scholarly journals Multivariate Alexander quandles, III. Sublinks

2019 ◽  
Vol 28 (14) ◽  
pp. 1950090
Author(s):  
Lorenzo Traldi
Keyword(s):  
Link Invariant ◽  
Classical Link

If [Formula: see text] is a classical link then the multivariate Alexander quandle, [Formula: see text], is a substructure of the multivariate Alexander module, [Formula: see text]. In the first paper of this series, we showed that if two links [Formula: see text] and [Formula: see text] have [Formula: see text], then after an appropriate re-indexing of the components of [Formula: see text] and [Formula: see text], there will be a module isomorphism [Formula: see text] of a particular type, which we call a “Crowell equivalence.” In this paper, we show that [Formula: see text] (up to quandle isomorphism) is a strictly stronger link invariant than [Formula: see text] (up to re-indexing and Crowell equivalence). This result follows from the fact that [Formula: see text] determines the [Formula: see text] quandles of all the sublinks of [Formula: see text], up to quandle isomorphisms.

IDEAL COSET INVARIANTS FOR SURFACE-LINKS IN ℝ4

2013 ◽  
Vol 22 (09) ◽  
pp. 1350052 ◽  
Author(s):  
YEWON JOUNG ◽  
JIEON KIM ◽  
SANG YOUL LEE
Keyword(s):  
Normal Form ◽  
Polynomial Ring ◽  
Gröbner Basis ◽  
Quotient Ring ◽  
Grobner Basis ◽  
Link Invariant ◽  
Link Invariants ◽  
Classical Link ◽  
Marked Graph

In [Towards invariants of surfaces in 4-space via classical link invariants, Trans. Amer. Math. Soc.361 (2009) 237–265], Lee defined a polynomial [[D]] for marked graph diagrams D of surface-links in 4-space by using a state-sum model involving a given classical link invariant. In this paper, we deal with some obstructions to obtain an invariant for surface-links represented by marked graph diagrams D by using the polynomial [[D]] and introduce an ideal coset invariant for surface-links, which is defined to be the coset of the polynomial [[D]] in a quotient ring of a certain polynomial ring modulo some ideal and represented by a unique normal form, i.e. a unique representative for the coset of [[D]] that can be calculated from [[D]] with the help of a Gröbner basis package on computer.

scholarly journals Applying Lipson's state models to marked graph diagrams of surface-links

2015 ◽  
Vol 24 (10) ◽  
pp. 1540003 ◽  
Author(s):  
Yewon Joung ◽  
Seiichi Kamada ◽  
Sang Youl Lee
Keyword(s):  
Link Invariant ◽  
Link Invariants ◽  
Classical Link ◽  
Marked Graph ◽  
Kauffman Polynomial ◽  
State Models

A. S. Lipson constructed two state models yielding the same classical link invariant obtained from the Kauffman polynomial F(a, u). In this paper, we apply Lipson's state models to marked graph diagrams of surface-links, and observe when they induce surface-link invariants.

scholarly journals AN INVARIANT FOR SINGULAR KNOTS

2009 ◽  
Vol 18 (06) ◽  
pp. 825-840 ◽  
Author(s):  
J. JUYUMAYA ◽  
S. LAMBROPOULOU
Keyword(s):  
Hecke Algebras ◽  
Quadratic Relation ◽  
Link Invariant ◽  
Topological Interpretation ◽  
Type Invariant ◽  
Singular Link ◽  
Markov Trace ◽  
Skein Relation ◽  
Singular Knots

In this paper we introduce a Jones-type invariant for singular knots, using a Markov trace on the Yokonuma–Hecke algebras Y d,n(u) and the theory of singular braids. The Yokonuma–Hecke algebras have a natural topological interpretation in the context of framed knots. Yet, we show that there is a homomorphism of the singular braid monoid SBn into the algebra Y d,n(u). Surprisingly, the trace does not normalize directly to yield a singular link invariant, so a condition must be imposed on the trace variables. Assuming this condition, the invariant satisfies a skein relation involving singular crossings, which arises from a quadratic relation in the algebra Y d,n(u).

