Narrow Results

In this paper, we define some polynomial invariants for virtual knots and links. In the first part we use Manturov's parity axioms [Parity in knot theory, Sb. Math.201 (2010) 693–733] to obtain a new polynomial invariant of virtual knots. This invariant can be regarded as a generalization of the odd writhe polynomial defined by the first author in [A polynomial invariant of virtual knots, preprint (2012), arXiv:math.GT/1202.3850v1]. The relation between this new polynomial invariant and the affine index polynomial [An affine index polynomial invariant of virtual knots, J. Knot Theory Ramification22 (2013) 1340007; A linking number definition of the affine index polynomial and applications, preprint (2012), arXiv:1211.1747v1] is discussed. In the second part we introduce a polynomial invariant for long flat virtual knots. In the third part we define a polynomial invariant for 2-component virtual links. This polynomial invariant can be regarded as a generalization of the linking number.

We consider knot theories possessing parity: each crossing is decreed odd or even according to some universal rule. If this rule satisfies some simple axioms concerning the behaviour under Reidemeister moves, then this leads to a possibility of constructing new invariants and proving minimality and non-triviality theorems for knots from these classes, and constructing maps from knots to knots.

Our main example is virtual knot theory and its simplification, free knot theory. By using Gauss diagrams, we show the existence of non-trivial free knots (counterexample to Turaev's conjecture) and construct simple yet deep invariants that rely on parity. Some invariants are valued in graph-like objects and some other are valued in groups.

We discuss applications of parity to virtual knots and ways of extending well-known invariants.

The existence of a non-trivial parity for classical knots remains an open problem.

Using the functional analysis method, we present the exact solutions of the double ring-shaped oscillator (DRSO) potential with certain parity in the cylindrical coordinates. Such a quantum system is separated to two differential equations, i.e. a one-dimensional harmonic oscillator plus an inverse square term and a two-dimensional harmonic oscillator plus an inverse square term. The key point is how to find the adapted symmetrical solutions of the one-dimensional harmonic oscillator plus an inverse square term at the singular point $z=0$. The obtained results are compared with those in the spherical coordinates. We also explore intimate connections $E_T(i\rho )=-E_T(\rho )$ and $E_z(iz)=-E_z(z)$ by substituting $\rho \to i\rho$ and $z\to iz$.

This paper studies rotational virtual knot theory and its relationship with quantum link invariants. Every quantum link invariant for classical knots and links extends to an invariant of rotational virtual knots and links. We give examples of non-trivial rotational virtuals that are undectable by quantum invariants.

In this paper, we study parity in planar and spherical knotoids in relation to virtual knots. We introduce a planar version of the parity bracket polynomial for planar knotoids. We show that the virtual closure map (a map from the set of knotoids in $S^2$ to the set of virtual knots of genus at most one) is not surjective, by utilizing the surface bracket polynomial of virtual knots. We give specific examples of virtual knots that are not in the image of the virtual closure map. Turaev conjectured that minimal diagrams of knot-type knotoids have zero height. We prove this conjecture by using the results of Nikonov and Manturov induced by parities of virtual knots.

This chapter introduces the system for crisis management in Sweden. Over the last century, Sweden has gone from being a poor European backwater to being among the countries with the highest human development in the world. The droughts and harsh winters that killed thousands and drove numerous Swedes to emigrate in the past are mere distractions today and the contemporary system for crisis management is designed to deal with a broader variety of crises than the ones triggered by natural hazards. The system is based on the principles of responsibility, parity and proximity, and distributes sector and area responsibility for crisis management to numerous actors. It is built to a great extent on collaboration between these actors, which is challenging but working relatively well in the cultural context of consensus-seeking and compliance to official guidelines and accepted rules of engagement. However, the system is in itself ambiguous in the sense of distributing responsibility to all kinds of actors and then focusing almost exclusively on public actors in legislation, guidelines and practice. There is also often a gap between policy and practice concerning how area responsibility is exercised, and a lack of clarity in current sector specific legislation.

We are investigating the dynamics of a new Poincaré gauge theory of gravity model, which has cross coupling between the spin-0⁺ and spin-0^- modes. To this end we here consider a very appropriate situation — homogeneous-isotropic cosmologies — which is relatively simple, and yet all the modes have nontrivial dynamics which reveals physically interesting and possibly observable results. More specifically we consider manifestly isotropic Bianchi class A cosmologies; for this case we find an effective Lagrangian and Hamiltonian for the dynamical system. The Lagrange equations for these models lead to a set of first-order equations that are compatible with those found for the FLRW models and provide a foundation for further investigations. Typical numerical evolution of these equations shows the expected effects of the cross parity coupling.

A left-right symmetric mirror model restoring parity at a high scale in a way such that the mirror fermions and mirror gauge sector simultaneously could exist at TeV scale is discussed. We also provide an ultraviolet completion of the model with vector-like fermions, and discuss some theoretical and phenomenological implications of this model.

We construct a D₄ flavor model based on SU(3)_C ⊗SU(3)_L ⊗U(1)_X gauge symmetry responsible for fermion masses and mixings. The neutrinos get small masses from antisextets which are in a singlet and a doublet under D₄. If the D₄ symmetry is violated as perturbation by a Higgs triplet under SU(3)_L and lying in of D₄, the corresponding neutrino mass mixing matrix gets the most general form. In this case, the model can fit the experimental data in 2012 on neutrino masses and mixing. Our results show that the neutrino masses are naturally small and a little deviation from the tribimaximal neutrino mixing form can be realized. The quark masses and mixing matrix are also discussed. In the model under consideration, the CKM matrix can be different from the unit matrix. The scalar potential of the model is more simpler than those of the model based on S₃ and S₄. Assignation of VEVs to antisextets leads to the mixing of the new gauge bosons and those in the Standard Model. The mixing in the charged gauge bosons as well as the neutral gauge boson is considered.

