An index definition of parity mappings of a virtual link diagram and Vassiliev invariants of degree one

2014 ◽  
Vol 23 (07) ◽  
pp. 1460010
Author(s):  
Kyeonghui Lee ◽  
Young Ho Im ◽  
Sunho Lee

H. Dye defined the parity mapping for a virtual knot diagram, which is a map from the set of real crossings of the diagram to ℤ. The notion generalizes the parity which is studied extensively by V. Manturov. The mapping induces the ith writhe (i ∈ ℤ\{0}) which is an invariant of the representing virtual knot. She applied the parity mapping to introduce a grade to the Henrich S-invariant for a virtual knot, and showed that the invariants are Vassiliev invariants of degree one. Following it, we define the parity mappings for a virtual link diagram, and define the similar invariants as above for a virtual link by using the parity mappings. We show that some of the invariants are Vassiliev invariants of degree one. We also checked necessary conditions for invertibility and amphicheirality via the invariants.

2003 ◽  
Vol 12 (06) ◽  
pp. 767-779 ◽  
Author(s):  
Jörg Sawollek

It is an open question whether there are Vassiliev invariants that can distinguish an oriented knot from its inverse, i.e., the knot with the opposite orientation. In this article, an example is given for a first order Vassiliev invariant that takes different values on a virtual knot and its inverse. The Vassiliev invariant is derived from the Conway polynomial for virtual knots. Furthermore, it is shown that the zeroth order Vassiliev invariant coming from the Conway polynomial cannot distinguish a virtual link from its inverse and that it vanishes for virtual knots.


Symmetry ◽  
2021 ◽  
Vol 14 (1) ◽  
pp. 15
Author(s):  
Amrendra Gill ◽  
Maxim Ivanov ◽  
Madeti Prabhakar ◽  
Andrei Vesnin

F-polynomials for virtual knots were defined by Kaur, Prabhakar and Vesnin in 2018 using flat virtual knot invariants. These polynomials naturally generalize Kauffman’s affine index polynomial and use smoothing in the classical crossing of a virtual knot diagram. In this paper, we introduce weight functions for ordered orientable virtual and flat virtual links. A flat virtual link is an equivalence class of virtual links with respect to a local symmetry changing a type of classical crossing in a diagram. By considering three types of smoothing in classical crossings of a virtual link diagram and suitable weight functions, there is provided a recurrent construction for new invariants. It is demonstrated by explicit examples that newly defined polynomial invariants are stronger than F-polynomials.


2017 ◽  
Vol 26 (12) ◽  
pp. 1750073 ◽  
Author(s):  
Kanako Oshiro ◽  
Ayaka Shimizu ◽  
Yoshiro Yaguchi

We introduce an up–down coloring of a virtual-link (or classical-link) diagram. The colorabilities give a lower bound of the minimum number of Reidemeister moves of type II which are needed between two [Formula: see text]-component virtual-link (or classical-link) diagrams. By using the notion of a quandle cocycle invariant, we give a method to detect the necessity of Reidemeister moves of type II between two given virtual-knot (or classical-knot) diagrams. As an application, we show that for any virtual-knot diagram [Formula: see text], there exists a diagram [Formula: see text] representing the same virtual-knot such that any sequence of generalized Reidemeister moves between them includes at least one Reidemeister move of type II.


2000 ◽  
Vol 09 (01) ◽  
pp. 93-106 ◽  
Author(s):  
Naoko KAMADA ◽  
Seiichi KAMADA

The notion of an abstract link diagram is re-introduced with a relationship with Kauffman's virtual knot theory. It is prove that there is a bijection from the equivalence classes of virtual link diagrams to those of abstract link diagrams. Using abstract link diagrams, we have a geometric interpretation of the group and the quandle of a virtual knot. A generalization to higher dimensional cases is introduced, and the state-sum invariants are treated.


Author(s):  
Robert D. Enright ◽  
Jacqueline Y. Song

The psychology of forgiveness originated from the creative and important work on the development of justice initiated by Piaget in 1932 and extended by Kohlberg in 1969. The scientific study of forgiveness is quite new, having emerged in print in 1989, with an examination of the developmental progression in children’s, adolescents’, and adults’ thinking about the necessary conditions for them to offer forgiveness to another person. In this chapter, the authors first review the definition of forgiveness, followed by this early cognitive work. They then turn to a discussion of the measurement of forgiveness correlates of forgiveness. The practical application of this construct is seen in the development of forgiveness therapy and forgiveness education, which the authors discuss in light of the empirical findings. Future directions for forgiveness studies are considered.


