arXiv subjects

Christian K. Zickert

Publications and source records attributed to Christian K. Zickert.

13 recordsLinked to original sources

arXiv2023

Holomorphic polylogarithms and Bloch complexes

For an integer n>2 we define a polylogarithm, which is a holomorphic function on the universal abelian cover of C-{0,1} defined modulo (2 pi i)^n/(n-1)!. We use the formal properties of its functional relations to define groups lifting Goncharov's Bloch groups of a field F, and show that they fit into a complex lifting Goncharov's Bloch complex. When F=C we show that the imaginary part (when n is even) or real part (when n is odd) of the holomorphic polylogarithm agrees with a real valued polylogarithm on the first cohomology group of the lifted Bloch complex. When n=2, this group is Neumann's extended Bloch group. Goncharov's complex conjecturally computes the rational motivic cohomology of F, and one may speculate whether the lifted complex computes the integral motivic cohomology. Finally, we construct a lift of Goncharov's real valued map on the 5th homology of SL(3,C) to a complex valued map. The lift makes use of the cluster ensemble structure on the Grassmannian Gr(3,6).

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arXiv2022

Hopf algebras of multiple polylogarithms, and holomorphic 1-forms

We associate to a multiple polylogarithm a holomorphic 1-form on the universal abelian cover of its domain. We relate the 1-forms to the symbol and variation matrix and show that the 1-forms naturally define a lift of the variation of mixed Hodge structure associated to a polylogarithm. The results are conveniently described in terms of a variant H of Goncharov's Hopf algebra of multiple polylogarithms. In particular, we show that the association of a 1-form to a multiple polylogarithm induces a map from the Chevalley-Eilenberg complex of the Lie coalgebra of indecomposables of H to the de Rham complex.

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arXiv2022

The Lie coalgebra of multiple polylogarithms

We use Goncharov's coproduct of multiple polylogarithms to define a Lie coalgebra over an arbitrary field. It is generated by symbols subject to inductively defined relations, which we think of as functional relations for multiple polylogarithms. In particular, we have inversion relations and shuffle relations. We relate our definition to Goncharov's Bloch groups, and to the concrete model in weight less than 5 by Goncharov and Rudenko.

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arXiv2019

On the Hikami-Inoue conjecture

Given a braid presentation $D$ of a hyperbolic knot, Hikami and Inoue consider a system of polynomial equations arising from a sequence of cluster mutations determined by $D$. They show that any solution gives rise to shape parameters and thus determines a boundary-parabolic $\mathrm{PSL}(2,\mathbb{C})$-representation of the knot group. They conjecture the existence of a solution corresponding to the geometric representation. In this paper, we show that a boundary-parabolic representation $ρ$ arises from a solution if and only if the length of $D$ modulo $2$ equals the obstruction to lifting $ρ$ to a boundary-parabolic $\mathrm{SL}(2,\mathbb{C})$-representation (as an element in $\mathbb{Z}_2$). In particular, the Hikami-Inoue conjecture holds if and only if the length of $D$ is odd. This can always be achieved by adding a kink to the braid if necessary. We also explicitly construct the solution corresponding to a boundary-parabolic representation given in the Wirtinger presentation of the knot group.

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arXiv2017

Triangulation Independent Ptolemy Varieties

The Ptolemy variety for 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 variety. The Ptolemy variety maps naturally to the set of conjugacy classes of boundary-unipotent SL(2,C)-representations, but (like the gluing equation variety) it depends on the triangulation, and may miss several components of representations. In this paper, we define a Ptolemy variety, which is independent of the choice of triangulation, and detects all boundary-unipotent irreducible SL(2,C)-representations. We also define variants of the Ptolemy variety for PSL(2,C)-representations, and representations that are not necessarily boundary-unipotent. In particular, we obtain an algorithm to compute all irreducible SL(2,C)-characters as well as the full A-polynomial. All the varieties are topological invariants of M.

