Polynomial Invariants Research Articles

The writhe and first intersection polynomials of virtual knots of supporting genus one

The supporting genus of a virtual knot is the minimal genus of all thickened surfaces where the virtual knot lies. In this paper, we study several polynomial invariants of a virtual knot of supporting genus one. We show that the first intersection polynomial of any virtual knot of supporting genus one is reciprocal. We also give a characterization of the writhe polynomial of a virtual knot of supporting genus one.

Universality for polynomial invariants for ribbon graphs with half-ribbons

In this paper, we analyze the Bollob\'as and Riordan polynomial $\mathcal{R}$ for ribbon graphs with half-ribbons introduced in [Combinatorics, Probability and Computing 31, 507-549, 2022]. We prove the universality property of a multivariate version of $\mathcal{R}$ whereas $\mathcal{R}$ itself turns out to be universal for a subclass of ribbon graphs with half-ribbons. We also show that $\mathcal{R}$ can be defined on some equivalence classes of ribbon graphs involving half-ribbons moves and that the new polynomial is universal on these classes.

Revisiting theN(N + 1)/2‐sites‐type Gaussian charge model for permutationally invariant polynomial fitting of global molecular tensor surfaces

AbstractThe exact expressions for the dipole, quadrupole, and octupoles of a collection ofNpoint charges involve summations of corresponding tensors over theNsites weighted by their charge magnitudes. When the point charges are atoms (in a molecule) theN‐site formula is an approximation, and one must integrate over the electron density to recover the exact multipoles. In the present work we revisit theN(N + 1)/2‐site point charge density model of Hall (Chem. Phys. Lett.6, 501, 1973) for the purpose of fitting ab initio derived multipole moment hypersurfaces using permutationally invariant polynomials (PIP). We examine new approaches in PIP‐fitting procedures for the dipole, quadrupole, octupole moments, and polarizability tensor surfaces (DMS, QMS, OMS and PTS, respectively) for a non‐polar CCl4and a polar CHCl3and show that compared to the primitiveN‐site model theN(N + 1)/2‐site model appreciably improves the relative RMSE of the DMS and does much more substantially so, by an order of magnitude, for the corresponding ones of QMS and OMS. Training datasets are obtained by sampling potential energies up to 18 000 cm−1above the global minima, generated by molecular dynamics simulations at the DFT B3LYP/aug‐cc‐pVDZ level of theory.

Index polynomial invariants of virtual knots using invariants for flat virtual knots

In this paper, we introduce new index polynomial invariants for virtual knots. They are defined by using invariants of flat virtual knots. And we give some properties of these invariants and examples.

Orientable maps and polynomial invariants of free-by-cyclic groups

We relate the McMullen polynomial of a free-by-cyclic group to its Alexander polynomial. To do so, we introduce the notion of an orientable fully irreducible outer automorphism φ and use it to characterize when the homological stretch factor of φ is equal to its geometric stretch factor.

Bilinear character correlators in superintegrable theory

We continue investigating the superintegrability property of matrix models, i.e. factorization of the matrix model averages of characters. This paper focuses on the Gaussian Hermitian example, where the role of characters is played by the Schur functions. We find a new intriguing corollary of superintegrability: factorization of an infinite set of correlators bilinear in the Schur functions. More exactly, these are correlators of products of the Schur functions and polynomials K_Delta that form a complete basis in the space of invariant matrix polynomials. Factorization of these correlators with a small subset of these K_Delta follow from the fact that the Schur functions are eigenfunctions of the generalized cut-an-join operators, but the full set of K_Delta is generated by another infinite commutative set of operators, which we manifestly describe.

PESPIP: Software to fit complex molecular and many-body potential energy surfaces with permutationally invariant polynomials.

We wish to describe a potential energy surface by using a basis of permutationally invariant polynomials whose coefficients will be determined by numerical regression so as to smoothly fit a dataset of electronic energies as well as, perhaps, gradients. The polynomials will be powers of transformed internuclear distances, usually either Morse variables, exp(-ri,j/λ), where λ is a constant range hyperparameter, or reciprocals of the distances, 1/ri,j. The question we address is how to create the most efficient basis, including (a) which polynomials to keep or discard, (b) how many polynomials will be needed, (c) how to make sure the polynomials correctly reproduce the zero interaction at a large distance, (d) how to ensure special symmetries, and (e) how to calculate gradients efficiently. This article discusses how these questions can be answered by using a set of programs to choose and manipulate the polynomials as well as to write efficient Fortran programs for the calculation of energies and gradients. A user-friendly interface for access to monomial symmetrization approach results is also described. The software for these programs is now publicly available.

