Related papers: Generating function polynomials for legendrian lin…
The main object of this paper is to construct a new generating function of the (q-) Bernstein type polynomials. We establish elementary properties of this function. By using this generating function, we derive recurrence relation and…
The altenating knots, links and twists projected on the S_2 sphere are identified with the phase Space of a Hamiltonian dynamic system of one degree of freedom. The saddles of the system correspond to the crossing points, the edges, to the…
As countless examples show, it can be fruitful to study a sequence of complicated objects all at once via the formalism of generating functions. We apply this point of view to the homology and combinatorics of orbit configuration spaces:…
In this short note we discuss certain examples of Legendrian submanifolds, whose linearized Legendrian contact (co)homology groups over integers have non-vanishing algebraic torsion. More precisely, for a given arbitrary finitely generated…
Fibonacci numbers can be expressed in terms of multinomial coefficients as sums over integer partitions into odd parts. We use this fact to introduce a family of double inequalities involving the generating function for the number of…
We prove that a version of the Thurston-Bennequin inequality holds for Legendrian and transverse links in a rational homology contact 3-sphere $(M,\xi)$, whenever $\xi$ is tight. More specifically, we show that the self-linking number of a…
Let $\Lambda$ be a link of Legendrian spheres in the boundary of a subcritical $2n$-dimensional Weinstein manifold $X$. We show that, under some geometrical assumptions, the computation of the Legendrian contact homology of $\Lambda$ can be…
We construct a 2-variable link polynomial, called $W_L$, for classical links by considering simultaneously the Kauffman state models for the Alexander and for the Jones polynomials. We conjecture that this polynomial is the product of two…
Trigonometric formulas are derived for certain families of associated Legendre functions of fractional degree and order, for use in approximation theory. These functions are algebraic, and when viewed as Gauss hypergeometric functions,…
Cut-diagrams are diagrammatic objects, defined in dimensions 1 and 2, that generalize links in 3-space and surface-links in 4-space; in dimension 1, this coincides with the theory of welded links. Using cut-diagrams, we introduce an…
Topological quantum field theories can be used as a powerful tool to probe geometry and topology in low dimensions. Chern-Simons theories, which are examples of such field theories, provide a field theoretic framework for the study of knots…
A solution is proposed for the problem of composition of ordinary generating functions. A new class of functions that provides a composition of ordinary generating functions is introduced; main theorems are presented; compositae are written…
It is shown that the Topological Massive and ``Self-dual'' theories, which are known to provide locally equivalent descriptions of spin 1 theories in 2+1 dimensions, have different global properties when formulated over topologically…
A new classification theorem for links by the authors and Roger Fenn leads to computable link invariants. As an illustration we distinguish the left and right trefoils and recover the result of Carter et al that the 2-twist-spun trefoil is…
Our present investigation is motivated essentially by the fact that, in Geometric Function Theory, one can find many interesting and fruitful usages of a wide variety of special functions and special polynomials. The main purpose of this…
We introduce new skein invariants of links based on a procedure where we first apply the skein relation only to crossings of distinct components, so as to produce collections of unlinked knots. We then evaluate the resulting knots using a…
In this paper it is shown how the generating functional for Green's functions in relativistic quantum field theory and in thermal field theory can be evaluated in terms of a standard quantum mechanical path integral. With this calculational…
Real Legendrian subvarieties are classical objects of differential geometry and classical mechanics and they have been studied since antiquity. However, complex Legendrian subvarieties are much more rigid and have more exceptional…
This paper contains the first knot polynomials which can distinguish the orientations of classical knots and which make no excplicit use of the knot group. But they make extensive use of the meridian and of the longitude in a geometric way.…
The spread between two lines in rational trigonometry replaces the concept of angle, allowing the complete specification of many geometrical and dynamical situations which have traditionally been viewed approximately. This paper…