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 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 A POLYNOMIAL INVARIANT OF VIRTUAL LINKS

2013 ◽  
Vol 22 (12) ◽  
pp. 1341002 ◽  
Author(s):  
ZHIYUN CHENG ◽  
HONGZHU GAO
Keyword(s):  
Knot Theory ◽  
Polynomial Invariant ◽  
Polynomial Invariants ◽  
Virtual Knots ◽  
The Third ◽  
Linking Number ◽  
Virtual Links ◽  
Definition Of ◽  
Knots And Links

In this paper, we define some polynomial invariants for virtual knots and links. In the first part we use Manturov's parity axioms [Parity in knot theory, Sb. Math.201 (2010) 693–733] to obtain a new polynomial invariant of virtual knots. This invariant can be regarded as a generalization of the odd writhe polynomial defined by the first author in [A polynomial invariant of virtual knots, preprint (2012), arXiv:math.GT/1202.3850v1]. The relation between this new polynomial invariant and the affine index polynomial [An affine index polynomial invariant of virtual knots, J. Knot Theory Ramification22 (2013) 1340007; A linking number definition of the affine index polynomial and applications, preprint (2012), arXiv:1211.1747v1] is discussed. In the second part we introduce a polynomial invariant for long flat virtual knots. In the third part we define a polynomial invariant for 2-component virtual links. This polynomial invariant can be regarded as a generalization of the linking number.

A LINK INVARIANT DOMINATING THE HOMFLY AND THE KAUFFMAN POLYNOMIALS

2010 ◽  
Vol 19 (11) ◽  
pp. 1507-1533
Author(s):  
YASUYUKI MIYAZAWA
Keyword(s):  
Polynomial Invariant ◽  
Link Invariant ◽  
Knots And Links

By using a graph diagram named a magnetic graph diagram, we construct a polynomial invariant for knots and links. We show that it is a generalization of both the HOMFLY and the Kauffman polynomials.

scholarly journals Identifying the Invariants for Classical Knots and Links from the Yokonuma–Hecke Algebras

2018 ◽  
Vol 2020 (1) ◽  
pp. 214-286 ◽  
Author(s):  
Maria Chlouveraki ◽  
Jesús Juyumaya ◽  
Konstantinos Karvounis ◽  
Sofia Lambropoulou
Keyword(s):  
Hecke Algebras ◽  
Hecke Algebra ◽  
Type A ◽  
Closed Formula ◽  
Link Invariant ◽  
Polynomial Invariants ◽  
Oriented Link ◽  
Linking Numbers ◽  
Markov Trace ◽  
Knots And Links

Abstract We announce the existence of a family of new 2-variable polynomial invariants for oriented classical links defined via a Markov trace on the Yokonuma–Hecke algebra of type A. Yokonuma–Hecke algebras are generalizations of Iwahori–Hecke algebras, and this family contains the HOMFLYPT polynomial, the famous 2-variable invariant for classical links arising from the Iwahori–Hecke algebra of type A. We show that these invariants are topologically equivalent to the HOMFLYPT polynomial on knots, but not on links, by providing pairs of HOMFLYPT-equivalent links that are distinguished by our invariants. In order to do this, we prove that our invariants can be defined diagrammatically via a special skein relation involving only crossings between different components. We further generalize this family of invariants to a new 3-variable skein link invariant that is stronger than the HOMFLYPT polynomial. Finally, we present a closed formula for this invariant, by W. B. R. Lickorish, that uses HOMFLYPT polynomials of sublinks and linking numbers of a given oriented link.

scholarly journals Alexander and writhe polynomials for virtual knots

2016 ◽  
Vol 25 (08) ◽  
pp. 1650050 ◽  
Author(s):  
Blake Mellor
Keyword(s):  
Knot Theory ◽  
Alexander Polynomial ◽  
Polynomial Invariant ◽  
Polynomial Invariants ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Virtual Links ◽  
New Interpretation ◽  
Polynomial Formula ◽  
Knots And Links

We give a new interpretation of the Alexander polynomial [Formula: see text] for virtual knots due to Sawollek [On Alexander–Conway polynomials for virtual knots and Links, preprint (2001), arXiv:math/9912173] and Silver and Williams [Polynomial invariants of virtual links, J. Knot Theory Ramifications 12 (2003) 987–1000], and use it to show that, for any virtual knot, [Formula: see text] determines the writhe polynomial of Cheng and Gao [A polynomial invariant of virtual links, J. Knot Theory Ramifications 22(12) (2013), Article ID: 1341002, 33pp.] (equivalently, Kauffman’s affine index polynomial [An affine index polynomial invariant of virtual knots, J. Knot Theory Ramifications 22(4) (2013), Article ID: 1340007, 30pp.]). We also use it to define a second-order writhe polynomial, and give some applications.

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.

A MULTI-VARIABLE POLYNOMIAL INVARIANT FOR UNORIENTED VIRTUAL KNOTS AND LINKS

2009 ◽  
Vol 18 (05) ◽  
pp. 625-649 ◽  
Author(s):  
YASUYUKI MIYAZAWA
Keyword(s):  
Crossing Number ◽  
Weighted Sum ◽  
Virtual Link ◽  
Polynomial Invariant ◽  
Virtual Knots ◽  
Virtual Links ◽  
Knots And Links

We construct a multi-variable polynomial invariant Y for unoriented virtual links as a certain weighted sum of polynomials, which are derived from virtual magnetic graphs with oriented vertices, on oriented virtual links associated with a given virtual link. We show some features of the Y-polynomial including an evaluation of the virtual crossing number of a virtual link.

GENERALIZED INDEX POLYNOMIALS FOR VIRTUAL LINKS

2012 ◽  
Vol 21 (14) ◽  
pp. 1250128
Author(s):  
KYEONGHUI LEE ◽  
YOUNG HO IM
Keyword(s):  
Knot Theory ◽  
Recursive Method ◽  
Polynomial Invariant ◽  
Polynomial Invariants ◽  
Virtual Knots ◽  
Virtual Links

We construct some polynomial invariants for virtual links by the recursive method, which are different from the index polynomial invariant defined in [Y. H. Im, K. Lee and S. Y. Lee, Index polynomial invariant of virtual links, J. Knot Theory Ramifications19(5) (2010) 709–725]. We show that these polynomials can distinguish whether virtual knots can be invertible or not although the index polynomial cannot distinguish the invertibility of virtual knots.

scholarly journals Introduction to disoriented knot theory

2018 ◽  
Vol 16 (1) ◽  
pp. 346-357
Author(s):  
İsmet Altıntaş
Keyword(s):  
Knot Theory ◽  
Jones Polynomial ◽  
Polynomial Invariants ◽  
Linking Number ◽  
Link Diagrams ◽  
New Ideas ◽  
Numerical Invariants ◽  
Bracket Polynomial ◽  
Knots And Links

AbstractThis paper is an introduction to disoriented knot theory, which is a generalization of the oriented knot and link diagrams and an exposition of new ideas and constructions, including the basic definitions and concepts such as disoriented knot, disoriented crossing and Reidemesiter moves for disoriented diagrams, numerical invariants such as the linking number and the complete writhe, the polynomial invariants such as the bracket polynomial, the Jones polynomial for the disoriented knots and links.

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