Sofia Lambropoulou

Knot theory related to generalized and cyclotomic Hecke algebras of type ℬ

In [12] is established that knot isotopy in a 3-manifold may be interpreted in terms of Markov braid equivalence and, also, that the braids related to the 3-manifold form algebraic structures. Moreover, the sets of braids related to the solid torus or to the lens spaces L(p, 1) form groups, which are in fact the Artin braid groups of type B. As a consequence, in [12, 13] appeared the first construction of a Jones-type invariant using Hecke algebras of type B, and this had a natural interpretation as an isotopy invariant for oriented knots in a solid torus. In a further ‘horizontal’ development and using a different technique we constructed in [8] all such solid torus knot invariants derived from the Hecke algebras of type B. Furthermore, in [7] all Markov traces related to the Hecke algebras of type D were consequently constructed.

Markov's theorem in 3-manifolds

Abstract In this paper we first give a one-move version of Markovs braid theorem for knot isotopy in S 3 that sharpens the classical theorem. Then we give a relative version of Markovs theorem concerning a fixed braided portion in the knot. We also prove an analogue of Markovs theorem for knot isotopy in knot complements. Finally we extend this last result to prove a Markov theorem for links in an arbitrary orientable 3-manifold.

VIRTUAL BRAIDS AND THE L-MOVE

In this paper we prove a Markov theorem for virtual braids and for analogs of this structure including flat virtual braids and welded braids. The virtual braid group is the natural companion to the category of virtual knots, just as the Artin braid group is the natural companion to classical knots and links. In this paper we follow L-move methods to prove the Virtual Markov theorems. One benefit of this approach is a fully local algebraic formulation of the theorems in each category.

AN INVARIANT FOR SINGULAR KNOTS

In this paper we introduce a Jones-type invariant for singular knots, using a Markov trace on the Yokonuma–Hecke algebras Yd,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 Yd,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 Yd,n(u).

THE YOKONUMA–HECKE ALGEBRAS AND THE HOMFLYPT POLYNOMIAL

We compare the invariants for classical knots and links defined using the Juyumaya trace on the Yokonuma–Hecke algebras with the HOMFLYPT polynomial. We show that these invariants do not coincide with the HOMFLYPT except in a few trivial cases.

An Adelic Extension of the Jones Polynomial

In this paper we represent the classical braids in the Yokonuma–Hecke and the adelic Yokonuma–Hecke algebras. More precisely, we define the completion of the framed braid group and we introduce the adelic Yokonuma–Hecke algebras, in analogy to the p-adic framed braids and the p-adic Yokonuma–Hecke algebras introduced in Juyumaya and Lambropoulou (Topol. Appl. 154:1804–1826, 2007; arXiv:0905.3626v1, 2009). We further construct an adelic Markov trace, analogous to the p-adic Markov trace constructed in Juyumaya and Lambropoulou (arXiv:0905.3626v1, 2009), and using the traces in Juyumaya (J. Knot Theory Ramif. 13:25–29, 2004) and the adelic Markov trace we define topological invariants of classical knots and links, upon imposing some condition. Each invariant satisfies a cubic skein relation coming from the Yokonuma–Hecke algebra.

The linking number and the writhe of uniform random walks and polygons in confined spaces

Random walks and polygons are used to model polymers. In this paper we consider the extension of the writhe, self-linking number and linking number to open chains. We then study the average writhe, self-linking and linking number of random walks and polygons over the space of configurations as a function of their length. We show that the mean squared linking number, the mean squared writhe and the mean squared self-linking number of oriented uniform random walks or polygons of length n, in a convex confined space, are of the form O(n2). Moreover, for a fixed simple closed curve in a convex confined space, we prove that the mean absolute value of the linking number between this curve and a uniform random walk or polygon of n edges is of the form . Our numerical studies confirm those results. They also indicate that the mean absolute linking number between any two oriented uniform random walks or polygons, of n edges each, is of the form O(n). Equilateral random walks and polygons are used to model polymers in θ-conditions. We use numerical simulations to investigate how the self-linking and linking number of equilateral random walks scale with their length.

KNOT THEORY IN HANDLEBODIES

We consider oriented knots and links in a handlebody of genus g through appropriate braid representatives in S3, which are elements of the braid groups Bg,n. We prove a geometric version of the Markov theorem for braid equivalence in the handlebody, which is based on the L-moves. Using this we then prove two algebraic versions of the Markov theorem. The first one uses the L-moves. The second one uses the Markov moves and conjugation in the groups Bg,n. We show that not all conjugations correspond to isotopies.

On the link invariants from the Yokonuma–Hecke algebras

In this paper we study properties of the Markov trace

A Study of the Entanglement in Systems with Periodic Boundary Conditions

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