Markov Trace on the Algebra of Braids and Ties (original) (raw)

4 Markov Trace on the Algebra of Braids and Ties

2016

We prove that the so-called algebra of braids and ties supports a Markov trace. Further, by using this trace in the Jones' recipe, we define invariant polynomials for classical knots and singular knots. Our invariants have three parameters. The invariant of classical knots is an extension of the Homflypt polynomial and the invariant of singular knots is an extension of an invariant of singular knots previously defined by S. Lambropoulou and the second author.

An Invariant for Singular Knots

Journal of Knot Theory and Its Ramifications, 2009

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).

An Algebra Involving Braids and Ties

In this note we study a family of algebras E n (u) with one parameter defined by generators and relations. The set of generators contains the generators of the usual braids algebra, and another set of generators which is interpreted as ties between consecutive strings. We also study the representations theory of the algebra when the parameter is specialized to 1. Foreword This article was written in the 2000 year and published as ICTP preprint [1]. The authors submitted it to a journal, but it was rejected since according to the referee report the considered algebra was the Hecke algebra with another vesture; moreover, this algebra should be not related to knot theory. However, in the last years the algebras E n (u) was revisited by Ryom-Hansen [5], who found explicit bases and classified the irreducible representations. Moreover, in 2013 Banjo [6] determined the complex generic representation theory, showing that a certain specialization of this algebra is isomorphic to the small ramified partition algebra. The authors of the article have successively proved that E n (u)E n (u) admits a trace [2] and introduced the tied links, defining different invariants for them [3]. More recently, they have considered Kauffman type invariants for tied links [4] and have introduced a new algebra (a generalization of the BMW algebra) which is related to E n (u). A Referee of [4] suggested to put in ArXiv the present article.

The HOMFLY polynomial of links in closed braid form

Discrete Mathematics

It is well known that any link can be represented by the closure of a braid. The minimum number of strings needed in a braid whose closure represents a given link is called the braid index of the link and the well known Morton-Frank-Williams inequality reveals a close relationship between the HOMFLY polynomial of a link and its braid index. In the case that a link is already presented in a closed braid form, Jaeger derived a special formulation of the HOMFLY polynomial. In this paper, we prove a variant of Jaeger's result as well as a dual version of it. Unlike Jaeger's original reasoning, which relies on representation theory, our proof uses only elementary geometric and combinatorial observations. Using our variant and its dual version, we provide a direct and elementary proof of the fact that the braid index of a link that has an n-string closed braid diagram that is also reduced and alternating, is exactly n. Until know this fact was only known as a consequence of a result due to Murasugi on fibered links that are star products of elementary torus links and of the fact that alternating braids are fibered.

Hecke algebra trace algorithm and some conjectures on weave knots

arXiv: Geometric Topology, 2019

Computing polynomial invariants for knots and links using braid representations relies heavily on finding the trace of Hecke algebra elements. There is no easy method known for computing the trace and hence it becomes difficult to compute the known polynomial invariants of knots using their braid representations. In this paper, we provide an algorithm to compute the trace of rho(alpha)\rho(\alpha)rho(alpha) for every braid alphainBn\alpha\in B_nalphainBn, where rho:BntoHn(q)\rho:B_n\to H_n(q)rho:BntoHn(q) is the representation determined by rho(sigmai)=Ti\rho(\sigma_i)=T_irho(sigmai)=Ti. We simplify this algorithm and write a Mathematica program to compute the invariants such as Alexander polynomial, Jones polynomial, HOMFLY-PT polynomial and Khovanov homology of a very special family of knots and links W(n,m)W(n,m)W(n,m) known as {\it weaving knots} by expressing them as closure of nnn-braids. We also explore on the relationship between the topological and geometric invariants of this family of alternating and hyperbolic knots (links) by generating data for subfamilies $W(3...

Polynomial invariants of singular knots and links

Journal of Knot Theory and Its Ramifications, 2021

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 New Polynomial Invariant of Knots and LINKS1

1985

The purpose of this note is to announce a new isotopy invariant of oriented links of tamely embedded circles in 3-space. We represent links by plane projections, using the customary conventions that the image of the link is a union of transversely intersecting immersed curves, each provided with an orientation, and undercrossings are indicated by broken lines. Following Conway [6], we use the symbols L+, Lo, L_ to denote links having plane projections which agree except in a small disk, and inside that disk are represented by the pictures of Figure 1. Conway showed that the one-variable Alexander polynomials of L+, Lo, L_ (when suitably normalized) satisfy the relation

A new polynomial invariant of knots and links

Bulletin of the American Mathematical Society, 1985

Minimum Braids: A Complete Invariant of Knots and Links

arXiv (Cornell University), 2004

Minimum braids are a complete invariant of knots and links. This paper defines minimum braids, describes how they can be generated, presents tables for knots up to ten crossings and oriented links up to nine crossings, and uses minimum braids to study graph trees, amphicheirality, unknotting numbers, and periodic tables.

Markov Trace on the Algebra of Braids and Ties (original) (raw)

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