In mathematics , the complex volume is an invariant of 3-dimensional manifolds. For complements of knots and links , the conjecture of volume establishes a connection between the complex volume and the asymptotics of quantum invariants .
definition
For a hyperbolic 3-manifold of finite volume , the complex volume is defined as
![M.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd)
-
,
where is the hyperbolic volume and the SO (3) - Chern-Simons invariant of the Levi-Civita relationship .
![\ operatorname {vol} (M)](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ecb41756f7701ba584afa8ccf1d8d5efdb53a49)
![\ operatorname {cs} (M)](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ff43a9f2066a22d2f9ea5630a7f97dcaac0f9f0)
More generally, one can define the complex volume for representations as
![{\ displaystyle \ rho \ colon \ pi _ {1} M \ to SL (n, \ mathbb {C})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3421a491f4ea064390bc3cd3b08db92f3a6bf41d)
-
,
where the flat bundle with holonomy ,
![\ rho](https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64)
![{\ displaystyle {\ hat {c}} _ {2} (E _ {\ rho}) \ in H ^ {3} (M; \ mathbb {C} / 4 \ pi ^ {2} \ mathbb {Z} i )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e9c4452a2f0def7e4a11639a684737f5c7c397f0)
his 2nd Cheeger-Chern-Simons class and
![\ left [M \ right] \ in H_ {3} (M; \ mathbb {Z})](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e6e807cf199c3c012669154464d256884218c33)
is the fundamental class of .
![M.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd)
The volume conjecture postulates the equation
for hyperbolic nodes ![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
-
,
where denotes the -th colored Jones polynomial of .
![J_ {N} (K, q)](https://wikimedia.org/api/rest_v1/media/math/render/svg/9cb75aa722b22dcdd65caa91468fe8abdb5eba2f)
![N](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3)
![K](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0)
literature
- WD Neumann: Extended Bloch group and the Cheeger-Chern-Simons class. Geom. Topol. 8: 413-474 (2004). pdf
- S. Garoufalidis, D. Thurston, C. Zickert: The complex volume of SL (n, C) -representations of 3-manifolds . Duke Math. J. 164, 2099-2160 (2015). pdf