scholarly journals Triple-crossing projections, moves on knots and links and their minimal diagrams

2020 ◽  
Vol 29 (04) ◽  
pp. 2050015 ◽  
Author(s):  
Michał Jabłonowski ◽  
Łukasz Trojanowski
Keyword(s):  
Triple Point ◽  
Crossing Number ◽  
Systematic Method ◽  
Generating Set ◽  
New Type ◽  
Knots And Links ◽  
Minimal Generating Set

In this paper, we present a systematic method to generate prime knot and prime link minimal triple-point projections, and then classify all classical prime knots and prime links with triple-crossing number at most four. We also extend the table of known knots and links with triple-crossing number equal to five. By introducing a new type of diagrammatic move, we reduce the number of generating moves on triple-crossing diagrams, and derive a minimal generating set of moves connecting triple-crossing diagrams of the same knot.

Stick numbers of Montesinos knots

2021 ◽  
pp. 2150013
Author(s):  
Hwa Jeong Lee ◽  
Sungjong No ◽  
Seungsang Oh
Keyword(s):  
Upper Bound ◽  
Crossing Number ◽  
Number Formula ◽  
Knots And Links

Negami found an upper bound on the stick number [Formula: see text] of a nontrivial knot [Formula: see text] in terms of the minimal crossing number [Formula: see text]: [Formula: see text]. Huh and Oh found an improved upper bound: [Formula: see text]. Huh, No and Oh proved that [Formula: see text] for a [Formula: see text]-bridge knot or link [Formula: see text] with at least six crossings. As a sequel to this study, we present an upper bound on the stick number of Montesinos knots and links. Let [Formula: see text] be a knot or link which admits a reduced Montesinos diagram with [Formula: see text] crossings. If each rational tangle in the diagram has five or more index of the related Conway notation, then [Formula: see text]. Furthermore, if [Formula: see text] is alternating, then we can additionally reduce the upper bound by [Formula: see text].

scholarly journals The Fundamental Group of Balanced Simplicial Complexes and Posets

10.37236/73 ◽  
2009 ◽  
Vol 16 (2) ◽  
Author(s):  
Steven Klee
Keyword(s):  
Fundamental Group ◽  
Upper Bound ◽  
Large Family ◽  
Simplicial Complexes ◽  
Generating Set ◽  
Minimal Generating Set

We establish an upper bound on the cardinality of a minimal generating set for the fundamental group of a large family of connected, balanced simplicial complexes and, more generally, simplicial posets.

A new type of a small cell for the realization of the triple point of mercury

2000 ◽  
Vol 11 (6) ◽  
pp. 738-742 ◽  
Author(s):  
Leszek Lipinski ◽  
Anna Szmyrka-Grzebyk ◽  
Henryk Manuszkiewicz
Keyword(s):  
Triple Point ◽  
Small Cell ◽  
New Type ◽  
Triple Point Of Mercury

A MULTI-VARIABLE POLYNOMIAL INVARIANT FOR UNORIENTED VIRTUAL KNOTS AND LINKS

2009 ◽  
Vol 18 (05) ◽  
pp. 625-649 ◽  
Author(s):  
YASUYUKI MIYAZAWA
Keyword(s):  
Crossing Number ◽  
Weighted Sum ◽  
Virtual Link ◽  
Polynomial Invariant ◽  
Virtual Knots ◽  
Virtual Links ◽  
Knots And Links

We construct a multi-variable polynomial invariant Y for unoriented virtual links as a certain weighted sum of polynomials, which are derived from virtual magnetic graphs with oriented vertices, on oriented virtual links associated with a given virtual link. We show some features of the Y-polynomial including an evaluation of the virtual crossing number of a virtual link.

The Fuzziness of a Minimal Generating Set of Fedorov Groups

2021 ◽  
Vol 66 (6) ◽  
pp. 913-919
Author(s):  
A. M. Banaru ◽  
V. R. Shiroky ◽  
D. A. Banaru
Keyword(s):  
Generating Set ◽  
Minimal Generating Set

The Number of Generators of a Linear p-Group

1972 ◽  
Vol 24 (5) ◽  
pp. 851-858 ◽  
Author(s):  
I. M. Isaacs
Keyword(s):  
Generating Set ◽  
Nilpotence Class ◽  
Number Of Generators ◽  
Minimal Generating Set

Let G be a finite p-group, having a faithful character χ of degree f. The object of this paper is to bound the number, d(G), of generators in a minimal generating set for G in terms of χ and in particular in terms of f. This problem was raised by D. M. Goldschmidt, and solved by him in the case that G has nilpotence class 2.

scholarly journals Triple-crossing number and moves on triple-crossing link diagrams

2019 ◽  
Vol 28 (11) ◽  
pp. 1940001 ◽  
Author(s):  
Colin Adams ◽  
Jim Hoste ◽  
Martin Palmer
Keyword(s):  
Triple Point ◽  
Crossing Number ◽  
Reidemeister Moves ◽  
Link Diagrams ◽  
Triple Points ◽  
Definition Of ◽  
Hopf Link

Every link in the 3-sphere has a projection to the plane where the only singularities are pairwise transverse triple points. The associated diagram, with height information at each triple point, is a triple-crossing diagram of the link. We give a set of diagrammatic moves on triple-crossing diagrams analogous to the Reidemeister moves on ordinary diagrams. The existence of [Formula: see text]-crossing diagrams for every [Formula: see text] greater than one allows the definition of the [Formula: see text]-crossing number. We prove that for any nontrivial, nonsplit link, other than the Hopf link, its triple-crossing number is strictly greater than its quintuple-crossing number.

Generalization of the basis property on finite groups

2020 ◽  
pp. 2150111
Author(s):  
Ibrahim Al-Dayel ◽  
Ahmad Al Khalaf
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Frobenius Group ◽  
Sufficient Conditions ◽  
Basis Property ◽  
Generating Set ◽  
Equivalent Basis ◽  
Necessary And Sufficient ◽  
Special Case ◽  
Minimal Generating Set

A group [Formula: see text] has the Basis Property if every subgroup [Formula: see text] of [Formula: see text] has an equivalent basis (minimal generating set). We studied a special case of the finite group with the Basis Property, when [Formula: see text]-group [Formula: see text] is an abelian group. We found the necessary and sufficient conditions on an abelian [Formula: see text]-group [Formula: see text] of [Formula: see text] with the Basis Property to be kernel of Frobenius group.

Minimal presentations for groups of order 2n, n ≤ 6

1973 ◽  
Vol 15 (4) ◽  
pp. 461-469 ◽  
Author(s):  
T. W. Saga ◽  
J. W. Wamsley
Keyword(s):  
Generating Set ◽  
Image Position ◽  
Minimal Presentations ◽  
Minimal Generating Set

Let G be a finite 2-group having a minimal generating set {x1, …, xr} so that r = d (G) is an invariant of G. Suppose further that G has a presentation then.

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