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 having 7 or fewer tetrahedra in their decompositions (see tetrahedral enumeration). Initial data for knots with 6 or fewer tetrahedra was taken from "The simplest hyperbolic knots" by by Callahan, Dean, and Weeks. Initial data for knots with 7 tetrahedra was taken from "The next simplest hyperbolic knots" by Champanerkar, Kofman, and Patterson.

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

References

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

P. Kirk and E. Klassen, 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.

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

W.D. Neumann, D. Zagier, Volumes of hyperbolic 3-manifolds, Topology 24 (1985) 307-332.