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arXiv2017

Fock-Goncharov coordinates for rank two Lie groups

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 action on triples by orientation preserving permutations corresponds to explicit quiver mutations, and that the same holds for the flip (changing the diagonal in a quadrilateral). This gives explicit coordinates on higher Teichmuller space, and also coordinates for boundary-unipotent representations of 3-manifold groups. As an application, we compute the (generic) boundary-unipotent representations in Sp(4,C) for the figure-eight knot complement.

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arXiv2014

Ptolemy coordinates, Dehn invariant, and the A-polynomial

We define Ptolemy coordinates for representations that are not necessarily boundary-unipotent. This gives rise to a new algorithm for computing the SL(2,C) A-polynomial, and more generally the SL(n,C) A-varieties. We also give a formula for the Dehn invariant of an SL(n,C)-representation.

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arXiv2014

The Ptolemy field of $3$-manifold-representations

The Ptolemy coordinates for boundary-unipotent SL(n,C)-representations of a 3-manifold group were introduced in Garoufalidis-Thurston-Zickert inspired by the A-coordinates on higher Teichmüller space due to Fock and Goncharov. In this paper, we define the Ptolemy field of a (generic) PSL(2,\C)-representation and prove that it coincides with the trace field of the representation. This gives an efficient algorithm to compute the trace field of a cusped hyperbolic manifold.

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arXiv2013

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 SL(n,C). Our parametrization uses Ptolemy coordinates, which are inspired by coordinates on higher Teichmueller spaces due to Fock and Goncharov. We show that a boundary-unipotent representation determines an element in Neumann's extended Bloch group, and use this to obtain an efficient formula for the Cheeger-Chern-Simons invariant, and in particular for the volume. Computations for the census manifolds show that boundary-unipotent representations are abundant, and numerical comparisons with census volumes, suggest that the volume of a representation is an integral linear combination of volumes of hyperbolic 3-manifolds. This is in agreement with a conjecture of Walter Neumann, stating that the Bloch group is generated by hyperbolic manifolds.

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arXiv2013

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 equations that play an important role in recent developments of exact and perturbative Chern-Simons theory. In this paper, we prove the symplectic properties of the PGL(n,C)-gluing equations for all ideal triangulations of compact oriented 3-manifolds.

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arXiv2012

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

In a previous paper, we parametrized boundary-unipotent representations of a 3-manifold group into SL(n,C) using Ptolemy coordinates, which were inspired by A-coordinates on higher Teichmüller space due to Fock and Goncharov. In this paper, we parametrize representations into PGL(n,C) using shape coordinates which are a 3-dimensional analogue of Fock and Goncharov's X-coordinates. These coordinates satisfy equations generalizing Thurston's gluing equations. These equations are of Neumann-Zagier type and satisfy symplectic relations with applications in quantum topology. We also explore a duality between the Ptolemy coordinates and the shape coordinates.

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arXiv2010

The extended Bloch group and algebraic K-theory

We define an extended Bloch group for an arbitrary field F, and show that this group is canonically isomorphic to K_3^ind(F) if F is a number field. This gives an explicit description of K_3^ind(F) in terms of generators and relations. We give a concrete formula for the regulator, and derive concrete symbol expressions generating the torsion. As an application, we show that a hyperbolic 3-manifold with finite volume and invariant trace field k has a fundamental class in K_3^ind(k) tensor Z[1/2].

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arXiv2009

The volume and Chern-Simons invariant of a representation

We give an efficient simplicial formula for the volume and Chern-Simons invariant of a boundary-parabolic PSL(2,C)-representation of a tame 3-manifold. If the representation is the geometric representation of a hyperbolic 3-manifold, our formula computes the volume and Chern-Simons invariant directly from an ideal triangulation with no use of additional combinatorial topology. In particular, the Chern-Simons invariant is computed just as easily as the volume.

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