Narrow Results

The quantum U(1) invariant of a closed 3-manifold M is defined from the linking matrix of a framed link of a surgery presentation of M. As an equivariant version of it, we formulate an invariant of a knot K from the equivariant linking matrix of a lift of a framed link of a surgery presentation of K. We show that this invariant is determined by the Blanchfield pairing of K, or equivalently, determined by the S-equivalent class of a Seifert matrix of K, and that the "product" of this invariant and its complex conjugation is presented by the Alexander module of K. We present some values of this invariant of some classes of knots concretely.

The intersecting kernel of a Heegaard splitting $H_1{\bigcup }_S{H}_2$ for a compact orientable 3-manifold $M$ is the subgroup $K=\mathrm{\text{Ker}}({i_1}_{\ast })\cap \mathrm{\text{Ker}}({i_2}_{\ast })$ of ${\pi }_1(S)$, where ${i_j}_{\ast }:{\pi }_1(S)\to {\pi }_1(H_j)$ is the homomorphism induced by the inclusion $i_j:S\hookrightarrow H_j$, $j=1,2$. In the paper, we obtain some invariants of 3-manifolds $M$ from certain quotient groups of the intersecting kernels of their Heegaard splittings. We also list two algebraic problems related to the new invariants, which might be interesting to study.

A virtual link diagram is called mod $m$ almost classical if it admits an Alexander numbering valued in integers modulo $m$, and a virtual link is called mod $m$ almost classical if it has a mod $m$ almost classical diagram as a representative. In this paper, we introduce a method of constructing a mod $m$ almost classical virtual link diagram from a given virtual link diagram, which we call an $m$-fold cyclic covering diagram. The main result is that $m$-fold cyclic covering diagrams obtained from two equivalent virtual link diagrams are equivalent. Thus, we have a well-defined map from the set of virtual links to the set of mod $m$ almost classical virtual links. Some applications are also given.

By using canonical and unitary transformations and the Lewis–Riesenfeld invariant method, the generalized invariant and the exact Schrödinger wave functions for a time-dependent parametric oscillator with and without an inverse quadratic potential are obtained.

We discuss the extension of the Lewis and Riesenfeld method of solving the time-dependent Schrödinger equation to cases where the invariant has continuous eigenvalues and apply it to the case of a generalized time-dependent inverted harmonic oscillator. As a special case, we consider a generalized inverted oscillator with constant frequency and exponentially increasing mass.

Invariance and low dimension of features are of crucial significance in pattern recognition. This paper proposes a novel orthonormal shell Fourier descriptor that satisfies all of these demands. This method first performs orthonormal shell decomposition on the line moment that is obtained from the 2-D pattern, then applies Fourier transform on each scale of the shell coefficients. Unlike other existing wavelet-based methods, our method allows applying common orthonormal wavelets, such as Daubechies, Symmlet and Coiflet, therefore it is simple to implement. We study the structure of the filter used and develop a fast algorithm to rapidly compute the spectra of orthonormal shell coefficients. The complexity of the proposed descriptor is O(n log n). We apply a coarse-to-fine strategy to search the image database; the matching is very quick because of the multiscale feature structure. The effectiveness of this new descriptor is demonstrated by a series of experiments as well as the comparison with other descriptors. The proposed descriptor is robust to white noise.

In this paper, we perform our algorithm developed in [Yu & Lee, 2001] to present the entire branches of quasiperiodic solutions starting from the bifurcation points in the branches of periodic solutions in an interval of parameters for the 2-mode damped, driven sine-Gordon ODE.

We first investigate the Lyapunov stability of the period-three solution of Todd's equation with a period-three coefficient:

Then for k = 2,3,… we extend our stability result to the k-order equation,

New existence conditions of chaos for a wide class of 3D autonomous quadratic dynamical systems are suggested. These conditions are based on the construction and study of properties of algebraic invariants for some 2D quadratic system differential equations.

Extracting the geometric characteristics of conic sections, such as their center, axes and foci, from their defining equations is required for various applications in computer graphics and geometric modeling. Although there exist standard techniques for computing the geometric characteristics for conics in implicit form, in shape modeling applications conic sections are often represented by rational quadratic parameterizations. Here we present closed formulas for computing the geometric characteristics of conics directly from their quadratic parameterizations without resorting to implicitization procedures. Our approach uses the invariants of rational quadratic parameterizations under rational linear reparameterizations. These invariants are also used to give a complete characterization of degenerate tonics represented by rational quadratic parameterizations.

Introducing a type of graph named a virtual magnetic graph diagram, we define a virtual link invariant, which generalizes the Jones-Kauffman polynomial and the 2-variable polynomial invariant defined by N. Kamada and the author. We show that this invariant can evaluate the number of virtual crossings of a virtual link.

We compute the third cohomology group of a dihedral quandle of odd order with finite field coefficient, generalizing our previous work. In particular, we obtain many new examples of non-trivial 3-cocycles. We check non-triviality of the associated invariant for knots.

