The reduced Dijkgraaf–Witten invariant of twist knots in the Bloch group of a finite field

2021 ◽  
pp. 2150014
Author(s):  
Hiroaki Karuo
Keyword(s):  
Finite Field ◽  
Discrete Group ◽  
Representation Formula ◽  
Witten Invariant ◽  
Fundamental Class ◽  
Hyperbolic Volume

Let [Formula: see text] be a closed oriented 3-manifold and let [Formula: see text] be a discrete group. We consider a representation [Formula: see text]. For a 3-cocycle [Formula: see text], the Dijkgraaf–Witten invariant is given by [Formula: see text], where [Formula: see text] is the map induced by [Formula: see text], and [Formula: see text] denotes the fundamental class of [Formula: see text]. Note that [Formula: see text], where [Formula: see text] is the map induced by [Formula: see text], we consider an equivalent invariant [Formula: see text], and we also regard it as the Dijkgraaf–Witten invariant. In 2004, Neumann described the complex hyperbolic volume of [Formula: see text] in terms of the image of the Dijkgraaf–Witten invariant for [Formula: see text] by the Bloch–Wigner map from [Formula: see text] to the Bloch group of [Formula: see text]. In this paper, by replacing [Formula: see text] with a finite field [Formula: see text], we calculate the reduced Dijkgraaf–Witten invariants of the complements of twist knots, where the reduced Dijkgraaf–Witten invariant is the image of the Dijkgraaf–Witten invariant for SL[Formula: see text] by the Bloch–Wigner map from [Formula: see text] to the Bloch group of [Formula: see text].

scholarly journals Quasi-flat representations of uniform groups and quantum groups

2019 ◽  
Vol 18 (08) ◽  
pp. 1950155 ◽  
Author(s):  
Teodor Banica ◽  
Alexandru Chirvasitu
Keyword(s):  
Quantum Groups ◽  
Discrete Group ◽  
Homogeneous Spaces ◽  
Model Space ◽  
Representation Formula ◽  
Discrete Groups ◽  
Universal Model ◽  
Induced Representation ◽  
Negative Results ◽  
Number Formula

Given a discrete group [Formula: see text] and a number [Formula: see text], a unitary representation [Formula: see text] is called quasi-flat when the eigenvalues of each [Formula: see text] are uniformly distributed among the [Formula: see text]th roots of unity. The quasi-flat representations of [Formula: see text] form altogether a parametric matrix model [Formula: see text]. We compute here the universal model space [Formula: see text] for various classes of discrete groups, notably with results in the case where [Formula: see text] is metabelian. We are particularly interested in the case where [Formula: see text] is a union of compact homogeneous spaces, and where the induced representation [Formula: see text] is stationary in the sense that it commutes with the Haar functionals. We present several positive and negative results on this subject. We also discuss similar questions for the discrete quantum groups, proving a stationarity result for the discrete dual of the twisted orthogonal group [Formula: see text].

scholarly journals Images of Galois representations in mod p Hecke algebras

2020 ◽  
pp. 1-21
Author(s):  
Laia Amorós
Keyword(s):  
Finite Field ◽  
Commutative Algebra ◽  
Galois Representations ◽  
Residue Field ◽  
Representation Formula ◽  
Galois Representation ◽  
Number Fields ◽  
Abelian Extensions ◽  
Hecke Eigenform ◽  
Coefficient Ring

Let [Formula: see text] denote the mod [Formula: see text] local Hecke algebra attached to a normalized Hecke eigenform [Formula: see text], which is a commutative algebra over some finite field [Formula: see text] of characteristic [Formula: see text] and with residue field [Formula: see text]. By a result of Carayol we know that, if the residual Galois representation [Formula: see text] is absolutely irreducible, then one can attach to this algebra a Galois representation [Formula: see text] that is a lift of [Formula: see text]. We will show how one can determine the image of [Formula: see text] under the assumptions that (i) the image of the residual representation contains [Formula: see text], (ii) [Formula: see text] and (iii) the coefficient ring is generated by the traces. As an application we will see that the methods that we use allow to deduce the existence of certain [Formula: see text]-elementary abelian extensions of big non-solvable number fields.

scholarly journals Cohomological invariants of representations of 3-manifold groups

2020 ◽  
pp. 2043003
Author(s):  
Haimiao Chen
Keyword(s):  
Abelian Group ◽  
Discrete Group ◽  
Representation Formula ◽  
Practical Method ◽  
Cohomological Invariants ◽  
Continuous Map ◽  
Definition Of ◽  
Manifolds With Corners ◽  
Chern Simons

Suppose [Formula: see text] is a discrete group, and [Formula: see text], with [Formula: see text] an abelian group. Given a representation [Formula: see text], with [Formula: see text] a closed 3-manifold, put [Formula: see text], where [Formula: see text] is a continuous map inducing [Formula: see text] which is unique up to homotopy, and [Formula: see text] is the pairing. We extend the definition of [Formula: see text] to manifolds with corners, and establish a gluing law. Based on these, we present a practical method for computing [Formula: see text] when [Formula: see text] is given by a surgery along a link [Formula: see text]. In particular, the Chern–Simons invariant can be computed this way.

