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 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 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.

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 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).

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.

Integrating a weight system of order n to an invariant of (n−1)-singular knots

1996 ◽  
Vol 05 (01) ◽  
pp. 23-35 ◽  
Author(s):  
MICHEL DOMERGUE ◽  
PAUL DONATO
Keyword(s):  
Weight System ◽  
Combinatorial Formula ◽  
Chord Diagrams ◽  
Reidemeister Moves ◽  
Singular Knots

Starting from a Weight-System denoted by P and defined on the n-Chord-Diagrams with values in an arbitrary Q–module, we give an explicit combinatorial formula for an invariant of (n–1)-singular knots which has P as its derivative. The formula is defined for regular knot projections. Its invariance under singular Reidemeister moves is then proved.

Virtual tribrackets

2019 ◽  
Vol 28 (04) ◽  
pp. 1950026
Author(s):  
Sam Nelson ◽  
Shane Pico
Keyword(s):  
Algebraic Structure ◽  
Link Diagram ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Knots And Links

We introduce virtual tribrackets, an algebraic structure for coloring regions in the planar complement of an oriented virtual knot or link diagram. We use these structures to define counting invariants of virtual knots and links and provide examples of the computation of the invariant; in particular, we show that the invariant can distinguish certain virtual knots.

scholarly journals Polynomial invariants of singular knots and links

2021 ◽  
pp. 2150003
Author(s):  
Jose Ceniceros ◽  
Indu R. Churchill ◽  
Mohamed Elhamdadi
Keyword(s):  
Polynomial Invariant ◽  
Link Invariant ◽  
Polynomial Invariants ◽  
Singular Link ◽  
Knots And Links ◽  
Singular Knots

We generalize the notion of the quandle polynomial to the case of singquandles. We show that the singquandle polynomial is an invariant of finite singquandles. We also construct a singular link invariant from the singquandle polynomial and show that this new singular link invariant generalizes the singquandle counting invariant. In particular, using the new polynomial invariant, we can distinguish singular links with the same singquandle counting invariant.

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.

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