Cyclotomic Research Articles

Lattice cohomology and q-series invariants of 3-manifolds

Abstract In this paper, an invariant is introduced for negative definite plumbed 3-manifolds equipped with a spin c {{}^{c}} -structure. It unifies and extends two theories with rather different origins and structures. One theory is lattice cohomology, motivated by the study of normal surface singularities, known to be isomorphic to the Heegaard Floer homology for certain classes of plumbed 3-manifolds. Another specialization gives BPS q-series which satisfy some remarkable modularity properties and recover SU ⁢ ( 2 ) {{\rm SU}(2)} quantum invariants of 3-manifolds at roots of unity. In particular, our work gives rise to a 2-variable refinement of the Z ^ {\widehat{Z}} -invariant.

Domestic violence as terrorism: Can control orders succeed where DV orders have failed?

Scholars have recently begun to explore the similarities between domestic violence and domestic terrorism. Both forms of offending rely on the use and threat of violence in conjunction with controlling behaviours to instil fear in their victims, and both forms of offending are predominantly committed by men. Yet despite these similarities, terrorism offences are treated as grave offences against the security of the nation-State, whilst domestic violence is not. The purpose of this paper therefore is to examine whether an amended control order framework – appropriately adapted to the unique criminological roots of domestic violence – might be a better mechanism for controlling these insidious crimes.

Learning knot invariants across dimensions

We use deep neural networks to machine learn correlations between knot invariants in various dimensions. The three-dimensional invariant of interest is the Jones polynomial J(q)J(q), and the four-dimensional invariants are the Khovanov polynomial \text{Kh}(q,t)Kh(q,t), smooth slice genus gg, and Rasmussen’s ss-invariant. We find that a two-layer feed-forward neural network can predict ss from \text{Kh}(q,-q^{-4})Kh(q,−q−4) with greater than 99%99% accuracy. A theoretical explanation for this performance exists in knot theory via the now disproven knight move conjecture, which is obeyed by all knots in our dataset. More surprisingly, we find similar performance for the prediction of ss from \text{Kh}(q,-q^{-2})Kh(q,−q−2), which suggests a novel relationship between the Khovanov and Lee homology theories of a knot. The network predicts gg from \text{Kh}(q,t)Kh(q,t) with similarly high accuracy, and we discuss the extent to which the machine is learning ss as opposed to gg, since there is a general inequality |s| ≤2g|s|≤2g. The Jones polynomial, as a three-dimensional invariant, is not obviously related to ss or gg, but the network achieves greater than 95%95% accuracy in predicting either from J(q)J(q). Moreover, similar accuracy can be achieved by evaluating J(q)J(q) at roots of unity. This suggests a relationship with SU(2)SU(2) Chern—Simons theory, and we review the gauge theory construction of Khovanov homology which may be relevant for explaining the network’s performance.

A fixed-point curve theorem for finite-orbits local diffeomorphisms

AbstractWe study local biholomorphisms with finite orbits in some neighborhood of the origin since they are intimately related to holomorphic foliations with closed leaves. We describe the structure of the set of periodic points in dimension 2. As a consequence we show that given a finite-orbits local biholomorphism F, in dimension 2, there exists an analytic curve passing through the origin and contained in the fixed-point set of some non-trivial iterate of $F.$ As an application we obtain that at least one eigenvalue of the linear part of F at the origin is a root of unity. Moreover, we show that such a result is sharp by exhibiting examples of finite-orbits local biholomorphisms such that exactly one of the eigenvalues is a root of unity. These examples are subtle since we show they cannot be embedded in one-parameter groups.

Type A DAHA and doubly periodic tableaux

Analogously to the construction of Suzuki and Vazirani, we construct representations of the GLm-type Double Affine Hecke Algebra at roots of unity. These representations are graded and the weight spaces for the X-variables are parametrized by the combinatorial objects we call doubly periodic tableaux. We show that our representations exhaust all graded X-semisimple representations, and the direct sum of all our representations is faithful. Analogously to the construction of Jordan and Vazirani of rectangular DAHA representations, we show that our representations can be interpreted in terms of ribbon fusion categories associated to Uq(glN) at roots of unity. Combining the ribbon structure with faithfulness we deduce a conjecture of Morton and Samuelson about realization of DAHA as a skein algebra of the torus with base string modulo certain local relations.

