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Computing and Understanding Representation Varieties Efficiently

Duality and invariants of representations of fundamental groups of 3-manifolds into PGL(3,C)
E. Falbel, Qingxue Wang
We determine the explicit transformation under duality of generic configurations of four flags in PGL(3,C) in cross-ratio coordinates. As an application we prove the invariance under duality of an invariant in the Bloch group obtained from decorated triangulations of 3-manifolds.
Fock-Goncharov coordinates for rank two Lie groups
Christian K. Zickert
This paper defines Ptolemy coordinates for all simple Lie groups of rank 2.
Ptolemy decoration
Triangulation independent Ptolemy varieties
Matthias Goerner, Christian K. Zickert
This paper defines triangulation independent refinements of all variants of the Ptolemy variety. Changing the triangulation changes the variety by a biregular isomorphism.
Ptolemy coordinates, Dehn invariant and the A-polynomial
Christian K. Zickert
This paper defines an enhanced Ptolemy variety for representations that are not necessarily boundary-unipotent, and gives a formula for the Dehn invariant. The enhanced Ptolemy variety projects to the A-polynomial curve.
The Ptolemy field of 3-manifold representations
Stavros Garoufalidis, Matthias Goerner, Christian K. Zickert
This paper defines the Ptolemy field of a representation, and proves that it is equal to the trace field.
Ptolemy decoration
The symplectic properties of the PGL(n,C)-gluing equations
Stavros Garoufalidis, Christian K. Zickert
This paper proves that the PGL(n,C)-gluing equations satisfy symplectic properties analogous to the properties satisfied by Thurston's gluing equations. Some results were independently proved by Antonin Guilloux.
Generalized chain complex
Local rigidity for PGL(3,C)-representations of 3-manifold groups
Nicolas Bergeron, Elisha Falbel, Antonin Guilloux, Pierre-Vincent Koseleff, Fabrice Rouillier
Let M be a non-compact hyperbolic 3-manifold that has a triangulation 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" representation 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.
Complex hyperbolic geometry of the figure eight knot
Martin Deraux, Elisha Falbel
To be written.
Gluing equations for PGL(n,C)-representations of 3-manifolds
Stavros Garoufalidis, Matthias Goerner, Christian K. Zickert
This paper defines shape coordinates for representations in PGL(n,C). The shape coordinates are 3-dimensional analogues of the Fock-Goncharov X-coordinates, and satisfy relations generalizing Thurston's gluing equations for representations in PGL(2,C).
Gluing Equations
The complex volume of SL(n,C)-representations of 3-manifolds
Stavros Garoufalidis, Dylan P. Thurston, Christian K. Zickert
This paper defines Ptolemy coordinates for boundary-unipotent representations in SL(n,C). These coordinates are 3-dimensional analogues of the Fock-Goncharov A-coordinates. The main result is that a Ptolemy assignment determines an element in the extended Bloch group, and that Neumann's extension of Rogers dilogarithm computes the complex volume. Explicit formulas for a representation are given in terms of the coordinates, and the behavior of the coordinates under the canonical irreducible representation of SL(2,C) in SL(n,C) are given. The paper also proves that the Ptolemy variety detects the geometric representation whenever all edges are essential.
Ptolemy variety
Tetrahedra of flags, volume and homology of SL(3)
Nicolas Bergeron, Elisha Falbel, Antonin Guilloux
In the paper we define a "volume" for simplicial complexes of flag tetrahedra. This generalizes and unifies the classical volume of hyperbolic manifolds and the volume of CR tetrahedra complexes considered by Falbel and Falbel-Wang. We describe when this volume belongs to the Bloch group and more generally describe a variation formula in terms of boundary data. In doing so, we recover and generalize results of Neumann-Zagier, Neumann, and Kabaya. Our approach is very related to the work of Fock and Goncharov.
The extended Bloch group and algebraic K-theory
Christian K. Zickert
The main result of this paper is that the extended Bloch group of a number field is isomorphic to indecomposable algebraic K-theory. The proof uses rudimentary versions of the shape and Ptolemy coordinates developed in later papers.
Isomorphism K3 and Extended Bloch group
The volume and Chern-Simons invariant of a representation
Christian K. Zickert
This paper gives an explicit algorithm for computing the volume and Chern-Simons invariant of a hyperbolic 3-manifold, and more generally, a parabolic representation in PSL(2,C). The formula uses shape and Ptolemy coordinates for n=2.
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