Narrow Results

Numerical modeling of the deuteron wave function in the coordinate representation for the phenomenological nucleon–nucleon potential Argonne v18 has been performed. For this purpose, the asymptotic behavior of the radial wave function has been taken into account near the origin of coordinates and at infinity. The charge deuteron form factor $G_C(p)$, depending on the transmitted momentums up to $p=22{\mathrm{\text{fm}}}^{-1}$, has been calculated employing five models for the deuteron wave function. A characteristic difference in calculations of $G_C$ is observed near the positions of the first and second zero. The difference between the obtained values for $G_C$ form factor has been analyzed using the values of the ratios and differences for the results. Obtained outcomes for charge deuteron form factor at large momentums may be a prediction for future experimental data.

Topological structure of vortex line in quantum mechanics is studied and classified by Hopf index, linking number in geometry. A mechanism of generation or annihilation of vortex line is given by method of phase singularity theory. The dynamic behavior of vortex line at the critical points is discussed in detail, and three kinds of length approximation relations are given.

In this paper, spinor and vector decomposition of SU(2) gauge potential are presented and their equivalence is constructed using a simple proposal. We also obtain the action of Skyrme–Faddeev model from the SU(2) massive gauge field theory which is proposed according to the gauge invariant principle. Then, the knot structure in Skyrme–Faddeev model is discussed in terms of the so-called ϕ-mapping topological current theory. The topological charge of the knot is characterized by the Hopf indices and the Brouwer degrees of ϕ-mapping, naturally. At last, we briefly discussed the topological invariant–Hopf invariant which describes the topology of these knots. It is shown that Hopf invariant is the total number of all the linking numbers and self-linking numbers of these knots.

This paper is an exposition of the relationship between Witten’s Chern–Simons functional integral and the theory of Vassiliev invariants of knots and links in three-dimensional space. We conceptualize the functional integral in terms of equivalence classes of functionals of gauge fields and we do not use measure theory. This approach makes it possible to discuss the mathematics intrinsic to the functional integral rigorously and without functional integration. Applications to loop quantum gravity are discussed.

After Dirac introduced the monopole, topological objects have played increasingly important roles in physics. In this review we discuss the role of the knot, the most sophisticated topological object in physics, and related topological objects in various areas in physics. In particular, we discuss how the knots appear in Maxwell’s theory, Skyrme theory, and multicomponent condensed matter physics.

In this paper, starting from Biot–Savart mechanics for entangled vortex-membranes, a new theory — knot physics — is developed to explore the underlying physics of quantum mechanics. Owning to the conservation conditions of the volume of knots on vortices in incompressible fluid, the shape of knots will never be changed and the corresponding Kelvin waves cannot evolve smoothly. Instead, the knot can only be split. The knot-pieces evolve following the equation of motion of Biot–Savart equation that becomes Schrödinger equation for probability waves of knots. The classical functions for Kelvin waves become wave-functions for knots. The effective theory of perturbative entangled vortex-membranes becomes a traditional model of relativistic quantum field theory — a massive Dirac model. As a result, this work would help researchers to understand the mystery in quantum mechanics.

In this paper, knot physics on entangled vortex-membranes are studied including classification, knot dynamics and effective theory. The physics objects in this paper are entangled vortex-membranes that are called composite knot-crystals. Under projection, a composite knot-crystal is reduced to coupled zero-lattices. In the continuum limit, the effective theories of coupled zero-lattices become quantum field theories. After considering the topological interplay between knots and different types of zero-lattices, gauge interactions emerge. Based on a particular composite knot-crystal (we call it a standard knot-crystal), the derived effective model becomes the Standard Model. As a result, the knot physics may provide an alternative interpretation on quantum field theory.

In this paper, we show that any $n$-dimensional autonomous systems can be regarded as subsystems of $2n$-dimensional Hamiltonian systems. One of the two subsystems is identical to the $n$-dimensional autonomous system, which is called the driving system. Another subsystem, called the response system, can exhibit interesting behaviors in the neighborhood of infinity. That is, the trajectories approach infinity with complicated nonperiodic (chaotic-like) behaviors, or periodic-like behavior. In order to show the above results, we project the trajectories of the Hamiltonian systems onto $n$-dimensional spheres, or $n$-dimensional balls by using the well-known central projection transformation. Another interesting behavior is that the transient regime of the subsystems can exhibit Chua corsage knots. We next show that generic memristors can be used to realize the above Hamiltonian systems. Finally, we show that the internal state of two-element memristor circuits can have the same dynamics as $n$-dimensional autonomous systems.