Exotic Lagrangian tori in Grassmannians

We describe an iterative construction of Lagrangian tori in the complex Grassmannian \operatorname{Gr}(k,n) , based on the cluster algebra structure of the coordinate ring of a mirror Landau–Ginzburg model proposed by Marsh and Rietsch (2020). Each torus comes with a Laurent polynomial, and local systems controlled by the k -variables Schur polynomials at the n -th roots of unity. We use this data to give examples of monotone Lagrangian tori that are neither displaceable nor Hamiltonian isotopic to each other, and that support nonzero objects in different summands of the spectral decomposition of the Fukaya category over \mathbb{C} .

Non-linearizability of power series over complete non-Archimedean fields of positive characteristic

In [7], Herman and Yoccoz prove that for any given locally analytic (at z=0) power series f(z)=z(λ+∑i=1∞aizi) over a complete non-Archimedean field of characteristic 0 if |λ|=1 and λ is not a root of unity, then f is locally linearizable at z=0. They ask the same question for power series over fields of positive characteristic.In this paper, we prove that, on opposite, most such power series in this case are more likely to be non-linearizable. More precisely, given a complete non-Archimedean field K of positive characteristic and a power series f(z)=z(λ+∑i=1∞aizi)∈K〚z〛 with λ not a root of unity and |1−λ|<1, we prove a sufficient condition (Criterion ★) for f to be non-linearizable. This phenomenon of prevalence for power series over fields of positive characteristic being non-linearizable was initially conjectured in [6, p 147] by Herman, and formulated into a concrete question by Lindahl as [9, Conjecture 2.2].As applications of Criterion ★, we prove the non-linearizability of three families of polynomials.

A note on the sum of finite multiple harmonic q-series on r-(r + 1) indices

We study the sum of the finite multiple harmonic q-series on r-(r + 1) indices at roots of unity with r = 1, 2, 3. Mainly, we give equivalent conditions of two conjectures regarding cyclic sums of finite multiple harmonic q-series on 1-2-3 indices at roots of unity, posed recently by Kh. Pilehrood, T. Pilehrood and R. Tauraso.

Analytical forms of the coefficients of the factored characteristic polynomials for zigzag SWCNT graphs

Factors of characteristic polynomials (CP) of zigzag single walled carbon nanotube (SWCNT) graphs have been constructed in generalized forms using rotational symmetry with respect to the tube axis. The CP coefficients in the factors have been expressed in analytical forms involving derivatives of different powers of pj, where pj=(1+ωj)(1+ωj*), ωj=0,1,2,…,(R−1), are the R-th roots of unity and R is the number of fused hexagonal rings per belt in the SWCNT graph. Some important results regarding total π-electron energy per electron, HOMO-LUMO energy gap and number of Kekulé structures for the SWCNTs have been derived from an analysis of the CPs and graph eigenspectra.

Mean square values of L-functions over subgroups for nonprimitive characters, Dedekind sums and bounds on relative class numbers

