# Chern-Simons Invariants

Associated to a hyperbolic knot there is a Chern-Simons Invariant.
The value is an element of the additive real numbers modulo the subgroup generated by 1/2.
Thus all Chern-Simons invariants are given as reals between 0 and 1/2.

There is a naturally defined complex
volume for hyperbolic manifolds, with the imaginary part given by
2π^{2} times the Chern-Simons invariant. The values of the
complex volumes are also available from KnotInfo.

The initial data is for knots with 7 or fewer tetrahedra in their
decompositions (see tetrahedral
enumeration). Initial data for knots with 6 or fewer tetrahedra was
taken from [1]. Initial data for knots with 7 tetrahedra was taken from [2].

Walter Neumann
provided us with a complete set of data for knots of 12 or fewer crossings,
using the program Snap.

## Specific Knots

[1] Callahan, P. J., Dean, J. C. and Weeks, J. R. , "The simplest hyperbolic knots,"
J. Knot Theory Ramifications, 8 (3), (1999), 279-297.

[2] Champanerkar, A., Kofman, I., and Patterson, E., "The next
simplest hyperbolic knots".

[3] Kirk, P. and Klassen, E., *Chern-Simons invariants of 3-manifolds
decomposed along tori and the circle bundle over the representation space
of T*^{2}, Comm. Math. Phys. **153** (1993), no. 3,
521-557.

[4] Meyerhoff, R., *Geometric invariants of 3-manifolds,* Math.
Intelligencer **14** (1992), 37-153.

[5] Neumann, W. D., *Extended Bloch group and the Cheeger-Chern-Simons
class,* Geom. Topol. **8** (2004), 413-474.

[6] Neumann, W. D. and Zagier, D., *Volumes of hyperbolic 3-manifolds,*
Topology **24** (1985) 307-332.

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