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 2π2 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 T2, 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.