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 A VOLUME FORM ON THE KHOVANOV INVARIANT

2012 ◽  
Vol 21 (04) ◽  
pp. 1250032
Author(s):  
JUAN ORTIZ-NAVARRO
Keyword(s):  
Chain Complex ◽  
Khovanov Homology ◽  
Volume Form ◽  
Reidemeister Torsion ◽  
Homology Groups ◽  
Numerical Invariant ◽  
Reidemeister Moves ◽  
Invariant Volume ◽  
Knots And Links

The Reidemeister torsion construction can be applied to the chain complex used to compute the Khovanov homology of a knot or a link. This defines a volume form on Khovanov homology. The volume form transforms correctly under Reidemeister moves to give an invariant volume on the Khovanov homology. In this paper, its construction and invariance under these moves is demonstrated. Also, some examples of the invariant are presented for particular choices for the bases of homology groups to obtain a numerical invariant of knots and links. In these examples, the algebraic torsion seen in the Khovanov chain complex when homology is computed over ℤ is recovered.

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).

Multi-tribrackets

2019 ◽  
Vol 28 (12) ◽  
pp. 1950075
Author(s):  
Sam Nelson ◽  
Evan Pauletich
Keyword(s):  
Single Component ◽  
Algebraic Structures ◽  
Future Directions ◽  
Knots And Links

We introduce multi-tribrackets, algebraic structures for region coloring of diagrams of knots and links with different operations at different kinds of crossings. In particular, we consider the case of component multi-tribrackets which have different tribracket operations at single-component crossings and multi-component crossings. We provide examples to show that the resulting counting invariants can distinguish links which are not distinguished by the counting invariants associated to the standard tribracket coloring. We reinterpret the results of [S. Nelson and S. Pico. Virtual tribrackets, preprint (2018), arXiv:1803.03210] in terms of multi-tribrackets and consider future directions for multi-tribracket theory.

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.

From neural dynamics to true combinatorial structures

2006 ◽  
Vol 29 (1) ◽  
pp. 88-104 ◽  
Author(s):  
Frank van der Velde ◽  
Marc de Kamps
Keyword(s):  
Neural Dynamics ◽  
Combinatorial Structures ◽  
Language And Vision ◽  
The Way

Various issues concerning the neural blackboard architectures for combinatorial structures are discussed and clarified. They range from issues related to neural dynamics, the structure of the architectures for language and vision, and alternative architectures, to linguistic issues concerning the language architecture. Particular attention is given to the nature of true combinatorial structures and the way in which information can be retrieved from them in a productive and systematic manner.

scholarly journals Involutory biquandles and singular knots and links

2018 ◽  
Vol 16 (1) ◽  
pp. 469-489
Author(s):  
Khaled Bataineh ◽  
Hadeel Ghaith
Keyword(s):  
Algebraic Structure ◽  
Reidemeister Moves ◽  
Knots And Links ◽  
Singular Knots

AbstractWe define a new algebraic structure for singular knots and links. It extends the notion of a bikei (or involutory biquandle) from regular knots and links to singular knots and links. We call this structure a singbikei. This structure results from the generalized Reidemeister moves representing singular isotopy. We give several examples on singbikei and we use singbikei to distinguish several singular knots and links.

scholarly journals Crossing change on Khovanov homology and a categorified Vassiliev skein relation

2020 ◽  
Vol 29 (07) ◽  
pp. 2050051
Author(s):  
Noboru Ito ◽  
Jun Yoshida
Keyword(s):  
Jones Polynomial ◽  
Khovanov Homology ◽  
Quantum Invariants ◽  
Quantum Invariant ◽  
Finite Type Invariants ◽  
Reidemeister Moves ◽  
Jones Polynomials ◽  
Genus One ◽  
Knots And Links ◽  
Skein Relation

Khovanov homology is a categorification of the Jones polynomial, so it may be seen as a kind of quantum invariant of knots and links. Although polynomial quantum invariants are deeply involved with Vassiliev (aka. finite type) invariants, the relation remains unclear in case of Khovanov homology. Aiming at it, in this paper, we discuss a categorified version of Vassiliev skein relation on Khovanov homology. More precisely, we will show that the “genus-one” operation gives rise to a crossing change on Khovanov complexes. Invariance under Reidemeister moves turns out, and it enables us to extend Khovanov homology to singular links. We then see that a long exact sequence of Khovanov homology groups categorifies Vassiliev skein relation for the Jones polynomials. In particular, the Jones polynomial is recovered even for singular links. We in addition discuss the FI relation on Khovanov homology.

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.

Tribracket Modules

2020 ◽  
Vol 31 (04) ◽  
pp. 2050028
Author(s):  
Deanna Needell ◽  
Sam Nelson ◽  
Yingqi Shi
Keyword(s):  
Link Diagram ◽  
Reidemeister Moves ◽  
Knots And Links

Niebrzydowski tribrackets are ternary operations on sets satisfying conditions obtained from the oriented Reidemeister moves such that the set of tribracket colorings of an oriented knot or link diagram is an invariant of oriented knots and links. We introduce tribracket modules analogous to quandle/biquandle/rack modules and use these structures to enhance the tribracket counting invariant. We provide examples to illustrate the computation of the invariant and show that the enhancement is proper.

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