Tribracket Modules

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.

Biquasiles and dual graph diagrams

10.1142/s0218216517500481 ◽

2017 ◽

Vol 26 (08) ◽

pp. 1750048 ◽

Cited By ~ 8

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.

Quandle coloring quivers

10.1142/s0218216519500019 ◽

2019 ◽

Vol 28 (01) ◽

pp. 1950001 ◽

Cited By ~ 1

Author(s):

Karina Cho ◽

Sam Nelson

Keyword(s):

Small Category ◽

Link Diagram ◽

Literal Sense ◽

Link Invariants ◽

Reidemeister Moves

We consider a quiver structure on the set of quandle colorings of an oriented knot or link diagram. This structure contains a wealth of knot and link invariants and provides a categorification of the quandle counting invariant in the most literal sense, i.e. giving the set of quandle colorings the structure of a small category which is unchanged by Reidemeister moves. We derive some new enhancements of the counting invariant from this quiver structure and show that the enhancements are proper with explicit examples.

A VOLUME FORM ON THE KHOVANOV INVARIANT

10.1142/s0218216511009844 ◽

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 ◽

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.

Reidemeister transformations of the potential function and the solution

10.1142/s0218216517500791 ◽

2017 ◽

Vol 26 (12) ◽

pp. 1750079 ◽

Cited By ~ 2

Author(s):

Jinseok Cho ◽

Jun Murakami

Keyword(s):

Potential Function ◽

Jones Polynomial ◽

Faithful Representation ◽

Link Diagram ◽

Explicit Formulas ◽

Colored Jones Polynomial ◽

Volume Formula ◽

Link Group ◽

Reidemeister Moves

The potential function of the optimistic limit of the colored Jones polynomial and the construction of the solution of the hyperbolicity equations were defined in the authors’ previous papers. In this paper, we define the Reidemeister transformations of the potential function and the solution by the changes of them under the Reidemeister moves of the link diagram and show the explicit formulas. These two formulas enable us to see the changes of the complex volume formula under the Reidemeister moves. As an application, we can simply specify the discrete faithful representation of the link group by showing a link diagram and one geometric solution.

ON A LOCAL MOVE FOR VIRTUAL KNOTS AND LINKS

10.1142/s0218216509007592 ◽

2009 ◽

Vol 18 (11) ◽

pp. 1577-1596 ◽

Cited By ~ 1

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 ◽

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).

Recurrent Generalization of F-Polynomials for Virtual Knots and Links

Symmetry ◽

10.3390/sym14010015 ◽

2021 ◽

Vol 14 (1) ◽

pp. 15

Author(s):

Amrendra Gill ◽

Maxim Ivanov ◽

Madeti Prabhakar ◽

Andrei Vesnin

Keyword(s):

Local Symmetry ◽

Weight Functions ◽

Virtual Link ◽

Link Diagram ◽

Polynomial Invariants ◽

Knot Invariants ◽

Virtual Knot ◽

Virtual Knots ◽

Virtual Links ◽

F-polynomials for virtual knots were defined by Kaur, Prabhakar and Vesnin in 2018 using flat virtual knot invariants. These polynomials naturally generalize Kauffman’s affine index polynomial and use smoothing in the classical crossing of a virtual knot diagram. In this paper, we introduce weight functions for ordered orientable virtual and flat virtual links. A flat virtual link is an equivalence class of virtual links with respect to a local symmetry changing a type of classical crossing in a diagram. By considering three types of smoothing in classical crossings of a virtual link diagram and suitable weight functions, there is provided a recurrent construction for new invariants. It is demonstrated by explicit examples that newly defined polynomial invariants are stronger than F-polynomials.

The self-smoothing number of knots and links

10.1142/s0218216518500700 ◽

2018 ◽

Vol 27 (12) ◽

pp. 1850070

Author(s):

Hideo Takioka

Keyword(s):

The Self ◽

Link Diagram ◽

Oriented Link ◽

Two Component ◽

Crossing Point ◽

We call smoothing a self-crossing point of an oriented link diagram self-smoothing. By self-smoothing repeatedly, we obtain an oriented link diagram without self-crossing points. In this paper, we show that every knot has an oriented diagram which becomes a two-component oriented link diagram without self-crossing points by a single self-smoothing.

Singular knots and involutive quandles

10.1142/s0218216517500997 ◽

2017 ◽

Vol 26 (14) ◽

pp. 1750099 ◽

Cited By ~ 5

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.

Linking invariants of even virtual links

10.1142/s0218216517500729 ◽

2017 ◽

Vol 26 (12) ◽

pp. 1750072 ◽

Cited By ~ 1

Author(s):

Haruko A. Miyazawa ◽

Kodai Wada ◽

Akira Yasuhara

Keyword(s):

Lower Bound ◽

The Other ◽

Virtual Link ◽

Link Diagram ◽

Linking Number ◽

Reidemeister Moves ◽

Link Diagrams ◽

Classical Link ◽

Virtual Links ◽

The Difference

A virtual link diagram is even if the virtual crossings divide each component into an even number of arcs. The set of even virtual link diagrams is closed under classical and virtual Reidemeister moves, and it contains the set of classical link diagrams. For an even virtual link diagram, we define a certain linking invariant which is similar to the linking number. In contrast to the usual linking number, our linking invariant is not preserved under the forbidden moves. In particular, for two fused isotopic even virtual link diagrams, the difference between the linking invariants of them gives a lower bound of the minimal number of forbidden moves needed to deform one into the other. Moreover, we give an example which shows that the lower bound is best possible.

GENERALIZED SPIN MODELS

10.1142/s0218216594000344 ◽

1994 ◽

Vol 03 (04) ◽

pp. 465-475 ◽

Cited By ~ 13

Author(s):

KENICHI KAWAGOE ◽

AKIHIRO MUNEMASA ◽

YASUO WATATANI

Keyword(s):

Partition Function ◽

Spin Model ◽

Symmetry Condition ◽

Type Ii ◽

Link Diagram ◽

Spin Models ◽

Oriented Link ◽

Reidemeister Moves

We introduce a generalization of spin models by dropping the symmetry condition. The partition function of a generalized spin model on a connected oriented link diagram is invariant under Reidemeister moves of type II and III, giving an invariant for oriented links.