Complex volume

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 ${\displaystyle M}$

${\displaystyle \operatorname {vol} _{\mathbb {C} }(M):=\operatorname {vol} (M)+i\,\operatorname {cs} (M)\in \mathbb {C} /4\pi ^{2}\mathbb {Z} i}$,

where is the hyperbolic volume and the SO (3) - Chern-Simons invariant of the Levi-Civita relationship . ${\displaystyle \operatorname {vol} (M)}$ ${\displaystyle \operatorname {cs} (M)}$

More generally, one can define the complex volume for representations as ${\displaystyle \rho \colon \pi _{1}M\to SL(n,\mathbb {C} )}$

${\displaystyle \operatorname {vol} _{\mathbb {C} }(\rho ):=i\left\langle {\hat {c}}_{2}(E_{\rho }),\left[M\right]\right\rangle \in \mathbb {C} /4\pi ^{2}\mathbb {Z} i}$,

where the flat bundle with holonomy , ${\displaystyle E_{\rho }}$ ${\displaystyle \rho }$

${\displaystyle {\hat {c}}_{2}(E_{\rho })\in H^{3}(M;\mathbb {C} /4\pi ^{2}\mathbb {Z} i)}$

his 2nd Cheeger-Chern-Simons class and

${\displaystyle \left[M\right]\in H_{3}(M;\mathbb {Z} )}$

is the fundamental class of . ${\displaystyle M}$

The volume conjecture postulates the equation for hyperbolic nodes ${\displaystyle K}$

${\displaystyle \operatorname {vol} _{\mathbb {C} }(S^{3}\setminus K)=2\pi \lim _{N\to \infty }{\frac {1}{N}}\log J_{N}(K,e^{\frac {2\pi i}{N}})}$,

where denotes the -th colored Jones polynomial of . ${\displaystyle J_{N}(K,q)}$ ${\displaystyle N}$ ${\displaystyle K}$

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