Anisotropic structure of two-dimensional linear Cosserat elasticity

In the present contribution the anisotropic structure of the two-dimensional linear Cosserat elasticity is investigated. The symmetry classes of this model are derived and detailed in a synthetic way. Particular attention is paid to specific features of Cosserat Elasticity which are the sensitivity to non-centrosymmetry as well as to chirality. These aspects are important for the application of this continuum theory to the mechanical modelling of lattices and metamaterials. In order to give a parameterisation to the Cosserat constitutive law, an explicit harmonic decomposition of its constitutive tensors is provided. Finally, using an algorithm introduced in a side paper, a minimal integrity basis, which is the minimal set of polynomial invariants generating the algebra of O(2)-invariant polynomials, is finally reported.

The separating variety for 2 × 2 matrix invariants

Let G be a linear algebraic group acting linearly on a vector space (or more generally, an affine variety) V , and let k [ V ] G be the corresponding algebra of invariant polynomial functions. A separating set S ⊆ k [ V ] G is a set of polynomials with the property that for all v , w ∈ V , if there exists f ∈ k [ V ] G separating $v$ and $w$, then there exists f ∈ S separating $v$ and $w$. In this article, we consider the action of G = GL 2 ⁡ ( C ) on the C -vector space M 2 n of n-tuples of 2 × 2 matrices by simultaneous conjugation. Minimal generating sets S n of C [ M 2 n ] G are well known and | S n | = 1 6 ( n 3 + 11 n ) . In recent work, Kaygorodov et al. [Kaygorodov I, Lopatin A, Popov Y. Separating invariants for 2 × 2 matrices. Linear Algebra Appl. 2018;559:114-124.] showed that for all n ≥ 1 , S n is a minimal separating set by inclusion, i.e. that no proper subset of S n is a separating set. This does not necessarily mean that S n has minimum cardinality among all separating sets for C [ M 2 n ] G . Our main result shows that any separating set for C [ M 2 n ] G has cardinality ≥ 5 n − 5 . In particular, there is no separating set of size dim ⁡ ( C [ M 2 n ] G ) = 4 n − 3 for n ≥ 3 . Further, S 3 has indeed minimum cardinality as a separating set, but for n ≥ 4 there may exist a smaller separating set than S n . We show that a smaller separating set does in fact exist for all n ≥ 5 . We also prove similar results for the left–right action of SL 2 ⁡ ( C ) × SL 2 ⁡ ( C ) on M 2 n .

A relation among tangle, 3-tangle, and von Neumann entropy of entanglement for three qubits

In this paper, we derive a general formula of the tangle for pure states of three qubits and present three explicit local unitary (LU) polynomial invariants. Our result goes beyond the classical work of tangle, 3-tangle, and von Neumann entropy of entanglement for Acín et al.’ Schmidt decomposition (ASD) of three qubits by connecting the tangle, 3-tangle, and von Neumann entropy for ASD with Acín et al.’s LU invariants. In particular, our result reveals a general relation among tangle, 3-tangle, and von Neumann entropy, together with a relation among their averages. The relations can help us find the entangled states satisfying distinct requirements for tangle, 3-tangle, and von Neumann entropy. Moreover, we obtain all the states of three qubits of which tangles, concurrence, 3-tangle, and von Neumann entropy do not vanish and these states are endurable when one of three qubits is traced out. We indicate that for the three-qubit W state, its average von Neumann entropy is maximal only within the W SLOCC class, and that under ASD the three-qubit GHZ state is the unique state of which the reduced density operator obtained by tracing any two qubits has the maximal von Neumann entropy.

Ring-Polymer Molecular Dynamics and Kinetics for the H- + C2H2 → H2 + C2H- Reaction Using the Full-Dimensional Potential Energy Surface.

