scholarly journals Niebrzydowski algebras and trivalent spatial graphs

2018 ◽  
Vol 29 (14) ◽  
pp. 1850102
Author(s):  
Paige Graves ◽  
Sam Nelson ◽  
Sherilyn Tamagawa
Keyword(s):  
Algebraic Structures ◽  
Generating Sets ◽  
Reidemeister Moves ◽  
Spatial Graphs ◽  
Ternary Operation

We introduce Niebrzydowski algebras, algebraic structures with a ternary operation and a partially defined multiplication, with axioms motivated by the Reidemeister moves for [Formula: see text]-oriented trivalent spatial graphs and handlebody-links. As part of this definition, we identify generating sets of [Formula: see text]-oriented Reidemeister moves. We give some examples to demonstrate that the counting invariant can distinguish some [Formula: see text]-oriented trivalent spatial graphs and handlebody-links.

scholarly journals Biquasiles and dual graph diagrams

2017 ◽  
Vol 26 (08) ◽  
pp. 1750048 ◽  
Author(s):  
Deanna Needell ◽  
Sam Nelson
Keyword(s):  
Dual Graph ◽  
Algebraic Structures ◽  
Combinatorial Structures ◽  
Reidemeister Moves ◽  
Knots And Links ◽  
The Way

We introduce dual graph diagrams representing oriented knots and links. We use these combinatorial structures to define corresponding algebraic structures which we call biquasiles whose axioms are motivated by dual graph Reidemeister moves, generalizing the Dehn presentation of the knot group analogously to the way quandles and biquandles generalize the Wirtinger presentation. We use these structures to define invariants of oriented knots and links and provide examples.

scholarly journals Singular knots and involutive quandles

2017 ◽  
Vol 26 (14) ◽  
pp. 1750099 ◽  
Author(s):  
Indu R. U. Churchill ◽  
Mohamed Elhamdadi ◽  
Mustafa Hajij ◽  
Sam Nelson
Keyword(s):  
Knot Theory ◽  
Algebraic Structures ◽  
Reidemeister Moves ◽  
Knots And Links ◽  
Singular Knots

The aim of this paper is to define certain algebraic structures coming from generalized Reidemeister moves of singular knot theory. We give examples, show that the set of colorings by these algebraic structures is an invariant of singular links. As an application, we distinguish several singular knots and links.

scholarly journals Involutory Quandles and Dichromatic Links

Symmetry ◽  
2020 ◽  
Vol 12 (1) ◽  
pp. 111
Author(s):  
Khaled Bataineh ◽  
Ilham Saidi
Keyword(s):  
Algebraic Structure ◽  
Algebraic Structures ◽  
Reidemeister Moves ◽  
Two Component

We define a new algebraic structure for two-component dichromatic links. This definition extends the notion of a kei (or involutory quandle) from regular links to dichromatic links. We call this structure a dikei that results from the generalized Reidemeister moves representing dichromatic isotopy. We give several examples on dikei and show that the set of colorings by these algebraic structures is an invariant of dichromatic links. As an application, we distinguish several pairs of dichromatic links that are symmetric as monochromatic links.

scholarly journals Minimal generating sets of directed oriented Reidemeister moves

2017 ◽  
Vol 26 (04) ◽  
pp. 1750016 ◽  
Author(s):  
Piotr Suwara
Keyword(s):  
Generating Set ◽  
Type Formula ◽  
Generating Sets ◽  
Reidemeister Moves ◽  
Different Types ◽  
Minimal Generating Set ◽  
Minimal Formula

Polyak proved that the set [Formula: see text] is a minimal generating set of oriented Reidemeister moves. One may distinguish between forward and backward moves, obtaining [Formula: see text] different types of moves, which we call directed oriented Reidemeister moves. In this paper, we prove that the set of eight directed Polyak moves [Formula: see text] is a minimal generating set of directed oriented Reidemeister moves. We also specialize the problem, introducing the notion of a [Formula: see text]-generating set for a link [Formula: see text]. The same set is proven to be a minimal [Formula: see text]-generating set for any link [Formula: see text] with at least two components. Finally, we discuss knot diagram invariants arising in the study of [Formula: see text]-generating sets for an arbitrary knot [Formula: see text], emphasizing the distinction between ascending and descending moves of type [Formula: see text].

scholarly journals Generating sets of Reidemeister moves of oriented singular links and quandles

2018 ◽  
Vol 27 (14) ◽  
pp. 1850064 ◽  
Author(s):  
Khaled Bataineh ◽  
Mohamed Elhamdadi ◽  
Mustafa Hajij ◽  
William Youmans
Keyword(s):  
Algebraic Structure ◽  
Abelian Groups ◽  
Generating Set ◽  
Generating Sets ◽  
Reidemeister Moves ◽  
Singular Knots

We give a generating set of the generalized Reidemeister moves for oriented singular links. We then introduce an algebraic structure arising from the axiomatization of Reidemeister moves on oriented singular knots. We give some examples, including some non-isomorphic families of such structures over non-abelian groups. We show that the set of colorings of a singular knot by this new structure is an invariant of oriented singular knots and use it to distinguish some singular links.

Main problems of import subsumption in domestic engineering, applicable to small-power electric generator installations

2018 ◽  
Vol 1 (1) ◽  
pp. 46-51 ◽  
Author(s):  
A. V. Shelgunov
Keyword(s):  
Low Power ◽  
Import Substitution ◽  
Power Generator ◽  
Small Power ◽  
Domestic Production ◽  
Power Generators ◽  
Electric Generator ◽  
Generating Sets ◽  
Current State ◽  
Generator Sets

Subject: the subject of the study are low-power generator sets with a power of up to 30 kW.Materials and methods: in this paper, the main domestic legislative documents regulating the requirements for products. An assessment is made of the current state of Russian engine building.Results: the detailed analysis of the modern domestic market of power generating units with a capacity of up to 30 kW is made, the main problems in the field of domestic production of  electric power generators in the range up to 30 kW are revealed, and the prospects for import substitution of gasoline and diesel engines are noted.Conclusions: almost complete absence of the market of domestic low-power generating sets is established, insufficient measures taken to support domestic producers are noted, measures are  proposed for the development of domestic production of power units in the range of up to 30 kW.

Generating the Mapping Class Group

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Exact Sequence ◽  
Simplicial Complex ◽  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Detailed Structure ◽  
Inductive Step ◽  
Closed Curves ◽  
Combinatorial Object ◽  
Generating Sets

This chapter considers the Dehn–Lickorish theorem, which states that when g is greater than or equal to 0, the mapping class group Mod(Sɡ) is generated by finitely many Dehn twists about nonseparating simple closed curves. The theorem is proved by induction on genus, and the Birman exact sequence is introduced as the key step for the induction. The key to the inductive step is to prove that the complex of curves C(Sɡ) is connected when g is greater than or equal to 2. The simplicial complex C(Sɡ) is a useful combinatorial object that encodes intersection patterns of simple closed curves in Sɡ. More detailed structure of C(Sɡ) is then used to find various explicit generating sets for Mod(Sɡ), including those due to Lickorish and to Humphries.

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