2017 ◽  
Vol 15 (19) ◽  
Author(s):  
Marek Volt

Artiklis käsitlen Lev Tolstoi kunstiteooria retseptsiooni anglo-ameerika esteetikas. Esiteks formuleerin Tolstoi kunstidefinitsiooni ja selle põhimõistete kanoonilise tõlgenduse. Seejärel analüüsin määratlust ekstensionaalse adekvaatsuse alusel, keskendudes nii tavapäraste kui võimalike uute etteheidete paikapidavuse uurimisele. Kolmandaks püstitan küsimuse, kas Tolstoi kunstidefinitsiooni kriitika ekstensionaalse adekvaatsuse alusel on üldse õigustatud. Väidan, et kuigi senistel Tolstoi meta-esteetilise rehabiliteerimise katsetel esineb puuduseid, paljastab Tolstoi kunstiteooria immanentne kriitika – teooria vaagimine eeldustelt, millelt see kritiseerib oponeerivaid teooriaid –, et ekstensionaalsest adekvaatsusest lähtuv kriitika on õigustatud. My article discusses Tolstoy’s theory of art in the context of Anglo-American aesthetics. Although Tolstoy’s What is Art touches upon a very wide spectrum of subjects (the place of art in the world, justification of sacrifices made for completing art works, criticism of previous theories of aesthetics, especially of the theory of beauty, defining of art as the expression of feelings, judging of art as such based on the religious knowledge of the era, action mechanisms of beauty/pleasure-centred art, consequences, conditions of the value of art, the relations between art and science, etc.), it has mainly been examined from the aspects of judging and defining of art.The article focuses on Tolstoy’s definition of art and consists of three notional parts. First, I present the canonical formulation of Tolstoy’s definition of art – something is a work of art if and only if the person, who lives through the feeling(s), causes by external signs that the recipients live through the same feelings. I also present the canonical interpretation of its main concepts – the conditions for creation, transmitting and reception.Second, I have an analytical insight into the criticism of the canonical treatment, displaying and commenting on, but also responding and complementing the presented arguments. The extensional adequacy-based analysis of Tolstoy’s definition of art shows that although it is possible to eliminate some of the typical criticisms, none of the three necessary conditions was necessary by itself, nor were all three of them sufficient when taken together.As Tolstoy’s definition of art has sometimes earned quite serious criticism, then, as my third point, I also examine some possibilities for rehabilitating Tolstoy’s theory of art: whether and in what sense can the extensional adequacy-based analysis of Tolstoy’s definition of art be justified at all? So far, the attempts of meta-aesthetic rehabilitation of Tolstoy (e.g., Mounce centrism) have not achieved the expected result. Furthermore, the immanent criticism of Tolstoy’s theory of art (criticism of the theory, based on the prerequisites it uses to criticize its opposing theories) reveals that the extensional adequacy-based criticism of Tolstoy’s definition of art is justified, but it is not necessarily the only yardstick for the theory.


1966 ◽  
Vol 18 ◽  
pp. 9-17
Author(s):  
Kulendra N. Majindar

In this paper, we give a connection between incidence matrices of affine resolvable balanced incomplete block designs and rectangular integer matrices subject to certain arithmetical conditions. The definition of these terms can be found in paper II of this series or in (2). For some necessary conditions on the parameters of affine resolvable balanced incomplete block designs and their properties see (2).


2006 ◽  
Vol 15 (03) ◽  
pp. 289-297 ◽  
Author(s):  
TERUHISA KADOKAMI

Any flat virtual link has a reduced diagram which satisfies a certain minimality, and reduced diagrams are related one another by a finite sequence of a certain Reidemeister move. The move preserves some numerical invariants of diagrams. So we can define numerical invariants for flat virtual links. One of them, the crossing number of a flat virtual knot K, coinsides with the self-intersection number of K as an essential geodesic loop on a hyperbolic closed surface. We also show an equation among these numerical invariants, basic properties by using the equation, and determine non-split flat virtual links with the crossing number up to three.


2019 ◽  
Vol 30 (14) ◽  
pp. 1950072 ◽  
Author(s):  
Naoko Kamada

A virtual link diagram is called mod [Formula: see text] almost classical if it admits an Alexander numbering valued in integers modulo [Formula: see text], and a virtual link is called mod [Formula: see text] almost classical if it has a mod [Formula: see text] almost classical diagram as a representative. In this paper, we introduce a method of constructing a mod [Formula: see text] almost classical virtual link diagram from a given virtual link diagram, which we call an [Formula: see text]-fold cyclic covering diagram. The main result is that [Formula: see text]-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 [Formula: see text] almost classical virtual links. Some applications are also given.


2003 ◽  
Vol 12 (06) ◽  
pp. 781-803 ◽  
Author(s):  
Teruhisa Kadokami

J. S. Carter, S. Kamada and M. Saito showed that there is one to one correspondence between the virtual Reidemeister equivalence classes of virtual link diagrams and the stable equivalence classes of link diagrams on compact oriented surfaces. Using the result, we show how to obtain the supporting genus of a projected virtual link by a geometric method. From this result, we show that a certain virtual knot which cannot be judged to be non-trivial by known algebraic invariants is non-trivial, and we suggest to classify the equivalence classes of projected virtual links by using the supporting genus.


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