (arXiv:hep-th/0306165)

• Per Kraus, Lectures on black holes and the $AdS_3/CFT_2$ correspondence, Lect. Notes Phys. 755: 193-247, 2008 (arXiv:hep-th/0609074)

• Micha Berkooz, Zohar Komargodski, Dori Reichmann, Thermal $AdS_3$, BTZ and competing winding modes condensation, JHEP 0712:020, 2007 (arXiv:0706.0610)

• M. Cadoni, M. Melis, Holographic entanglement entropy of the BTZ black hole, Found. Phys. 40: 638-657, 2010 (arXiv:0907.1559)