scholarly journals On the character variety of the four-holed sphere

2015 ◽  
Vol 9 (3) ◽  
pp. 737-782 ◽  
Author(s):  
Sara Maloni ◽  
Frédéric Palesi ◽  
Ser Peow Tan
Keyword(s):  
Character Variety

scholarly journals A METHOD TO FIND IDEAL POINTS FROM IDEAL TRIANGULATIONS

2010 ◽  
Vol 19 (04) ◽  
pp. 509-524
Author(s):  
YUICHI KABAYA
Keyword(s):  
Linear Algebra ◽  
Character Variety ◽  
Ideal Triangulations ◽  
Ideal Points

We give a method to find ideal points of the character variety of a 3-manifold with toral boundary. This can be easily carried out by using linear algebra.

Toric Geometry ofSL2(ℂ) Free Group Character Varieties from Outer Space

2018 ◽  
Vol 70 (2) ◽  
pp. 354-399 ◽  
Author(s):  
Christopher Manon
Keyword(s):  
Free Group ◽  
Outer Space ◽  
Integrable Hamiltonian System ◽  
Outer Automorphism ◽  
Character Variety ◽  
Toric Geometry ◽  
Character Varieties ◽  
Simplicial Space ◽  
Proper Map ◽  
Divisorial Valuations

AbstractCuller and Vogtmann defined a simplicial spaceO(g), calledouter space, to study the outer automorphism group of the free groupFg. Using representation theoretic methods, we give an embedding ofO(g) into the analytification of X(Fg,SL2(ℂ)), theSL2(ℂ) character variety ofFg, reproving a result of Morgan and Shalen. Then we show that every pointvcontained in a maximal cell ofO(g) defines a flat degeneration of X(Fg,SL2(ℂ)) to a toric varietyX(PΓ). We relate X(Fg,SL2(ℂ)) andX(v) topologically by showing that there is a surjective, continuous, proper map Ξv:X(Fg,SL2(ℂ)) →X(v). We then show that this map is a symplectomorphism on a dense open subset of X(Fg, SL2(ℂ)) with respect to natural symplectic structures on X(Fg, SL2(ℂ)) andX(v). In this way, we construct an integrable Hamiltonian system in X(Fg, SL2(ℂ)) for each point in a maximal cell ofO(g), and we show that eachvdefines a topological decomposition of X(Fg, SL2(ℂ)) derived from the decomposition ofX(PΓ) by its torus orbits. Finally, we show that the valuations coming from the closure of a maximal cell inO(g) all arise as divisorial valuations built from an associated projective compactification of X(Fg, SL2(ℂ)).

scholarly journals Quantum computing and the brain: quantum nets, dessins d’enfants and neural networks

2019 ◽  
Vol 198 ◽  
pp. 00014
Author(s):  
Torsten Asselmeyer-Maluga
Keyword(s):  
Neural Networks ◽  
Quantum Computing ◽  
Feedback Loops ◽  
Character Variety ◽  
General Transformation ◽  
Closed Loops ◽  
Dessins D’Enfants ◽  
Topological Quantum ◽  
The Neural Network ◽  
Dessins D'enfants

In this paper, we will discuss a formal link between neural networks and quantum computing. For that purpose we will present a simple model for the description of the neural network by forming sub-graphs of the whole network with the same or a similar state. We will describe the interaction between these areas by closed loops, the feedback loops. The change of the graph is given by the deformations of the loops. This fact can be mathematically formalized by the fundamental group of the graph. Furthermore the neuron has two basic states |0〉 (ground state) and |1〉 (excited state). The whole state of an area of neurons is the linear combination of the two basic state with complex coefficients representing the signals (with 3 Parameters: amplitude, frequency and phase) along the neurons. If something changed in this area, we need a transformation which will preserve this general form of a state (mathematically, this transformation must be an element of the group S L(2; C)). The same argumentation must be true for the feedback loops, i.e. a general transformation of states along the feedback loops is an assignment of this loop to an element of the transformation group. Then it can be shown that the set of all signals forms a manifold (character variety) and all properties of the network must be encoded in this manifold. In the paper, we will discuss how to interpret learning and intuition in this model. Using the Morgan-Shalen compactification, the limit for signals with large amplitude can be analyzed by using quasi-Fuchsian groups as represented by dessins d’enfants (graphs to analyze Riemannian surfaces). As shown by Planat and collaborators, these dessins d’enfants are a direct bridge to (topological) quantum computing with permutation groups. The normalization of the signal reduces to the group S U(2) and the whole model to a quantum network. Then we have a direct connection to quantum circuits. This network can be transformed into operations on tensor networks. Formally we will obtain a link between machine learning and Quantum computing.