AbstractAn explicit formula for the mean value of $\vert L(1,\chi )\vert ^2$ is known, where $\chi $ runs over all odd primitive Dirichlet characters of prime conductors p. Bounds on the relative class number of the cyclotomic field ${\mathbb Q}(\zeta _p)$ follow. Lately, the authors obtained that the mean value of $\vert L(1,\chi )\vert ^2$ is asymptotic to $\pi ^2/6$ , where $\chi $ runs over all odd primitive Dirichlet characters of prime conductors $p\equiv 1\ \ \pmod {2d}$ which are trivial on a subgroup H of odd order d of the multiplicative group $({\mathbb Z}/p{\mathbb Z})^*$ , provided that $d\ll \frac {\log p}{\log \log p}$ . Bounds on the relative class number of the subfield of degree $\frac {p-1}{2d}$ of the cyclotomic field ${\mathbb Q}(\zeta _p)$ follow. Here, for a given integer $d_0&gt;1$ , we consider the same questions for the nonprimitive odd Dirichlet characters $\chi '$ modulo $d_0p$ induced by the odd primitive characters $\chi $ modulo p. We obtain new estimates for Dedekind sums and deduce that the mean value of $\vert L(1,\chi ')\vert ^2$ is asymptotic to $\frac {\pi ^2}{6}\prod _{q\mid d_0}\left (1-\frac {1}{q^2}\right )$ , where $\chi $ runs over all odd primitive Dirichlet characters of prime conductors p which are trivial on a subgroup H of odd order $d\ll \frac {\log p}{\log \log p}$ . As a consequence, we improve the previous bounds on the relative class number of the subfield of degree $\frac {p-1}{2d}$ of the cyclotomic field ${\mathbb Q}(\zeta _p)$ . Moreover, we give a method to obtain explicit formulas and use Mersenne primes to show that our restriction on d is essentially sharp.

Unification of Chowla’s Problem and Maillet–Demyanenko Determinants

Chowla’s (inverse) problem (CP) is to mean a proof of linear independence of cotangent-like values from non-vanishing of L(1,χ)=∑n=1∞χ(n)n. On the other hand, we refer to determinant expressions for the (relative) class number of a cyclotomic field as the Maillet–Demyanenko determinants (MD). Our aim is to develop the theory of discrete Fourier transforms (DFT) with parity and to unify Chowla’s problem and Maillet–Demyanenko determinants (CPMD) as different-looking expressions of the relative class number via the Dedekind determinant and the base change formula.

Quantum invariants of three-manifolds obtained by surgeries along torus knots

We study the asymptotic behavior of the Witten–Reshetikhin–Turaev invariant associated with the square of the n -th root of unity with odd n for a Seifert fibered space obtained by an integral Dehn surgery along a torus knot. We show that it can be described as a sum of the Chern–Simons invariants and the twisted Reidemeister torsions both associated with representations of the fundamental group to the two-dimensional complex special linear group.

Ternary algebras associated with irreducible tensor representations of SO(3) and the quark model

We show that each irreducible tensor representation of weight 2 of the rotation group in the space of rank 3 tensors over three-dimensional space gives rise to an associative algebra with unity. We find the algebraic conditions to be satisfied by the generators of these algebras. Part of these relations is of binary, and another part is of ternary type. The structure of the latter one is based on the use of the cyclic group [Formula: see text] generated by the primitive cubic root of unity given by [Formula: see text]. The subspace of each algebra spanned by triple products of generators is five-dimensional (5D) and is identical with the space of irreducible tensor representation of weight 2 of the rotation group [Formula: see text]. We define a Hermitian scalar product in this 5D subspace and construct its orthonormal basis in terms of triple products of generators. Then we find an explicit formula for the Lie algebra homomorphism [Formula: see text]. We suggest that the algebras constructed in this way, with binary and ternary constitutive relations, may find applications in the quark model and the Grand Unified Theories.

Not Always Helpful: Linking Intrateam Helping Types to Team Effectiveness From a Role Theory Perspective

Existing research on intrateam helping has predominantly taken a positive view on its impact on teams, overlooking the potential negative consequences that helping may elicit. To account for divergent implications of intrateam helping, we differentiate two types of helping that can manifest during teamwork: team autonomous helping and team dependent helping. Integrating this dual-type view of helping with a role theory perspective on teamwork, we propose a theoretical model that delineates the relationships among team goal orientations, two types of intrateam helping, team role-based functions (i.e., team role overload, team role coordination, and team role breadth self-efficacy), and team effectiveness. To test our model, we conducted two multi-wave survey studies, including 110 student project teams (Study 1) and 80 manufacturing teams in a pharmaceutical company (Study 2). Overall, the results showed that team autonomous helping benefited team role-based functions and ultimately team effectiveness, whereas team dependent helping hindered them. We also found that team learning goal orientation drove the occurrence of team autonomous helping in both studies, while team performance goal orientation drove team dependent helping in Study 1. By distinguishing between autonomous and dependent types of intrateam helping, and examining their unique motivational roots and divergent implications for teams, this research advances the extant literature by providing a more balanced account of the nature of helping in teams.

