Ptolemy coordinates, Dehn invariant and the A-polynomial

@article{Zickert2014PtolemyCD,
  title={Ptolemy coordinates, Dehn invariant and the A-polynomial},
  author={Christian K. Zickert},
  journal={Mathematische Zeitschrift},
  year={2014},
  volume={283},
  pages={515-537}
}
  • C. Zickert
  • Published 30 April 2014
  • Mathematics
  • Mathematische Zeitschrift
We define Ptolemy coordinates for representations that are not necessarily boundary-unipotent. This gives rise to a new algorithm for computing the $${{\mathrm{SL}}}(2,{\mathbb {C}})\;A$$SL(2,C)A-polynomial, and more generally the $${{\mathrm{SL}}}(n,{\mathbb {C}})\;A$$SL(n,C)A-varieties. We also give a formula for the Dehn invariant of an $${{\mathrm{SL}}}(n,{\mathbb {C}})$$SL(n,C)-representation. 

Triangulation independent Ptolemy varieties

The Ptolemy variety for $${{\mathrm{SL}}}(2,{\mathbb {C}})$$SL(2,C) is an invariant of a topological ideal triangulation of a compact 3-manifold M. It is closely related to Thurston’s gluing equation

On SL(3,$${\varvec{\mathbb {C}}}$$C)-representations of the Whitehead link group

We describe a family of representations in SL(3,$$\mathbb {C}$$C) of the fundamental group $$\pi $$π of the Whitehead link complement. These representations are obtained by considering pairs of

Fock-Goncharov coordinates for rank two Lie groups

  • C. Zickert
  • Mathematics
    Mathematische Zeitschrift
  • 2019
Let G be a simply connected, simple, complex Lie group of rank 2. We give explicit Fock-Goncharov coordinates for configurations of triples and quadruples of affine flags in G . We show that the

A-polynomials, Ptolemy equations and Dehn filling

TLDR
This work compute A-polynomials by starting with a triangulation of a manifold, then using symplectic properties of the NeumannZagier matrix encoding the gluings to change the basis of the computation, and the result is a simplicifation of the defining equations.

A-polynomials, Ptolemy varieties and Dehn filling

TLDR
This work shows how to compute A-polynomials by starting with a triangulation of a manifold, then using symplectic properties of the Neumann-Zagier matrix encoding the gluings to change the basis of the computation, and shows that the result is a simplicifation of the defining equations.

The volume and Chern–Simons invariant of a Dehn-filled manifold

Twisted Neumann--Zagier matrices

The Neumann–Zagier matrices of an ideal triangulation are integer matrices with symplectic properties whose entries encode the number of tetrahedra that wind around each edge of the triangulation.

Character varieties for SL(3,C): the figure eight knot

We 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

Character Varieties For : The Figure Eight Knot

TLDR
A description of several representation varieties of the fundamental group of the complement of the figure eight knot in PGL or PSL and the projection of the representation variety into the character variety of the boundary torus into SL.

On the volume and the Chern-Simons invariant for the hyperbolic alternating knot orbifolds

We extend the Neumann's methods and give the explicit formulae for the volume and the Chern-Simons invariant for hyperbolic alternating knot orbifolds.

References

SHOWING 1-10 OF 15 REFERENCES

Representations of fundamental groups of 3-manifolds into $$\mathrm{PGL}(3,\mathbb {C})$$PGL(3,C): exact computations in low complexity

In this paper we are interested in computing representations of the fundamental group of SL 3-manifold into $$\mathrm{PGL}(3,\mathbb {C})$$PGL(3,C) (in particular in $$\mathrm{PGL}(2,\mathbb {C}),

K-decompositions and 3d gauge theories

A bstractThis paper combines several new constructions in mathematics and physics. Mathematically, we study framed flat PGL(K, ℂ)-connections on a large class of 3-manifolds M with boundary. We

A-polynomial and Bloch invariants of hyperbolic 3-manifolds

Let N be a complete, orientable, finite-volume, one-cusped hyperbolic 3-manifold with an ideal triangulation. Using combinatorics of the ideal triangulation of N we construct a plane curve in C×C

Gluing equations for PGL(n, ℂ)–representations of 3–manifolds

Garoufalidis, Thurston and Zickert parametrized boundary-unipotent representations of a 3‐manifold group into SL.n;C/ using Ptolemy coordinates, which were inspired by A‐coordinates on higher

Plane curves associated to character varieties of 3-manifolds

Consider a compact 3-manifold M with boundary consisting of a single torus. The papers [CS1, CS2, CGLS] discuss the variety of characters of SL2(C) representations of zl(M), and some of the ways in

The complex volume of SL(n,C)-representations of 3-manifolds

For a compact 3-manifold M with arbitrary (possibly empty) boundary, we give a parametrization of the set of conjugacy classes of boundary-unipotent representations of the fundamental group of M into

REPRESENTATIONS OF 3-MANIFOLDS GROUPS IN PGL(n, C) AND THEIR RESTRICTION TO THE BOUNDARY

Let M be a cusped 3-manifold – e.g. a knot complement – and note ∂M the collection of its peripheral tori. Thurston [Thu79] gave a combinatorial way to produce hyperbolic structures via triangulation

Tetrahedra of flags, volume and homology of SL.3/

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 tetrahedral complexes

Moduli spaces of local systems and higher Teichmüller theory

Let G be a split semisimple algebraic group over Q with trivial center. Let S be a compact oriented surface, with or without boundary. We define positive representations of the fundamental group of S

THE SYMPLECTIC PROPERTIES OF THE PGL(n;C)-GLUING EQUATIONS

In a previous article we studied PGL(n,C)-representations of a 3-manifold via a generalization of Thurston's gluing equations. Neumann has proved some symplectic properties of Thurston's gluing