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Francesca Aicardi

Publications and source records attributed to Francesca Aicardi.

At least 19 recordsLinked to original sources
arXiv2025

Framization and Deframization

Starting from the geometric construction of the framed braid group, we define and study the framization of several Brauer-type monoids and also the set partition monoid, all of which appear in knot theory. We introduce the concept of deframization, which is a procedure to obtain a tied monoid from a given framed monoid. Furthermore, we show in detail how this procedure works on the monoids mentioned above. We also discuss the framization and deframization of some algebras, which are deformations, respectively, of the framized and deframized monoids discussed here.

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arXiv2024

Fuss-Catalan Triangles

For each $p>0$ we define by recurrence a triangle $T^p(n,k)$ whose rows sum to the Fuss-Catalan numbers $ \frac{1}{p n+1}\binom{pn+1}{n}$, generalizing the known Catalan triangle corresponding to the case $p=2$. (In fact, $T^p(n,k)$ has an explicit formula counting simple lattice paths). Moreover, for some small values of $p$, the signed sums turn out to be known sequences. \end{abstract}

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arXiv2023

Ramified inverse and planar monoids

Ramified monoids are a class of monoids introduced by the authors. The main motivation for considering these monoids comes from knot theory, see [3, 4, 5]. Thus, in [2] we have studied the ramified monoids of the symmeytric group and of the Brauer monoid, among others. This paper study the ramified of the inverse symmetric monoid, which plays a notable role in knot theory as well, see [10]. Here is also introduced the notion of planar ramified monoid. In particular, we give presentations for some planar ramified monoids arising from noncrossing set partitions.

math.RT↗
arXiv2021

Brauer and Jones tied monoids

We introduce a ramified monoid, attached to each Brauer--type monoid, that is, to the symmetric group, to the Jones and Brauer monoids among others. Ramified monoids correspond to a class of tied monoids which arise from knot theory and are interesting in itself. The ramified monoid attached to the symmetric group is the Coxeter-like version of the so--called tied braid monoid. We give a presentation of the ramified monoid attached to the Brauer monoid. Also, we introduce and studied two tied-like monoids that cannot be described as ramified monoids. However, these monoids can also be regarded as tied versions of the Jones and Brauer monoids.

math.RT↗
arXiv2021

Tied links and invariants for singular links

Tied links and the tied braid monoid were introduced recently by the authors and used to define new invariants for classical links. Here, we give a version purely algebraic-combinatoric of tied links. With this new version we prove that the tied braid monoid has a decomposition like a semi--direct group product. By using this decomposition we reprove the Alexander and Markov theorem for tied links; also, we introduce the tied singular knots, the tied singular braid monoid and certain families of Homflypt type invariants for tied singular links; these invariants are five-variables polynomials. Finally, we study the behavior of these invariants; in particular, we show that our invariants distinguish non isotopic singular links indistinguishable by the Paris-Rabenda invariant.

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arXiv2020

Catalan triangles and tied arc diagrams

The Catalan triangle, as well as a Fuss-Catalan triangle, enter a problem of counting particular tied arc diagrams. This setting allows us to prove some combinatorial properties of these triangles.

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arXiv2020

Two parameters bt-algebra and invariants for links and tied links

We introduce a two-parameters bt-algebra which, by specialization, becomes the one-parameter bt-algebra, introduced by the authors, as well as another one-parameter presentation of it; the invariant for links and tied links, associated to this two-parameter algebra via Jones recipe, contains as specializations the invariants obtained from these two presentations of the bt-algebra and then is more powerful than each of them. Also, a new non Homflypt polynomial invariant is obtained for links, which is related to the linking matrix.