scholarly journals Link invariant and $G_2$ web space

2017 ◽  
Vol 47 (1) ◽  
pp. 19-41
Author(s):  
Takuro Sakamoto ◽  
Yasuyoshi Yonezawa
Keyword(s):  
Link Invariant ◽  
Web Space

scholarly journals ALL LINK INVARIANTS FOR TWO DIMENSIONAL SOLUTIONS OF YANG–BAXTER EQUATION AND DRESSINGS

2006 ◽  
Vol 15 (10) ◽  
pp. 1279-1301
Author(s):  
N. AIZAWA ◽  
M. HARADA ◽  
M. KAWAGUCHI ◽  
E. OTSUKI
Keyword(s):  
Jones Polynomial ◽  
Alexander Polynomial ◽  
Baxter Equation ◽  
Two Dimensional ◽  
Link Invariant ◽  
Three Solutions ◽  
Polynomial Invariants ◽  
Link Invariants ◽  
Higher Dimensional

All polynomial invariants of links for two dimensional solutions of Yang–Baxter equation is constructed by employing Turaev's method. As a consequence, it is proved that the best invariant so constructed is the Jones polynomial and there exist three solutions connecting to the Alexander polynomial. Invariants for higher dimensional solutions, obtained by the so-called dressings, are also investigated. It is observed that the dressings do not improve link invariant unless some restrictions are put on dressed solutions.

Making Links Between Solutions to an Unstructured Problem

2017 ◽  
pp. 152-175
Author(s):  
Sheila Evans
Keyword(s):  
Link Invariant ◽  
Teaching Resources ◽  
Student Responses ◽  
Solution Strategies ◽  
Invariant Properties

In the study described here, teaching resources have been developed to provide students with explicit opportunities to link invariant properties across a range of different solution strategies, and make comparative judgments about the same solutions. After tackling an unstructured problem, students complete, compare and critique pre-designed student responses to the same problem. The framework used to analyze the data focuses on the types of links students may make between responses. The findings indicate students made varied links when completing them. The outcome of these links appeared to be influenced by how students perceived the representation being completed. Students made further assorted links that focused on invariant properties and the comparative validity of the completed responses.

A generalized skein relation for Khovanov homology and a categorification of the θ-invariant

2020 ◽  
pp. 1-27
Author(s):  
M. Chlouveraki ◽  
D. Goundaroulis ◽  
A. Kontogeorgis ◽  
S. Lambropoulou
Keyword(s):  
Jones Polynomial ◽  
Khovanov Homology ◽  
Spectral Sequences ◽  
Link Invariant ◽  
Type Relation ◽  
Skein Relation

The Jones polynomial is a famous link invariant that can be defined diagrammatically via a skein relation. Khovanov homology is a richer link invariant that categorifies the Jones polynomial. Using spectral sequences, we obtain a skein-type relation satisfied by the Khovanov homology. Thanks to this relation, we are able to generalize the Khovanov homology in order to obtain a categorification of the θ-invariant, which is itself a generalization of the Jones polynomial.

An enhanced bracket polynomial for knotoids

2019 ◽  
Vol 28 (10) ◽  
pp. 1950061
Author(s):  
Yasuyuki Miyazawa
Keyword(s):  
Virtual Link ◽  
Polynomial Invariant ◽  
Link Invariant ◽  
Pole Diagram ◽  
Bracket Polynomial

A multi-variable polynomial invariant for knotoids and linkoids, which is an enhancement of the bracket polynomial for knotoids introduced by Turaev, is given by using the concept of a pole diagram which originates in constructing a virtual link invariant. Several features of the polynomial are revealed.

CLASSICAL LINK INVARIANTS AND THE HAWAIIAN EARRINGS SPACE

1992 ◽  
Vol 01 (04) ◽  
pp. 327-342
Author(s):  
TIM D. COCHRAN
Keyword(s):  
Infinite Number ◽  
Number Of Components ◽  
Link Invariants ◽  
Classical Link

We show that, in search of link invariants more discriminating than Milnor's [Formula: see text]-invariants, one is naturally led to consider seemingly pathological objects such as links with an infinite number of components and the join of an infinite number of circles (Hawaiian earrings space). We define an infinite homology boundary link, and show that any finite sublink of an infinite homology boundary link has vanishing Milnor's invariants. Moreover, all links known to have vanishing Milnor's invariants are finite sublinks of infinite homology boundary links. We show that the exterior of an infinite homology boundary link admits a map to the Hawaiian earrings space, and that this may be employed to get a factorization of K. E. Orr's omega-invariant through a rather simple space.

Sign in / Sign up

Export Citation Format

Share Document