The H- + C2H2 → H2 + C2H- reaction is important in understanding the production mechanisms of anionic molecules in interstellar environments. Herein, the rate coefficients for the H- + C2H2 → H2 + C2H- reaction were calculated using ring-polymer molecular dynamics (RPMD), classical molecular dynamics (MD), and quasi-classical trajectory (QCT) approaches on a newly developed ab initio potential energy surface (PES) in full dimensions. PES was constructed by fitting a large number of ab initio energy points and their gradients using the permutationally invariant polynomial basis set method. There was no barrier in the reaction coordinates, which was a collinear-dominated reaction, and the reaction proceeded exothermically. It is found that the fitted PES provides the appropriate thermal rate coefficients based on all RPMD, classical MD, and QCT simulations at higher temperatures. The evaluation of the rate coefficients at lower temperatures should be conducted carefully because the fitting of the PES associated with the long-range interaction should be further improved. The spatial distribution of the nucleus allows a more effective attraction between the reactants.

Boost invariant polynomials for efficient jet tagging

Given the vast amounts of data generated by modern particle detectors, computational efficiency is essential for many data-analysis jobs in high-energy physics. We develop a new class of physically interpretable boost invariant polynomial (BIP) features for jet tagging that achieves such efficiency. We show that, for both supervised and unsupervised tasks, integrating BIPs with conventional classification techniques leads to models achieving high accuracy on jet tagging benchmarks while being orders of magnitudes faster to train and evaluate than contemporary deep learning systems.

Spectral data for parabolic projective symplectic/orthogonal Higgs bundles

Hitchin [Duke Math. J. 54(1), 91–114 (1987)] introduced a proper morphism from the moduli space of stable G-Higgs bundles [G=GL(n,C),Sp(2m,C) and SO(n,C)] over a curve to a vector space of invariant polynomials, and he described the generic fibers of that morphism. In this paper, we first describe the generic Hitchin fibers for the moduli space of stable parabolic projective symplectic/orthogonal Higgs bundles without fixing the determinant. We also describe the generic fibers when the determinant is trivial.

On the expected number of real roots of polynomials and exponential sums

The expected number of real projective roots of orthogonally invariant random homogeneous real polynomial systems is known to be equal to the square root of the Bézout number. A similar result is known for random multi-homogeneous systems, invariant through a product of orthogonal groups. In this note, those results are generalized to certain families of sparse polynomial systems, with no orthogonal invariance assumed.

Twists of rational Cherednik algebras

Abstract We show that braided Cherednik algebras introduced by Bazlov and Berenstein are cocycle twists of rational Cherednik algebras of the imprimitive complex reflection groups $G(m,p,n)$, when m is even. This gives a new construction of mystic reflection groups which have Artin–Schelter regular rings of quantum polynomial invariants. As an application of this result, we show that a braided Cherednik algebra has a finite-dimensional representation if and only if its rational counterpart has one.

All-orders asymptotics of tensor model observables from symmetries of restricted partitions

The counting of the dimension of the space of polynomial invariants of a complex 3-index tensor as a function of degree n is known in terms of a sum of squares of Kronecker coefficients. For , the formula can be expressed in terms of a sum of symmetry factors of partitions of n denoted , which also counts the number of bipartite ribbon graphs with n edges. We derive the large n all-orders asymptotic formula for making contact with high order results previously obtained numerically. We explain how the different terms in the asymptotics are associated with probability distributions over ribbon graphs. The derivation of the asymptotics relies on the dominance in the sum, of partitions with many parts of length 1. This large n dominance of small parts also leads to conjectured formulae for the asymptotics of invariants for general d-index tensors. The coefficients of powers of in these expansions involve Stirling numbers of the second kind along with restricted partition sums.

Twisting Kuperberg invariants via Fox calculus and Reidemeister torsion

We study Kuperberg invariants for sutured manifolds in the case of a semidirect product of an involutory Hopf superalgebra $H$ with its automorphism group $\text{Aut}(H)$. These are topological invariants of balanced sutured 3-manifolds endowed with a homomorphism of the fundamental group into $\text{Aut}(H)$ and possibly with a $\text{Spin}^c$ structure and a homology orientation. We show that these invariants are computed via a form of Fox calculus and that, if $H$ is $\mathbb{N}$-graded, they can be extended in a canonical way to polynomial invariants. When $H$ is an exterior algebra, we show that this invariant specializes to a refinement of the twisted relative Reidemeister torsion of sutured 3-manifolds. We also give an explanation of our Fox calculus formulas in terms of a particular Hopf group-algebra.