ON THE NUMBER OF IRREDUCIBLE COMPONENTS OF THE REPRESENTATION VARIETY OF A FAMILY OF ONE-RELATOR GROUPS

2010 ◽  
Vol 20 (01) ◽  
pp. 77-87 ◽  
Author(s):  
JORGE MARTÍN-MORALES ◽  
ANTONIO M. OLLER-MARCÉN
Keyword(s):  
Algebraic Variety ◽  
Character Variety ◽  
Representation Variety ◽  
Irreducible Components ◽  
One Relator Groups

Let us consider the group G = 〈x, y | xm = yn〉 with m and n nonzero integers. The set R(G) of representations of G over SL(2, ℂ) is a four-dimensional algebraic variety which is an invariant of G. In this paper the number of irreducible components of R(G) together with their dimensions are computed. We also study the set of metabelian representations of this family of groups. Finally, the behavior of the projection t : R(G) → X(G), where X(G) is the character variety of the group, and some combinatorial aspects of R(G) are investigated.

scholarly journals The mapping class group action on -character varieties

2020 ◽  
pp. 1-15
Author(s):  
WILLIAM M. GOLDMAN ◽  
SEAN LAWTON ◽  
EUGENE Z. XIA
Keyword(s):  
Group Action ◽  
Mapping Class Group ◽  
Boundary Component ◽  
Class Group ◽  
Orientable Surface ◽  
Mapping Class ◽  
Character Variety ◽  
Character Varieties ◽  
Compact Orientable Surface

Let $\unicode[STIX]{x1D6F4}$ be a compact orientable surface of genus $g=1$ with $n=1$ boundary component. The mapping class group $\unicode[STIX]{x1D6E4}$ of $\unicode[STIX]{x1D6F4}$ acts on the $\mathsf{SU}(3)$ -character variety of $\unicode[STIX]{x1D6F4}$ . We show that the action is ergodic with respect to the natural symplectic measure on the character variety.

THE CHARACTER VARIETY OF A FAMILY OF ONE-RELATOR GROUPS

2012 ◽  
Vol 23 (01) ◽  
pp. 1250015 ◽  
Author(s):  
KHALED QAZAQZEH
Keyword(s):  
Character Variety ◽  
Skein Module ◽  
Kauffman Bracket ◽  
Kauffman Bracket Skein Module ◽  
One Relator Groups

We prove that the character variety of a family of one-relator groups has only one defining polynomial and we provide the means to compute it. Consequently, we give a basis for the Kauffman bracket skein module of the exterior of the rational link Lp/q of two components modulo the (A + 1)-torsion.

scholarly journals Non-commutative trigonometry and the A-polynomial of the trefoil knot

2002 ◽  
Vol 133 (2) ◽  
pp. 311-323 ◽  
Author(s):  
RĂZVAN GELCA
Keyword(s):  
Character Variety ◽  
Skein Modules ◽  
Skein Module ◽  
Kauffman Bracket ◽  
Trefoil Knot ◽  
Sum Formula ◽  
Left And Right ◽  
Trefoil Knots ◽  
The Relationship ◽  
The Ideal

The non-commutative generalization of the A-polynomial of a knot of Cooper, Culler, Gillet, Long and Shalen [4] was introduced in [6]. This generalization consists of a finitely generated left ideal of polynomials in the quantum plane, the non- commutative A-ideal, and was defined based on Kauffman bracket skein modules, by deforming the ideal generated by the A-polynomial with respect to a parameter. The deformation was possible because of the relationship between the skein module with the variable t of the Kauffman bracket evaluated at −1 and the SL(2, C)-character variety of the fundamental group, which was explained in [2]. The purpose of the present paper is to compute the non-commutative A-ideal for the left- and right- handed trefoil knots. As will be seen below, this reduces to trigonometric operations in the non-commutative torus, the main device used being the product-to-sum formula for non-commutative cosines.

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