VIRTUAL KNOT GROUPS AND THEIR PERIPHERAL STRUCTURE

2000 ◽  
Vol 09 (06) ◽  
pp. 797-812 ◽  
Author(s):  
SE-GOO KIM
Keyword(s):  
Sufficient Conditions ◽  
Natural Generalization ◽  
Fundamental Groups ◽  
Arbitrary Group ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Classical Setting ◽  
Peripheral Structure ◽  
Necessary And Sufficient ◽  
Natural Way

Virtual knots, defined by Kauffman, provide a natural generalization of classical knots. Most invariants of knots extend in a natural way to give invariants of virtual knots. In this paper we study the fundamental groups of virtual knots and observe several new and unexpected phenomena. In the classical setting, if the longitude of a knot is trivial in the knot group then the group is infinite cyclic. We show that for any classical knot group there is a virtual knot with that group and trivial longitude. It is well known that the second homology of a classical knot group is trivial. We provide counterexamples of this for virtual knots. For an arbitrary group G, we give necessary and sufficient conditions for the existence of a virtual knot group that maps onto G with specified behavior on the peripheral subgroup. These conditions simplify those that arise in the classical setting.

ON A LOCAL MOVE FOR VIRTUAL KNOTS AND LINKS

2009 ◽  
Vol 18 (11) ◽  
pp. 1577-1596 ◽  
Author(s):  
TOSHIYUKI OIKAWA
Keyword(s):  
Sufficient Conditions ◽  
Virtual Link ◽  
Link Invariant ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Linking Number ◽  
Reidemeister Moves ◽  
Link Diagrams ◽  
Necessary And Sufficient ◽  
Knots And Links

We define a local move called a CF-move on virtual link diagrams, and show that any virtual knot can be deformed into a trivial knot by using generalized Reidemeister moves and CF-moves. Moreover, we define a new virtual link invariant n(L) for a virtual 2-component link L whose virtual linking number is an integer. Then we give necessary and sufficient conditions for two virtual 2-component links to be deformed into each other by using generalized Reidemeister moves and CF-moves in terms of a virtual linking number and n(L).

scholarly journals CROWELL'S DERIVED GROUP AND TWISTED POLYNOMIALS

2006 ◽  
Vol 15 (08) ◽  
pp. 1079-1094 ◽  
Author(s):  
DANIEL S. SILVER ◽  
SUSAN G. WILLIAMS
Keyword(s):  
Knot Theory ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Permutation Representation ◽  
Virtual Knot ◽  
Alexander Polynomials ◽  
Necessary And Sufficient ◽  
New Applications ◽  
Extended Group

The derived group of a permutation representation, introduced by Crowell, unites many notions of knot theory. We survey Crowell's construction, and offer new applications. The twisted Alexander group of a knot is defined. Using it, we obtain twisted Alexander polynomials. Also, we extend a well-known theorem of Neuwirth and Stallings giving necessary and sufficient conditions for a knot to be fibered. Virtual Alexander polynomials provide obstructions for a virtual knot that must vanish if the knot has a diagram with an Alexander numbering. The extended group of a virtual knot is defined, and using it a more sensitive obstruction is obtained.

The Supremum of a Family of Additive Functions

1952 ◽  
Vol 4 ◽  
pp. 463-479 ◽  
Author(s):  
Israel Halperin
Keyword(s):  
Vector Space ◽  
Commutative Semigroup ◽  
Sufficient Conditions ◽  
Boolean Ring ◽  
Additive Functions ◽  
Semi Group ◽  
Additive Functionals ◽  
Banach Theorem ◽  
Necessary And Sufficient ◽  
Natural Way

Any system S in which an addition is defined for some, but not necessarily all, pairs of elements can be imbedded in a natural way in a commutative semi-group G, although different elements in S need not always determine different elements in G (see §2). Theorem 2.1 gives necessary and sufficient conditions in order that a functional p(x) on S can be represented as the su prémuni of some family of additive functionals on S, and one such set of conditions is in terms of possible extensions of p(x) to G. This generalizes the case with 5 a Boolean ring treated by Lorentz [4], Lorentz imbeds the Boolean ring in a vector space and this could be done for the general S; but we prefer to imbed S in a commutative semi-group and to give a proof (see § 1) generalizing the classical Hahn-Banach theorem to the case of an arbitrary commutative semigroup.

scholarly journals Two-variable polynomial invariants of virtual knots arising from flat virtual knot invariants

2018 ◽  
Vol 27 (13) ◽  
pp. 1842015 ◽  
Author(s):  
Kirandeep Kaur ◽  
Madeti Prabhakar ◽  
Andrei Vesnin
Keyword(s):  
Sufficient Conditions ◽  
Polynomial Invariants ◽  
Knot Invariants ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Knot Invariant ◽  
Index Value

We introduce two sequences of two-variable polynomials [Formula: see text] and [Formula: see text], expressed in terms of index value of a crossing and [Formula: see text]-dwrithe value of a virtual knot [Formula: see text], where [Formula: see text] and [Formula: see text] are variables. Basing on the fact that [Formula: see text]-dwrithe is a flat virtual knot invariant, we prove that [Formula: see text] and [Formula: see text] are virtual knot invariants containing Kauffman affine index polynomial as a particular case. Using [Formula: see text] we give sufficient conditions when virtual knot does not admit cosmetic crossing change.