This paper examines mispricing, volatility and parity on the Hang Seng Index (HSI) options and futures market. Most of the previous research has focused on futures contracts; we update this research and extend it by considering also option contracts. It is also important to examine these issues post 1997 Asian crisis. We find mispricing of HSI futures and option contracts if no transaction costs were considered. However, by incorporating transaction costs, the HSI futures are bounded within the arbitrage free region and most of the mispricing of the HSI options disappears. Additional tests on the mispricing series reveals that most of the derivative HSI contracts are positively autocorrelated and that the mispricing series for both derivative contracts are not identical among the different contract months. From our results we cannot conclude that there is causal relationship between the mispricing and the spot index volatility. Finally, our empirical results show that for HSI derivative contracts future and option parity holds, supporting our mispricing test that the HSI derivative market is efficient and has not been adversely affected by the Asian economic crisis.

We analyze the parity violation issue in the Poincaré gauge theory of gravity for the two classes of models which are built as natural extensions of the Einstein–Cartan theory. The conservation laws of the matter currents are revisited and we clarify the derivation of the effective Einstein field equation and the structure of the effective energy–momentum current for arbitrary matter sources.

A right–right–left extension of the Standard Model is proposed. In this model, SM gauge group $\mathrm{\text{SU}}{(2)}_L\otimes \mathrm{\text{U}}{(1)}_Y$ is extended to $\mathrm{\text{SU}}{(2)}_L\otimes \mathrm{\text{SU}}{(2)}_R\otimes \mathrm{\text{SU}}{(2)}_R^{\prime }\otimes \mathrm{\text{SU}}{(2)}_L^{\prime }\otimes \mathrm{\text{U}}{(1)}_Y$. The gauge symmetries $\mathrm{\text{SU}}{(2)}_R^{\prime }$, $\mathrm{\text{SU}}{(2)}_L^{\prime }$ are the mirror counterparts of the $\mathrm{\text{SU}}{(2)}_L$ and $\mathrm{\text{SU}}{(2)}_R$, respectively. Parity is spontaneously broken when the scalar Higgs fields acquire vacuum expectation values (VEVs) in a certain pattern. Parity is restored at the scale of $\mathrm{\text{SU}}{(2)}_L^{\prime }$. The gauge sector has a unique pattern. The scalar sector of the model is optimum, elegant and unique.

Yang's theorem states that an initial J = 1 state cannot decay into two photons. Because of this result some reactions relating to elementary particles or atomic transitions can be ruled out. The theorem is not valid in the presence of background electric or magnetic fields. In this work we show that the decay of a J = 1 particle into two photons is permitted by the Bose symmetry and rotational invariance when the background of the decay process is not pure vacuum but contains an external classical magnetic/electric field. We also discuss constraints on these amplitudes from the CP invariance.

We investigate an alternative CPT-odd Lorentz-breaking QED which includes the Carroll–Field–Jackiw (CFJ) term of the Standard Model Extension (SME), writing the gauge sector in the action in a Palatini-like form, in which the vectorial field and the field-strength tensor are treated as independent entities. Interestingly, this naturally induces a Lorentz-violating mass term in the classical action. We study physical consistency aspects of the model both at classical and quantum levels.

The aim of the present paper is to prove that the minimal number of virtual crossings for some families of virtual knots grows quadratically with respect to the minimal number of classical crossings. All previously known estimates for virtual crossing number ([D. M. Afanasiev, Refining the invariants of virtual knots by using parity, Sb. Math.201(6) (2010) 3–18; H. A. Dye and L. H. Kauffman, Virtual crossing numbers and the arrow polynomial, in The Mathematics of Knots. Theory and Applications, eds. M. Banagl and D. Vogel (Springer-Verlag, 2010); S. Satoh and Y. Tomiyama, On the crossing number of a virtual knot, Proc. Amer. Math. Soc.140 (2012) 367–376] and so on) were principally no more than linear in the number of classical crossings (or, what is the same, in the number of edges of a virtual knot diagram) and no virtual knot was found with virtual crossing number greater than the classical crossing number.

We investigate an application of crossing parity for the bracket expansion of the Jones polynomial for virtual knots. In addition we consider an application of parity for the arrow polynomial as well as for the categorifications of both polynomials. We present a number of examples found through our calculations. We provide tables of calculations for these invariants on virtual knots with at most four real crossings.

We consider a problem of realizability of Gauss diagrams by closed plane curves where the plane curves have only double points of transversal self-intersection. We formulate the necessary and sufficient conditions for realizability. These conditions are based only on the parity of double and triple intersections of the chords in the Gauss diagram.

A parity is a labeling of the crossings of knot diagrams which is compatible with Reidemeister moves. We define the notion of parity for based matrices — algebraic objects are introduced by Turaev in his research on virtual strings. We present the reduced stable parity on based matrix which gives a new example of a parity of virtual knots.

A parity is a rule to assign labels to the crossings of knot diagrams in a way compatible with the Reidemeister moves. Parity functors can be viewed as parities which provide to each knot diagram its own coefficient group that contains parities of the crossings. In the paper we describe the universal oriented parity functors for free knots and for knots in a fixed surface.

We prove that the Gaussian parity on free two-dimensional knots is universal.