A Fermi resonance and a parallel-proton-transfer overtone in the Raman spectrum of linear centrosymmetric N4H+: A polarizability-driven first principles molecular dynamics study.

We present molecular dynamics (MD), polarizability driven MD (α-DMD), and pump-probe simulations of Raman spectra of the protonated nitrogen dimer N4H+, and some of its isotopologues, using the explicitly correlated coupled-cluster singles and doubles with perturbative triples [CCSD(T)]-F12b/aug-cc-pVTZ based potential energy surface in permutationally invariant polynomials (PIPs) of Yu et al. [J. Phys. Chem. A 119, 11623 (2015)] and a corresponding PIP-derived CCSD(T)/aug-cc-pVTZ-tr (N:spd, H:sp) polarizability tensor surface (PTS), the latter reported here for the first time. To represent the PTS in terms of a PIP basis, we utilize a recently described formulation for computing the polarizability using a many-body expansion in the orders of dipole-dipole interactions while generating a training set using a novel approach based on linear regression for potential energy distributions. The MD/α-DMD simulations reveal (i) a strong Raman activity at 260 and 2400cm-1, corresponding to the symmetric N-N⋯H bend and symmetric N-N stretch modes, respectively; (ii) a very broad spectral region in the 500-2000cm-1 range, assignable to the parallel N⋯H+⋯N proton transfer overtone; and (iii) the presence of a Fermi-like resonance in the Raman spectrum near 2400cm-1 between the Σg + N-N stretch fundamental and the Πu overtone corresponding to perpendicular N⋯H+⋯N proton transfer.

Accurate Modeling of Bromide and Iodide Hydration with Data-Driven Many-Body Potentials.

Ion-water interactions play a central role in determining the properties of aqueous systems in a wide range of environments. However, a quantitative understanding of how the hydration properties of ions evolve from small aqueous clusters to bulk solutions and interfaces remains elusive. Here, we introduce the second generation of data-driven many-body energy (MB-nrg) potential energy functions (PEFs) representing bromide-water and iodide-water interactions. The MB-nrg PEFs use permutationally invariant polynomials to reproduce two-body and three-body energies calculated at the coupled cluster level of theory, and implicitly represent all higher-body energies using classical many-body polarization. A systematic analysis of the hydration structure of small Br-(H2O)n and I-(H2O)n clusters demonstrates that the MB-nrg PEFs predict interaction energies in quantitative agreement with "gold standard" coupled cluster reference values. Importantly, when used in molecular dynamics simulations carried out in the isothermal-isobaric ensemble for single bromide and iodide ions in liquid water, the MB-nrg PEFs predict extended X-ray absorption fine structure (EXAFS) spectra that accurately reproduce the experimental spectra, which thus allows for characterizing the hydration structure of the two ions with a high level of confidence.

Data-Driven Many-Body Potential Energy Functions for Generic Molecules: Linear Alkanes as a Proof-of-Concept Application.

We present a generalization of the many-body energy (MB-nrg) theoretical/computational framework that enables the development of data-driven potential energy functions (PEFs) for generic covalently bonded molecules, with arbitrary quantum mechanical accuracy. The "nearsightedness of electronic matter" is exploited to define monomers as "natural building blocks" on the basis of their distinct chemical identity. The energy of generic molecules is then expressed as a sum of individual many-body energies of incrementally larger subsystems. The MB-nrg PEFs represent the low-order n-body energies, with n = 1-4, using permutationally invariant polynomials derived from electronic structure data carried out at an arbitrary quantum mechanical level of theory, while all higher-order n-body terms (n > 4) are represented by a classical many-body polarization term. As a proof-of-concept application of the general MB-nrg framework, we present MB-nrg PEFs for linear alkanes. The MB-nrg PEFs are shown to accurately reproduce reference energies, harmonic frequencies, and potential energy scans of alkanes, independently of their length. Since, by construction, the MB-nrg framework introduced here can be applied to generic covalently bonded molecules, we envision future computer simulations of complex molecular systems using data-driven MB-nrg PEFs, with arbitrary quantum mechanical accuracy.