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 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 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.

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.

ON THE G2 LINK INVARIANT

1993 ◽  
Vol 02 (04) ◽  
pp. 431-451 ◽  
Author(s):  
EFSTRATIA KALFAGIANNI
Keyword(s):  
Lie Algebra ◽  
Link Invariant ◽  
Markov Trace ◽  
Knots And Links

We construct a polynomial link invariant as Markov trace on certain one parameter algebras and we prove that it is equal to the invariant corresponding to the exeptional Lie algebra of type G2. We use braid representatives to calculate the invariant for several knots and links.

CLASPER-MOVES AMONG RIBBON 2-KNOTS CHARACTERIZING THEIR FINITE TYPE INVARIANTS

2006 ◽  
Vol 15 (09) ◽  
pp. 1163-1199 ◽  
Author(s):  
TADAYUKI WATANABE
Keyword(s):  
Finite Type ◽  
Finite Type Invariants ◽  
Topological Interpretation ◽  
Type Invariant

Habiro found in his thesis a topological interpretation of finite type invariants of knots in terms of local moves called Habiro's Ck-moves. Ck-moves are defined by using his claspers. In this paper we define "oriented" claspers and RCk-moves among ribbon 2-knots as modifications of Habiro's notions to give a similar interpretation of Habiro–Kanenobu–Shima's finite type invariants of ribbon 2-knots. It works also for ribbon 1-knots. Furthermore, by using oriented claspers for ribbon 1-knots, we can prove Habiro–Shima's conjecture in the case of ℚ-valued invariants, saying that ℚ-valued Habiro–Kanenobu–Shima finite type invariant and ℚ-valued Vassiliev–Goussarov finite type invariant are the same thing.

scholarly journals ON THE MARKOV TRACE FOR TEMPERLEY–LIEB ALGEBRAS OF TYPE En

2009 ◽  
Vol 18 (02) ◽  
pp. 237-264 ◽  
Author(s):  
R. M. GREEN
Keyword(s):  
Hecke Algebras ◽  
Markov Trace ◽  
Diagram Representation ◽  
Lusztig Polynomials

We show that there is a unique Markov trace on the tower of Temperley–Lieb type quotients of Hecke algebras of Coxeter type En (for all n ≥ 6). We explain in detail how this trace may be computed easily using tom Dieck's calculus of diagrams. As applications, we show how to use the trace to show that the diagram representation is faithful, and to compute leading coefficients of certain Kazhdan–Lusztig polynomials.

scholarly journals G-LINKS INVARIANTS, MARKOV TRACES AND THE SEMI-CYCLIC Uq𝔰𝔩(2)-MODULES

2013 ◽  
Vol 22 (11) ◽  
pp. 1350063 ◽  
Author(s):  
NATHAN GEER ◽  
BERTRAND PATUREAU-MIRAND
Keyword(s):  
Quantum Group ◽  
Natural Extension ◽  
Alexander Polynomial ◽  
Flat Connection ◽  
Link Invariant ◽  
Polynomial Invariants ◽  
Markov Trace

Kashaev and Reshetikhin proposed a generalization of the Reshetikhin–Turaev link invariant construction to tangles with a flat connection in a principal G-bundle of the complement of the tangle. The purpose of this paper is to adapt and renormalize their construction to define invariants of G-links using the semi-cyclic representations of the non-restricted quantum group associated to 𝔰𝔩(2), defined by De Concini and Kac. Our construction uses a modified Markov trace. In our main example, the semi-cyclic invariants are a natural extension of the generalized Alexander polynomial invariants defined by Akutsu, Deguchi and Ohtsuki. Surprisingly, direct computations suggest that these invariants are actually equal.

scholarly journals Weights of Markov traces on Hecke algebras

1999 ◽  
Vol 1999 (508) ◽  
pp. 157-178 ◽  
Author(s):  
R. C Orellana
Keyword(s):  
Hecke Algebras ◽  
Hecke Algebra ◽  
Markov Trace

Abstract We compute the weights, i.e. the values at the minimal idempotents, for the Markov trace on the Hecke algebra of type Bn and type Dn. In order to prove the weight formula, we define representations of the Hecke algebra of type B onto a reduced Hecke algebra of type A. To compute the weights for type D we use the inclusion of the Hecke algebra of type D into the Hecke algebra of type B.

scholarly journals A STATE-SUM FORMULA FOR THE ALEXANDER POLYNOMIAL

2012 ◽  
Vol 21 (03) ◽  
pp. 1250008
Author(s):  
SAMSON BLACK
Keyword(s):  
Hecke Algebras ◽  
Type A ◽  
Alexander Polynomial ◽  
Trace Formulas ◽  
Sum Formula ◽  
Markov Trace

We develop a diagrammatic formalism for calculating the Alexander polynomial of the closure of a braid as a state-sum. Our main tools are the Markov trace formulas for the HOMFLY-PT polynomial and Young's semi-normal representations of the Iwahori–Hecke algebras of type A.

scholarly journals On the link invariants from the Yokonuma–Hecke algebras

2016 ◽  
Vol 25 (09) ◽  
pp. 1641004 ◽  
Author(s):  
S. Chmutov ◽  
S. Jablan ◽  
K. Karvounis ◽  
S. Lambropoulou
Keyword(s):  
Trace Formula ◽  
Hecke Algebras ◽  
Computer Programs ◽  
Link Invariants ◽  
Connected Sums ◽  
Markov Trace ◽  
Knots And Links

In this paper, we study properties of the Markov trace tr[Formula: see text] and the specialized trace [Formula: see text] on the Yokonuma–Hecke algebras, such as behavior under inversion of a word, connected sums and mirror imaging. We then define invariants for framed, classical and singular links through the trace [Formula: see text] and also invariants for transverse links through the trace tr[Formula: see text]. In order to compare the invariants for classical links with the Homflypt polynomial, we develop computer programs and we evaluate them on several Homflypt-equivalent pairs of knots and links. Our computations lead to the result that these invariants are topologically equivalent to the Homflypt polynomial on knots. However, they do not demonstrate the same behavior on links.

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