In this paper we consider the generalized Smith conjecture of codimension greater than two, which says that no periodic tranformation of S^l can have the tame knotted S^h as fixed point set if l-h>2 and h>3. Using the Brieskorn spheres, this paper gives the explicit counterexamples to show that the conjecture is false in the DIFF category for the following cases: (i) l-h is even more than 2 and l is odd; (ii) 2l≤3(h+1) and h+1≡0 (mod 4).

An uncountable collection of arcs in S³ is constructed, each member of which is wild precisely at its endpoints, such that the fundamental groups of their complements are non-trivial, pairwise non-isomorphic, and indecomposable with respect to free products. The fundamental group of the complement of a certain Fox-Artin arc is also shown to be indecomposable.

A double-torus knot is a knot embedded in a genus two Heegaard surface in S³. We consider double-torus knots L such that is connected, and consider fibred knots in various classes.

For a finite graph G, let Γ(G) be the set of all cycles of G. Suppose that for each γ∈Γ(G), an embedding ϕ_γ:γ→S³ is given. A set {ϕ_γ|γ∈Γ(G)} of embeddings is said to be realizable if there is an embedding f:G→S³ such that the restriction map f|γ is ambient isotopic to ϕ_γ for any γ∈Γ(G). In this paper on seven specified graphs G, we give a necessary and sufficient condition for the set {ϕ_γ|γ∈Γ(G)} to be realizable by using the second coefficients of Conway polynomials of knots.

We give a basis for the space of the Vassiliev knot invariants of order 6, which implies a sixth order Vassiliev knot invariant is determined by its HOMFLY and Kauffman polynomials. Furthermore, we show that there is a seventh order invariant which cannot be obtained from these polynomials of a knot.

In the present paper, we will study the creation of Klein bottles by Dehn surgery on knots in the 3-sphere, and we will give an upper bound for slopes creating Klein bottles for non-cabled knots by using the genera of knots. In particular, it is shown that if a Klein bottle is created by Dehn surgery on a genus one knot then the knot is a doubled knot. As a corollary, we obtain that genus one, cross-cap number two knots are doubled knot.

A quandle is a set with a binary operation satisfying some properties. A quandle homomorphism is a map between quandles preserving the structure of their binary operations. A knot determines a quandle called a knot quandle. We show that the number of all quandle homomorphisms of a knot quandle of a knot to an Alexander quandle is completely determined by Alexander polynomials of the knot. Further we show that the set of all quandle homomorphisms of a knot quandle to an Alexander quandle has a module structure.

We show that the integer-valued knot invariants appearing as the nontrivial coefficients of the HOMFLY polynomial, the Kauffman polynomial and the Q-polynomial are not of finite type.

We show that any tunnel number one knot group has a two generator one relator presentation in which the relator is a palindrome in the generators. We use this fact to compute the character variety for this knot groups and we show that it is an affine algebraic set .

We characterize the knot types in a spatial embedding of a graph which is obtained from the 5-cycle by duplicating each edge. It is characterized in terms of the second coefficients of the Conway polynomials and the third derivatives at 1 of the Jones polynomials of knots.

We introduce a family of cyclic presentations of groups depending on a finite set of integers. This family contains many classes of cyclic presentations of groups, previously considered by several authors. We prove that, under certain conditions on the parameters, the groups defined by our presentations cannot be fundamental groups of closed connected hyperbolic 3–dimensional orbifolds (in particular, manifolds) of finite volume. We also study the split extensions and the natural HNN extensions of these groups, and determine conditions on the parameters for which they are groups of 3–orbifolds and high–dimensional knots, respectively.

Two knots in three-space are S-equivalent if they are indistinguishable by Seifert matrices. We show that S-equivalence is generated by the doubled-delta move on knot diagrams. It follows as a corollary that a knot has trivial Alexander polynomial if and only if it can be undone by doubled-delta moves.