scholarly journals Quandle coloring quivers

2019 ◽  
Vol 28 (01) ◽  
pp. 1950001 ◽  
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.

RACKS AND LINKS IN CODIMENSION TWO

1992 ◽  
Vol 01 (04) ◽  
pp. 343-406 ◽  
Author(s):  
ROGER FENN ◽  
COLIN ROURKE
Keyword(s):  
Fundamental Group ◽  
Binary Operation ◽  
Codimension Two ◽  
Link Invariants ◽  
Reidemeister Moves ◽  
Algebraic Framework

A rack, which is the algebraic distillation of two of the Reidemeister moves, is a set with a binary operation such that right multiplication is an automorphism. Any codimension two link has a fundamental rack which contains more information than the fundamental group. Racks provide an elegant and complete algebraic framework in which to study links and knots in 3–manifolds, and also for the 3–manifolds themselves. Racks have been studied by several previous authors and have been called a variety of names. In this first paper of a series we consolidate the algebra of racks and show that the fundamental rack is a complete invariant for irreducible framed links in a 3–manifold and for the 3–manifold itself. We give some examples of computable link invariants derived from the fundamental rack and explain the connection of the theory of racks with that of braids.

scholarly journals Reidemeister transformations of the potential function and the solution

2017 ◽  
Vol 26 (12) ◽  
pp. 1750079 ◽  
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.

scholarly journals Writhe-like invariants of alternating links

2021 ◽  
Vol 30 (01) ◽  
pp. 2150004
Author(s):  
Yuanan Diao ◽  
Van Pham
Keyword(s):  
Jones Polynomial ◽  
Link Diagram ◽  
Link Invariant ◽  
Link Invariants ◽  
Link Diagrams ◽  
Alternating Links ◽  
Alternating Link

It is known that the writhe calculated from any reduced alternating link diagram of the same (alternating) link has the same value. That is, it is a link invariant if we restrict ourselves to reduced alternating link diagrams. This is due to the fact that reduced alternating link diagrams of the same link are obtainable from each other via flypes and flypes do not change writhe. In this paper, we introduce several quantities that are derived from Seifert graphs of reduced alternating link diagrams. We prove that they are “writhe-like” invariants, namely they are not general link invariants, but are invariants when restricted to reduced alternating link diagrams. The determination of these invariants are elementary and non-recursive so they are easy to calculate. We demonstrate that many different alternating links can be easily distinguished by these new invariants, even for large, complicated knots for which other invariants such as the Jones polynomial are hard to compute. As an application, we also derive an if and only if condition for a strongly invertible rational link.

Linking invariants of even virtual links

2017 ◽  
Vol 26 (12) ◽  
pp. 1750072 ◽  
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

1994 ◽  
Vol 03 (04) ◽  
pp. 465-475 ◽  
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.

ON THE NULLIFICATION WRITHE, THE SIGNATURE AND THE CHIRALITY OF ALTERNATING LINKS

2001 ◽  
Vol 10 (04) ◽  
pp. 537-545 ◽  
Author(s):  
CAM VAN QUACH HONGLER
Keyword(s):  
Link Diagram ◽  
Link Invariants ◽  
Link Diagrams ◽  
The Difference ◽  
Alternating Links ◽  
Alternating Link

In this paper, we relate the nullification writhe and the remaining writhe defined by C. Cerf to other link invariants. We prove that the nullification writhe of an oriented reduced alternating link diagram is equal, up to sign, to the signature of the link. Moreover, we relate the difference between the nullication writhe and the remaining writhe to invariants obtained from chessboard-coloured link diagrams such as their numbers of shaded and unshaded regions.

scholarly journals Up–down colorings of virtual-link diagrams and the necessity of Reidemeister moves of type II

2017 ◽  
Vol 26 (12) ◽  
pp. 1750073 ◽  
Author(s):  
Kanako Oshiro ◽  
Ayaka Shimizu ◽  
Yoshiro Yaguchi
Keyword(s):  
Type Ii ◽  
Virtual Link ◽  
Link Diagram ◽  
Virtual Knot ◽  
Reidemeister Moves ◽  
Link Diagrams ◽  
Classical Link ◽  
Minimum Number ◽  
Knot Diagrams ◽  
Text Component

We introduce an up–down coloring of a virtual-link (or classical-link) diagram. The colorabilities give a lower bound of the minimum number of Reidemeister moves of type II which are needed between two [Formula: see text]-component virtual-link (or classical-link) diagrams. By using the notion of a quandle cocycle invariant, we give a method to detect the necessity of Reidemeister moves of type II between two given virtual-knot (or classical-knot) diagrams. As an application, we show that for any virtual-knot diagram [Formula: see text], there exists a diagram [Formula: see text] representing the same virtual-knot such that any sequence of generalized Reidemeister moves between them includes at least one Reidemeister move of type II.

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.

scholarly journals BIRACK DYNAMICAL COCYCLES AND HOMOMORPHISM INVARIANTS

2013 ◽  
Vol 12 (08) ◽  
pp. 1350049 ◽  
Author(s):  
SAM NELSON ◽  
EMILY WATTERBERG
Keyword(s):  
Algebraic Structures ◽  
Link Diagram ◽  
Link Invariants ◽  
Knots And Links

Biracks are algebraic structures related to knots and links. We define a new enhancement of the birack counting invariant for oriented knots and links via algebraic structures called birack dynamical cocycles. The new invariants can also be understood in terms of partitions of the set of birack labelings of a link diagram determined by a homomorphism p : X → Y between finite labeling biracks. We provide examples to show that the new invariant is stronger than the unenhanced birack counting invariant and examine connections with other knot and link invariants.

Knot invariants with multiple skein relations

2018 ◽  
Vol 27 (02) ◽  
pp. 1850017 ◽  
Author(s):  
Zhiqing Yang
Keyword(s):  
Link Diagram ◽  
Oriented Link ◽  
Knot Invariants ◽  
Link Invariants ◽  
Kauffman Polynomial ◽  
Skein Relation

Given any oriented link diagram, one can construct knot invariants using skein relations. Usually such a skein relation contains three or four terms. In this paper, the author introduces several new ways to smooth a crossings, and uses a system of skein equations to construct link invariants. This invariant can also be modified by writhe to get a more powerful invariant. The modified invariant is a generalization of both the HOMFLYPT polynomial and the two-variable Kauffman polynomial. Using the diamond lemma, a simplified version of the modified invariant is given. It is easy to compute and is a generalization of the two-variable Kauffman polynomial.

Sign in / Sign up

Export Citation Format

Share Document