Parity mappings (weights) from the chords of a Gauss diagram to the integers are defined. The parity of the chords is used to construct families of invariants of Gauss diagrams and consequently, virtual knots. Each family forms a set of degree n Vassiliev invariants for n ≥ 1.

The Kuperberg bracket is a well-known invariant of classical links. Recently, the second named author and Kauffman constructed the graph-valued generalization of the Kuperberg bracket for the case of virtual links: unlike the classical case, the invariant in the virtual case is valued in graphs which carry a significant amount of information about the virtual knot. The crucial difference between virtual knot theory and classical knot theory is the rich topology of the ambient space for virtual knots. In a paper by Chrisman and the second named author, two-component classical links with one fibered component were considered; the complement to the fibered component allows one to get highly non-trivial ambient topology for the other component. In this paper, we combine the ideas of the above mentioned papers and construct the "virtual" Kuperberg bracket for two-component links L = J ⊔ K with one component (J) fibered. We consider a new geometrical complexity for such links and establish minimality of diagrams in a strong sense. Roughly speaking, every other "diagram" of the knot in question contains the initial diagram as a subdiagram. We prove a sufficient condition for minimality in a strong sense where minimality cannot be established as introduced in the paper by Chrisman and the second named author.

We construct graph-valued analogues of the Kuperberg sl(3) and G₂ invariants for virtual knots. The restriction of the sl(3) and G₂ invariants for classical knots coincides with the usual Homflypt sl(3) invariant and G₂ invariants. For virtual knots and graphs these invariants provide new graphical information that allows one to prove minimality theorems and to construct new invariants for free knots (unoriented and unlabeled Gauss codes taken up to abstract Reidemeister moves). A novel feature of this approach is that some knots are of sufficient complexity that they evaluate themselves in the sense that the invariant is the knot itself seen as a combinatorial structure. The paper generalizes these structures to virtual braids and discusses the relationship with the original Penrose bracket for graph colorings.

In this paper, we describe a two-dimensional rectangular-cell-complex derived from a surface-knot diagram of a surface-knot. We define a pseudo-cycle for a quandle colored surface-knot diagram. We show that the maximal number of pseudo-cycles is a surface-knot invariant.

This paper studies cobordism and concordance for virtual knots. We define the affine index polynomial, prove that it is a concordance invariant for knots and links (explaining when it is defined for links), show that it is also invariant under certain forms of labeled cobordism and study a number of examples in relation to these phenomena. Information on determinations of the four-ball genus of some virtual knots is obtained by via the affine index polynomial in conjunction with results on the genus of positive virtual knots using joint work with Dye and Kaestner.

In the present paper, we develop a picture formalism which gives rise to an invariant that dominates several known invariants of classical and virtual knots: the Jones polynomial ${𝒳}\left(\cdot \right)$, the Kuperberg bracket ${\left|\cdot \right.〉}_{A_2}$, and the normalized arrow polynomial ${𝒲}\left[\cdot \right]$.

We provide a physical definition of new homological invariants ${{\mathscr{H}}}_a(M_3)$ of 3-manifolds (possibly, with knots) labeled by abelian flat connections. The physical system in question involves a 6d fivebrane theory on $M_3$ times a 2-disk, $D^2$, whose Hilbert space of BPS states plays the role of a basic building block in categorification of various partition functions of 3d ${𝒩}=2$ theory $T[M_3]$: $D^2\times S^1$ half-index, $S^2\times S^1$ superconformal index, and $S^2\times S^1$ topologically twisted index. The first partition function is labeled by a choice of boundary condition and provides a refinement of Chern–Simons (WRT) invariant. A linear combination of them in the unrefined limit gives the analytically continued WRT invariant of $M_3$. The last two can be factorized into the product of half-indices. We show how this works explicitly for many examples, including Lens spaces, circle fibrations over Riemann surfaces, and plumbed 3-manifolds.

In V. O. Manturov, On free knots, preprint (2009), arXiv:math.GT/0901.2214], the second named author constructed the bracket invariant $[\cdot ]$ of virtual knots valued in pictures (linear combinations of virtual knot diagrams with some crossing information omitted), such that for many diagrams $K$, the following formula holds: $[K]=\tilde{K}$, where $\tilde{K}$ is the underlying graph of the diagram, i.e. the value of the invariant on a diagram equals the diagram itself with some crossing information omitted. This phenomenon allows one to reduce many questions about virtual knots to questions about their diagrams. In [S. Nelson, M. E. Orrison and V. Rivera, Quantum enhancements and biquandle brackets, preprint (2015), arXiv:math.GT/1508.06573], the authors discovered the following phenomenon: having a biquandle coloring of a certain knot, one can enhance various state-sum invariants (say, Kauffman bracket) by using various coefficients depending on colors. Taking into account that the parity can be treated in terms of biquandles, we bring together the two ideas from these papers and construct the picture-valued parity-biquandle bracket for classical and virtual knots. This is an invariant of virtual knots valued in pictures. Both the parity bracket and Nelson–Orrison–Rivera invariants are partial cases of this invariant, hence this invariant enjoys many properties of various kinds. Recently, the authors together with E. Horvat and S. Kim have found that the picture-valued phenomenon works in the classical case.