## Algorithms for finding representations of the fundamental group of a 3-manifold into PSL(3, C)

With unipotent boundary tori.We consider all the triangulations (of hyperbolic manifolds with cusps) with at most 3 tetrahedra (there are 11 cases), our input being computed by Snappea. For each case, we compute :

• The face, edge, cross-ratio and holonomy equations imposing that the boundary tori are unipotent.

• A discrimination of solutions that lead to PSL(3,R), PSL(2,C) PSL(2,R) and PU(2,1) representations.

• For each solution, a numerical approximation of the solutions of the initial system, of the related matrix generators and boundary holonomy.

All the data are stored here.

Two tetrahedra: m003 (4-1 sister) m004 (4-1 Knot complement)

Three tetrahedra: m006 m007 m009 m010 m011 m015 (5-2 Knot complement) m016 m017 m019

Four Tetrahedra: Whitehead Knot complement.

At the neighborhood of unipotent solutions.The 4-1 knot complement : maple worksheet

## Publications describing the computations

LOCAL RIGIDITY FOR SL (3,C) REPRESENTATIONS OF 3-MANIFOLDS GROUPSLet M be a non-compact hyperbolic 3-manifold that has a tri- angulation by positively oriented ideal tetraedra. We explain how to produce local coordinates for the variety defined by the gluing equations for SL(3, C)- representations. In particular we prove local rigidity of the “geometric” rep- resentation in SL(3, C), recovering a recent result of Menal-Ferrer and Porti. More generally we give a criterion for local rigidty of SL(3, C)-representations and provide detailed analysis of the figure eight knot sister manifold exhibiting the different possibilities that can occur.

Illustration in the database : the 4-1 sister example (m003)

Article on ArXiv : 1307.8343

REPRESENTATIONS OF FUNDAMENTAL GROUPS OF 3-MANIFOLDS INTO PSL(3,C): EXACT COMPUTATIONS IN LOW COMPLEXITY.We are interested in computing representations of the fundamental group of a 3-manifold into PSL(3,C). The representations are obtained by gluing decorated tetrahedra of flags as in [7, 2]. We list complete computations (giving 0-dimensional or 1-dimensional solution sets) in a number of examples with a description of the computer methods used to find them.

Illustrations in the database : all the examples where one imposes that the boundary tori are unipotent.

Article on ArXiv : 1307.6697

CHARACTER VARIETIES FOR SL(3,C): THE FIGURE EIGHT KNOTWe give a description of several representation varieties of the fundamental group of the complement of the figure eight knot in PGL(3,C) or SL(3,C). We moreover obtain an explicit parametrization of matrices generating the representation and a description of the projection of the representation variety into the character variety of the boundary torus into SL(3,C).

Illustration : maple worksheet that illustrates the computations done.

Article on ArXiv : 1412.4711