Ramification filtration and differential forms

Let $L$ be a complete discrete valuation field of prime characteristic $p$ with finite residue field. Denote by $\Gamma_{L}^{(v)}$ the ramification subgroups of $\Gamma_{L}=\operatorname{Gal}(L^{\mathrm{sep}}/L)$. We consider the category $\operatorname{M\Gamma}_{L}^{\mathrm{Lie}}$ of finite $\mathbb{Z}_p[\Gamma_{L}]$-modules $H$, satisfying some additional (Lie)-condition on the image of $\Gamma_L$ in $\operatorname{Aut}_{\mathbb{Z}_p}H$. In the paper it is proved that all information about the images of the groups $\Gamma_L^{(v)}$ in $\operatorname{Aut}_{\mathbb{Z}_p}H$ can be explicitly extracted from some differential forms $\widetilde{\Omega} [N]$ on the Fontaine etale $\phi $-module $M(H)$ associated with $H$. The forms $\widetilde{\Omega}[N]$ are completely determined by a canonical connection $\nabla $ on $M(H)$. In the case of fields $L$ of mixed characteristic, which contain a primitive $p$th root of unity, we show that a similar problem for $\mathbb{F}_p[\Gamma_L]$-modules also admits a solution. In this case we use the field-of-norms functor to construct the corresponding $\phi $-module together with the action of the Galois group of a cyclic extension $L_1$ of $L$ of degree $p$. Then our solution involves the characteristic $p$ part (provided by the field-of-norms functor) and the condition for a "good" lift of a generator of $\operatorname{Gal}(L_1/L)$. Apart from the above differential forms the statement of this condition uses the power series coming from the $p$-adic period of the formal group $\mathbb{G}_m$.

Ramification filtration and differential forms

Let $L$ be a complete discrete valuation field of prime characteristic $p$ with finite residue field. Denote by $\Gamma_{L}^{(v)}$ the ramification subgroups of $\Gamma_{L}=\operatorname{Gal}(L^{\mathrm{sep}}/L)$. We consider the category $\operatorname{M\Gamma}_{L}^{\mathrm{Lie}}$ of finite $\mathbb{Z}_p[\Gamma_{L}]$-modules $H$, satisfying some additional (Lie)-condition on the image of $\Gamma_L$ in $\operatorname{Aut}_{\mathbb{Z}_p}H$. In the paper it is proved that all information about the images of the groups $\Gamma_L^{(v)}$ in $\operatorname{Aut}_{\mathbb{Z}_p}H$ can be explicitly extracted from some differential forms $\widetilde{\Omega} [N]$ on the Fontaine etale $\phi $-module $M(H)$ associated with $H$. The forms $\widetilde{\Omega}[N]$ are completely determined by a canonical connection $\nabla $ on $M(H)$. In the case of fields $L$ of mixed characteristic, which contain a primitive $p$th root of unity, we show that a similar problem for $\mathbb{F}_p[\Gamma_L]$-modules also admits a solution. In this case we use the field-of-norms functor to construct the corresponding $\phi $-module together with the action of the Galois group of a cyclic extension $L_1$ of $L$ of degree $p$. Then our solution involves the characteristic $p$ part (provided by the field-of-norms functor) and the condition for a "good" lift of a generator of $\operatorname{Gal}(L_1/L)$. Apart from the above differential forms the statement of this condition uses the power series coming from the $p$-adic period of the formal group $\mathbb{G}_m$. Bibliography: 21 titles.

