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Introduction
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October 2018 - October 2019
Education
August 1999 - May 2003
Publications
Publications (31)
We characterize Vassiliev invariants of type two for knots with zero winding number in the solid torus.
In this paper we study the ℤ-module A 2 of two-chord diagrams for knots with zero winding number in the solid torus KST 0 , which is needed in studying the type-two invariants for knots in KST 0 . We show that this module (or abelian group), which is given as a presentation with an infinite number of generators and an infinite number of relations,...
In [Proc. Am. Math. Soc. 105, No.4, 1003–1007 (1989; Zbl 0687.57002)] J. Hoste and J. Przytycki defined a two variable polynomial invariant for 1-trivial dichromatic links by a method similar to that of Kauffman in defining the Jones polynomial. In this paper we view the invariant of Hoste and Przytycki as an invariant for knots and links in the so...
In Bataineh (2003) [2] we studied the type one invariants for knots in the solid torus. In this research we study the type one invariants for n-component links in the solid torus by generalizing Aicardi's invariant for knots in the solid torus to n-component links in the solid torus. We show that the generalized Aicardi's invariant is the universal...
We relate equivalences of loops and arcs in some three manifolds to equivalences of various kinds of 1-polar knot diagrams on some subsets of a plane.
In this paper, we introduce a new polynomial invariant of planar (but not spherical) knotoids, which we call the winding signed sum polynomial. This Laurent polynomial invariant of planar knotoids denoted by [Formula: see text] is a type-one Vassiliev invariant. This invariant might tell whether a planar knotoid is a zero-height or nonzero-height p...
We introduce new polynomial invariants for both planar knotoids and spherical knotoids. To introduce these invariants, we first introduce what we call polar knots, and give the fundamentals of their theory. Polar knots are knots with a finite number of points that do not allow strands slide over or under them. We focus on polar knots with two poles...
We introduce new polynomial invariants for both planar knotoids and spherical knotoids. To introduce these invariants we first introduce what we call polar knots, which are knots with a finite number of points that do not allow strands slide over or under them. We focus on polar knots with two poles, which we call 2-polar knots. We show that knotoi...
Singular knots and links have projections involving some usual crossings and some four-valent rigid vertices. Such vertices are symmetric in the sense that no strand overpasses the other. In this research we introduce stuck knots and links to represent physical knots and links with projections involving some stuck crossings, where the physical stra...
We develop a method for computing Kauffman bracket and Jones polynomial for algebraic tangles and their numerator closures. We also introduce the notion of connectivity type of pretzel tangles and give a way of computing it. Several examples are given.
We develop a method for computing Kauffman bracket and Jones polynomial for algebraic tangles and their numerator closures. We also introduce the notion of connectivity type of pretzel tangles and give a way of computing it. Several examples are given.
We extend the axioms of Kauffman bracket to cover singular knots and links in a way that leads to a generalization of the Jones polynomial in the three-space. We also give a state sum formula to calculate this generalized Kauffman bracket. For singular knots and links in the solid torus and handlebodies, we extend the axioms so that the homotopy ty...
We define a new algebraic structure for two-component dichro-matic links. This definition extends the notation of a kei (or involutory quan-dle) from regular links to dichromatic links. We call this structure a dikei. This structure results from the generalized Reidemeister moves representing dichromatic isotopy. We give several examples on dikei,...
We introduce labeled singular knots and equivalently labeled 4-valent rigid vertex spatial graphs. Labeled singular knots are singular knots with labeled singularities. These knots are considered subject to isotopies preserving the labelings. We provide a topological invariant schema similar to that of Henrich and Kauffman in [A. Henrich and L. H....
We introduce the notion of rational links in the solid torus. We show that rational links in the solid torus are fully characterized by rational tangles, and hence by the continued fraction of the rational tangle. Furthermore, we generalize this by giving an infinite family of ambient isotopy invariants of colored diagrams in the Kauffman bracket s...
We define a new algebraic structure for singular knots and links. It extends the notion of a bikei (or involutory biquandle) from regular knots and links to singular knots and links. We call this structure a singbikei. This structure results from the generalized Reidemeister moves representing singular isotopy. We give several examples on singbikei...
In [Dichromatic link invariants, Trans. Amer. Math. Soc. 321(1) (1990) 197–229], Hoste and Kidwell investigated the skein theory of oriented dichromatic links in (Formula presented.). They introduced a multi-variable polynomial invariant (Formula presented.). We use special substitutions for some of the parameters of the invariant (Formula presente...
We give a generating set of the generalized Reidemeister moves for oriented singular links. We use it to introduce an algebraic structure arising from the study of oriented singular knots. We give some examples, including some non-isomorphic families of such structures over non-abelian groups. We show that the set of colorings of a singular knot by...
We give a generating set of the generalized Reidemeister moves for oriented singular links. We use it to introduce an algebraic structure arising from the study of oriented singular knots. We give some examples, including some non-isomorphic families of such structures over non-abelian groups. We show that the set of colorings of a singular knot by...
In 1989 Hoste and Przytycki in [7] introduced a two-variable Laurent polynomial invariant for 1-trivial dichromatic links with oriented 2-sublink in ℝ³, which we view in [4] as an invariant YL (A, t) of oriented links in the solid torus. We show that ∂ⁿ/∂Aⁿ YL(A, t) at A = 1 is a Vassiliev link invariant of order less than or equal to n. Then we gi...
We introduce the notion of rational links in the solid torus. We show that rational links in the solid torus are fully characterized by rational tangles, and hence by the continued fraction of the rational tangle. Furthermore, we generalize this by giving an infinite family of ambient isotopy invariants of colored diagrams in the Kauffman bracket s...
We introduce a natural filtration in the space of knots and singular knots in the solid torus, and start the study of the type-two Vassiliev invariants with respect to this filtration. The main result of the work states that any such invariant within the second term of this filtration in the space of knots with zero winding number is a linear combi...
The Kauffman-Vogel polynomial is an invariant of unoriented four-valent graphs embedded in three dimensional space. In this article we give a one variable generalization of this invariant. This generalization is given as a sequence of invariants in which the first term is the Kauffman-Vogel polynomial. We use the invariant we construct to give a se...
We generalize the colored Jones polynomial to 4-valent graphs. This generalization is given as a sequence of invariants in which the first term is a one variable specialization of the Kauffman-Vogel polynomial. We use the invariant we construct to give a sequence of singular braid group representations.
We define some new numerical invariants for knots with zero winding number in the solid torus. These
invariants explore some geometric features of knots embedded in the solid torus. We characterize these invariants
and interpret them on the level of the knot projection. We also find some relations among some of these invariants.
Moreover, we give l...
We show that the coefficients of a reformulation of a Taylor series expansion of the Hoste and Pryzyticki polynomial are Vassiliev invariants. We also show that many other reformulations of the Taylor series expansion have coefficients that are Vassiliev invariants. We charecterize the first two coefficients b 1 0 (t) and b L 1 (t) for one of these...
We use Pascal matrices to show that every coefficient of Jones polynomial of a knot is the pointwise limit of a sequence of Vassiliev invariants. We also find an explicit finite sum formula expressing the coefficients of Jones polynomials bounded by a given degree as integral linear combinations of Vassiliev invariants.
We introduce the notion of an essential copy in a topological space. Then we present a classification of topological spaces based on this notion. In addition, we obtain some results regarding this classification.