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arXiv2017

Kauffman type invariants for tied links

We define two new invariants for tied links. One of them can be thought as an extension of the Kauffman polynomial and the other one as an extension of the Jones polynomial which is constructed via a bracket polynomial for tied links. These invariants are more powerful than both the Kauffman and the bracket polynomials when evaluated on classical links. Further, the extension of the Kauffman polynomial is more powerful of the Homflypt polynomial, as well as of certain new invariants introduced recently. Also we propose a new algebra which plays in the case of tied links the same role as the BMW algebra for the Kauffman polynomial in the classical case. Moreover, we prove that the Markov trace on this new algebra can be recovered from the extension of the Kauffman polynomial defined here.

math.GT↗
arXiv2017

An algebra involving braids and ties

In this note we study a family of algebras 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.

math.GN↗
arXiv2016

Tied Links

In this paper we introduce the tied links, i.e. ordinary links provided with some ties between strands. The motivation for introducing such objects originates from a diagrammatical interpretation of the defining generators of the so-called algebra of braids and ties; indeed, one half of such generators can be interpreted as the usual generators of the braid algebra, and the remaining generators can be interpreted as ties between consecutive strands; this interpretation leads the definition of tied braids. We define an invariant polynomial for the tied links via a skein relation. Furthermore, we introduce the monoid of tied braids and we prove the corresponding theorems of Alexander and Markov for tied links. Finally, we prove that the invariant of tied links that we have defined can be obtained also by using the Jones recipe.

math.GT↗
arXiv2016

Markov trace on the algebra of braids and ties

We prove that the so-called t 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.

math.GT↗
arXiv2010

Projective geometry from Poisson algebras

In analogy with the Poisson algebra of the quadratic forms on the symplectic plane, and the notion of duality in the projective plane introduced by Arnold in \cite{Arn}, where the concurrence of the triangle altitudes is deduced from the Jacobi identity, we consider the Poisson algebras of the first degree harmonics on the sphere, the pseudo-sphere and on the hyperboloid, to obtain analogous duality notions and similar results for the spherical, pseudo-spherical and hyperbolic geometry. Such algebras, including the algebra of quadratic forms, are isomorphic, as Lie algebras, either to the Lie algebra of the vectors in $\R^3$, with vector product, or to algebra $sl_2(\R)$. The Tomihisa identity, introduced in \cite{Tom} for the algebra of quadratic forms, holds for all these Poisson algebras and has a geometrical interpretation. The relation between the different definitions of duality in projective geometry inherited by these structures is shown.

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arXiv2009

Rational tangles and the modular group

There is a natural way to associate with a transformation of an isotopy class of rational tangles to another, an element of the modular group. The correspondence between the isotopy classes of rational tangles and rational numbers follows, as well as the relation with the braid group $B_3$.

math.GT↗
arXiv2008

Symmetries of quadratic forms classes and of quadratic surds continued fractions. Part I: A Poincaré model for the de Sitter world

The problem of the classification of the indefinite binary quadratic forms with integer coefficients is solved introducing a special partition of the de Sitter world, where the coefficients of the forms lie, into separate domains. Every class of indefinite forms, under the action of the special linear group acting on the integer plane lattice, has a finite and well defined number of representatives inside each one of such domains. This property belongs exclusively to rational points on the one-sheeted hyperboloid. In the second part we will show how to obtain the symmetry type of a class as well as its number of points in all domains from a sole representative of that class.

math.NT↗
arXiv2008

Symmetries of quadratic forms classes and of quadratic surds continued fractions. Part II: Classification of the periods' palindromes

The continue fractions of quadratic surds are periodic, according to a theorem by Lagrange. Their periods may have differing types of symmetries. This work relates these types of symmetries to the symmetries of the classes of the corresponding indefinite quadratic forms. This allows to classify the periods of quadratic surds and at the same time to find, for an arbitrary indefinite quadratic form, the symmetry type of its class and the number of integer points, for that class, contained in each domain of the Poincare' model of the de Sitter world, introduced in Part I. Moreover, we obtain the same information for every class of forms representing zero, by the finite continue fraction related to a special representative of that class. We will see finally the relation between the reduction procedure for indefinite quadratic forms, defined by the continued fractions, and the classical reduction theory, which acquires a geometrical description by the results of Part I.

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