The reduced Dijkgraaf–Witten invariant of double twist knots in the Bloch group of 𝔽p

2021 ◽  
Author(s):  
Hiroaki Karuo
Keyword(s):  
Finite Fields ◽  
Geometric Method ◽  
Witten Invariant ◽  
Hyperbolic Volume ◽  
Knot Complements ◽  
Free Fields

In 2004, Neumann showed that the complex hyperbolic volume of a hyperbolic 3-manifold [Formula: see text] can be obtained as the image of the Dijkgraaf–Witten invariant of [Formula: see text] by a certain 3-cocycle. After that, Zickert gave an analogue of Neumann’s work for free fields containing finite fields. The author formulated a geometric method to calculate a weaker version of Zickert’s analogue, called the reduced Dijkgraaf–Witten invariant, for finite fields and gave a formula for twist knot complements and [Formula: see text] in his previous work. In this paper, we show concretely how to calculate the reduced Dijkgraaf–Witten invariants of double twist knot complements and [Formula: see text], and give a formula of them for [Formula: see text].

scholarly journals A NOTE ON QUANTUM 3-MANIFOLD INVARIANTS AND HYPERBOLIC VOLUME

2009 ◽  
Vol 18 (01) ◽  
pp. 33-39
Author(s):  
EFSTRATIA KALFAGIANNI
Keyword(s):  
Witten Invariant ◽  
Hyperbolic Volume

For a closed oriented 3-manifold M and an integer r > 0, let τr(M) denote the SU(2) Reshetikhin–Turaev–Witten invariant of M at level r. We show that for every n > 0, and for r1,…, rn > 0 sufficiently large integers, there exist infinitely many non-homeomorphic hyperbolic 3-manifolds M, all of which have different hyperbolic volume, such that τri(M) = 1, for i = 1,…, n.

Beta dynamical systems over the formal fields

2014 ◽  
Vol 51 (4) ◽  
pp. 454-465
Author(s):  
Lu-Ming Shen ◽  
Huiping Jing
Keyword(s):  
Dynamical Systems ◽  
Finite Field ◽  
Haar Measure ◽  
Laurent Series ◽  
Formal Laurent Series ◽  
Almost All

Let \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{F}_q ((X^{ - 1} ))$$ \end{document} denote the formal field of all formal Laurent series x = Σ n=ν∞anX−n in an indeterminate X, with coefficients an lying in a given finite field \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{F}_q$$ \end{document}. For any \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\beta \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document} with deg β > 1, it is known that for almost all \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$x \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document} (with respect to the Haar measure), x is β-normal. In this paper, we show the inverse direction, i.e., for any x, for almost all \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\beta \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document}, x is β-normal.

Valued Field Graph and Some Related Parameters

2020 ◽  
pp. 389-395
Author(s):  
G. Suresh Singh ◽  
P. K. Prasobha
Keyword(s):  
Finite Field ◽  
Independence Number ◽  
Valued Field ◽  
Vertex Set ◽  
Adic Valuation

Let $K$ be any finite field. For any prime $p$, the $p$-adic valuation map is given by $\psi_{p}:K/\{0\} \to \R^+\bigcup\{0\}$ is given by $\psi_{p}(r) = n$ where $r = p^n \frac{a}{b}$, where $p,a,b$ are relatively prime. The field $K$ together with a valuation is called valued field. Also, any field $K$ has the trivial valuation determined by $\psi{(K)} = \{0,1\}$. Through out the paper K represents $\Z_q$. In this paper, we construct the graph corresponding to the valuation map called the valued field graph, denoted by $VFG_{p}(\Z_{q})$ whose vertex set is $\{v_0,v_1,v_2,\ldots, v_{q-1}\}$ where two vertices $v_i$ and $v_j$ are adjacent if $\psi_{p}(i) = j$ or $\psi_{p}(j) = i$. Here, we tried to characterize the valued field graph in $\Z_q$. Also we analyse various graph theoretical parameters such as diameter, independence number etc.

Image encryption based on the finite field cosine transform

2013 ◽  
Vol 28 (10) ◽  
pp. 1537-1547 ◽  
Author(s):  
J.B. Lima ◽  
E.A.O. Lima ◽  
F. Madeiro
Keyword(s):  
Finite Field ◽  
Image Encryption ◽  
Cosine Transform
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