Irreducible Radical Field Extensions

This paper is devoted to irreducible radical field extensions, i.e., extensions that can be obtained by adjunction of roots of irreducible binomials. We find a criterion for cyclotomic extensions to be irreducible radical. This criterion developes the Gauss–Wantzel theorem about constructible polygons. We also prove that any normal solvable extension of some field K is irreducible radical until K has all roots of unity. This generalizes Abel’s theorem, which fills the gap in the uncomplete Ruffini proof of the impossibility theorem for the general equation of degree five or higher. Finally, we prove that the root set of any irreducible solvable polynomial coincides with the value set of some radical formula using irreducible radical extensions.

Bloch groups, algebraic $K$-theory, units, and Nahm's Conjecture

Given an element of the Bloch group of a number field~$F$ and a natural number~$n$, we construct an explicit unit in the field $F_n=F(e^{2 \pi i/n})$, well-defined up to $\nn$-th powers of nonzero elements of~$F_n$. The construction uses the cyclic quantum dilogarithm, and under the identification of the Bloch group of~$F$ with the $K$-group $K_3(F)$ gives \changed{(up to an unidentified invertible scalar)} a \changed{formula} for a certain abstract Chern class from~$K_3(F)$. The units we define are conjectured to coincide with numbers appearing in the quantum modularity conjecture for the Kashaev invariant of knots (which was the original motivation for our investigation), and also appear in the radial asymptotics of Nahm sums near roots of unity. This latter connection is used to prove Nahm's conjecture relating the modularity of certain $q$-hypergeometric series to the vanishing of the associated elements in the Bloch group of~$\overline{\mathbb Q}$.

Intertextuality in the contemporary Kabyle novel: The state-of-the-art

This article delves into the phenomenon of intertextuality in the Kabyle novel, focusing on its genesis as a literary genre and the dialogic strategies employed by Kabyle authors. The 20th century witnessed a decline in Kabyle literature due to societal changes. However, a resurgence with the emergence of Kabyle novelists who skillfully incorporated oral cultural heritage into their literary works. Notably, the first written works in prose, exemplified by Belaid At-Ali's Lwali n Udrar, signified a poetic rejuvenation in Kabyle literature. Subsequently, the 1970s witnessed the rise of the Kabyle novel, influenced by the establishment of publishing houses and the imperative to express Amazigh identity. Distinguished by its unique themes, forms, geographical specificity, and cultural roots, the Kabyle novel ingeniously incorporates oral traditions, such as proverbs and popular tales, through the art of intertextuality. The integration of oral traditions serves multiple purposes, including the preservation of Kabyle society's values, the promotion of the aesthetic significance of the Kabyle language, and the expression of personal viewpoints. Kabyle novelists employ diverse approaches to incorporate proverbs, either seamlessly blending them into the narrative fabric or demarcating them with quotation marks. Likewise, popular tales are artfully interwoven into the novels, with each author offering their distinctive interpretation. The interaction with oral tradition manifests itself through the use of narrative anachronism, references to legendary figures, and the incorporation of traditional symbolism. These elements contribute significantly to the cultural and aesthetic dimensions of Kabyle novels.

Generic Gelfand-Tsetlin modules of quantized and classical orthogonal algebras

We construct infinite-dimensional analogues of finite-dimensional simple modules of the nonstandard q-deformed enveloping algebra Uq′(son) defined by Gavrilik and Klimyk, and we do the same for the classical universal enveloping algebra U(son). In this paper we only consider the case when q is not a root of unity, and q→1 for the classical case. Extending work by Mazorchuk on son, we provide rational matrix coefficients for these infinite-dimensional modules of both Uq′(son) and U(son). We use these modules with rationalized formulas to embed the respective algebras into skew group algebras of shift operators. Casimir elements of Uq′(son) were given by Gavrilik and Iorgov, and we consider the commutative subalgebra Γ⊂Uq′(son) generated by these elements and the corresponding subalgebra Γ1⊂U(son). The images of Γ and Γ1 under their respective embeddings into skew group algebras are equal to invariant algebras under certain group actions. We use these facts to show that Γ is a Harish-Chandra subalgebra of Uq′(son) and Γ1 is a Harish-Chandra subalgebra of U(son).