Generalization of the Hausdorff Moment Problem

1981 ◽  
Vol 33 (4) ◽  
pp. 946-960 ◽  
Author(s):  
David Borwein ◽  
Amnon Jakimovski
Keyword(s):  
Moment Problem ◽  
Sufficient Conditions ◽  
Natural Generalization ◽  
Necessary And Sufficient Conditions ◽  
Real Numbers ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Hausdorff Moment Problem ◽  
Positive Integers ◽  
Classical Moment Problem

Suppose throughout that {kn} is a sequence of positive integers, thatthat k0 = 1 if l0 = 1, and that {un(r)}; (r = 0, 1, …, kn – 1, n = 0, 1, …) is a sequence of real numbers. We shall be concerned with the problem of establishing necessary and sufficient conditions for there to be a function a satisfying(1)and certain additional conditions. The case l0 = 0, kn = 1 for n = 0, 1, … of the problem is the version of the classical moment problem considered originally by Hausdorff [5], [6], [7]; the above formulation will emerge as a natural generalization thereof.

scholarly journals The ordering on permutations induced by continuous maps of the real line

1987 ◽  
Vol 7 (2) ◽  
pp. 155-160 ◽  
Author(s):  
Chris Bernhardt
Keyword(s):  
Real Line ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Partial Ordering ◽  
The Real ◽  
Continuous Maps ◽  
Necessary And Sufficient ◽  
Natural Way

AbstractContinuous maps from the real line to itself give, in a natural way, a partial ordering of permutations. This ordering restricted to cycles is studied.Necessary and sufficient conditions are given for a cycle to have an immediate predecessor. When a cycle has an immediate predecessor it is unique; it is shown how to construct it. Every cycle has immediate successors; it is shown how to construct them.

scholarly journals On the topology of a boolean representable simplicial complex

2017 ◽  
Vol 27 (01) ◽  
pp. 121-156 ◽  
Author(s):  
Stuart Margolis ◽  
John Rhodes ◽  
Pedro V. Silva
Keyword(s):  
Simplicial Complex ◽  
Homotopy Type ◽  
Sufficient Conditions ◽  
Simplicial Complexes ◽  
Connected Components ◽  
Fundamental Groups ◽  
Complexity Bounds ◽  
Dimension Formula ◽  
Dimension 2 ◽  
Necessary And Sufficient

It is proved that the fundamental groups of boolean representable simplicial complexes (BRSC) are free and the rank is determined by the number and nature of the connected components of their graph of flats for dimension [Formula: see text]. In the case of dimension 2, it is shown that BRSC have the homotopy type of a wedge of spheres of dimensions 1 and 2. Also, in the case of dimension 2, necessary and sufficient conditions for shellability and being sequentially Cohen–Macaulay are determined. Complexity bounds are provided for all the algorithms involved.

Characterization of small polygroups by their fundamental groups

2019 ◽  
Vol 11 (04) ◽  
pp. 1950047 ◽  
Author(s):  
D. Heidari ◽  
B. Davvaz
Keyword(s):  
Fundamental Group ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Fundamental Groups ◽  
Trivial Group ◽  
Necessary And Sufficient

In this paper, we consider polygroup [Formula: see text] and prove necessary and sufficient conditions such that [Formula: see text] is non-commutative. Then by using the Maple programming, we obtain all polygroups of order less than five up to isomorphism. In fact, we determine all 115 non-isomorphic polygroups of order less than five and characterize them by their fundamental groups, i.e., polygroups with same fundamental group, say [Formula: see text], classifies in the class [Formula: see text]. Finally, we obtain that the fundamental groups of 94 polygroups are the trivial group. The numbers of polygroups in classes [Formula: see text] and [Formula: see text] are 16 and 3, respectively, and the classes [Formula: see text] and [Formula: see text] are singleton.

scholarly journals On the Construction of Substitutes

2020 ◽  
Vol 45 (1) ◽  
pp. 272-291
Author(s):  
Eric Balkanski ◽  
Renato Paes Leme
Keyword(s):  
Greedy Algorithms ◽  
Sufficient Conditions ◽  
Natural Generalization ◽  
Negative Answer ◽  
Linear Combinations ◽  
Constructive Description ◽  
Discrete Convex Analysis ◽  
Discrete Convex ◽  
Necessary And Sufficient ◽  
Rank Functions

Gross substitutability is a central concept in economics and is connected to important notions in discrete convex analysis, number theory, and the analysis of greedy algorithms in computer science. Many different characterizations are known for this class, but providing a constructive description remains a major open problem. The construction problem asks how to construct all gross substitutes from a class of simpler functions using a set of operations. Because gross substitutes are a natural generalization of matroids to real-valued functions, matroid rank functions form a desirable such class of simpler functions. Shioura proved that a rich class of gross substitutes can be expressed as sums of matroid rank functions, but it is open whether all gross substitutes can be constructed this way. Our main result is a negative answer showing that some gross substitutes cannot be expressed as positive linear combinations of matroid rank functions. En route, we provide necessary and sufficient conditions for the sum to preserve substitutability, uncover a new operation preserving substitutability, and fully describe all substitutes with at most four items.

scholarly journals Conditions under which a lattice is isomorphic to the subgroup lattice of an abelian group

2011 ◽  
Vol 27 (2) ◽  
pp. 193-199
Author(s):  
CAROLINA CONTIU ◽  
Keyword(s):  
Abelian Group ◽  
Normal Subgroup ◽  
Sufficient Conditions ◽  
Subgroup Lattice ◽  
Necessary And Sufficient Conditions ◽  
Arbitrary Group ◽  
Necessary And Sufficient ◽  
Arbitrary Abelian Group

In this paper, we provide necessary and sufficient conditions under which a lattice is isomorphic to the subgroup lattice of an arbitrary abelian group. We also give necessary and sufficient conditions for a lattice L, to be isomorphic to the normal subgroup lattice of an arbitrary group.

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