Abstract:
Turning the skein relation for HOMFLY into a Fibonacci recurrence, we prove that there are only three rational specializations of HOMFLY polynomial: Alexander-Conway, Jones, and a new one. Using the recurrence relation, we find general and relative expansion formulae and rational generating functions for Alexander-Conway polynomial and the new polynomial, which reduce the computations to closure of simple braids, a subset of square free braids; HOMFLY polynomials of these simple braids are also computed. Algebraic independence of these three polynomials is proved.

Abstract:
In this paper, we obtain a characterizations of the recurrence of a continuous vector field $w$ of a closed connected surface $M$ as follows. The following are equivalent: 1) $w$ is pointwise recurrent. 2)$w$ is pointwise almost periodic. 3) $w$ is minimal or pointwise periodic. Moreover, if $w$ is regular, then the following are equivalent: 1) $w$ is pointwise recurrent. 2)$w$ is minimal or the orbit space $M/w$ is either $[0,1]$, or $S^1$. 3) $R$ is closed (where $R := \{(x,y) \in M \times M \mid y \in \bar{O(x)} \}$ is the orbit closure relation). On the other hand, we show that the following are equivalent for a codimension one foliation $\mathcal{F}$ on a compact manifold: 1) $\mathcal{F}$ is pointwise almost periodic. 2) $\mathcal{F}$ is minimal or compact. 3) $\mathcal{F}$ is $R$-closed. Also we show that if a foliated space on a compact metrizable space is either minimal or is both compact and without infinite holonomy, then it is $R$-closed.

Abstract:
Using a simple recurrence relation we give a new method to compute Jones polynomials of closed braids: we find a general expansion formula and a rational generating function for Jones polynomials. The method is used to estimate degree of Jones polynomials for some families of braids and to obtain general qualitative results.

Abstract:
We investigate a variant of the octahedron recurrence which lives in a 3-dimensional lattice contained in [0,n] x [0,m] x R. Generalizing results of David Speyer math.CO/0402452, we give an explicit non-recursive formula for the values of this recurrence in terms of perfect matchings. We then use it to prove that the octahedron recurrence is periodic of period n+m. This result is reminiscent of Fomin and Zelevinsky's theorem about the periodicity of Y-systems.

Abstract:
Roughly speaking, a recurrence relation is nested if it contains a subexpression of the form ... A(...A(...)...). Many nested recurrence relations occur in the literature, and determining their behavior seems to be quite difficult and highly dependent on their initial conditions. A nested recurrence relation A(n) is said to be undecidable if the following problem is undecidable: given a finite set of initial conditions for A(n), is the recurrence relation calculable? Here calculable means that for every n >= 0, either A(n) is an initial condition or the calculation of A(n) involves only invocations of A on arguments in {0,1,...,n-1}. We show that the recurrence relation A(n) = A(n-4-A(A(n-4)))+4A(A(n-4)) +A(2A(n-4-A(n-2))+A(n-2)). is undecidable by showing how it can be used, together with carefully chosen initial conditions, to simulate Post 2-tag systems, a known Turing complete problem.

Abstract:
Consider the celebrated Lyness recurrence $x_{n+2}=(a+x_{n+1})/x_{n}$ with $a\in\Q$. First we prove that there exist initial conditions and values of $a$ for which it generates periodic sequences of rational numbers with prime periods $1,2,3,5,6,7,8,9,10$ or $12$ and that these are the only periods that rational sequences $\{x_n\}_n$ can have. It is known that if we restrict our attention to positive rational values of $a$ and positive rational initial conditions the only possible periods are $1,5$ and $9$. Moreover 1-periodic and 5-periodic sequences are easily obtained. We prove that for infinitely many positive values of $a,$ positive 9-period rational sequences occur. This last result is our main contribution and answers an open question left in previous works of Bastien \& Rogalski and Zeeman. We also prove that the level sets of the invariant associated to the Lyness map is a two-parameter family of elliptic curves that is a universal family of the elliptic curves with a point of order $n, n\ge5,$ including $n$ infinity. This fact implies that the Lyness map is a universal normal form for most birrational maps on elliptic curves.

Abstract:
The strong recurrence is equivalent to the Riemann hypothesis. On the other hand, the generalized strong recurrence holds for any irrational number. In this paper, we show the generalized strong recurrence for all non-zero rational numbers. Moreover, we prove that the generalized strong recurrence in the region of absolute convergence holds for any real number.

Abstract:
In this paper we propose a definition of a recurrence relation homomorphism and illustrate our definition with a few examples. We then define the period of a k-th order of linear recurrence relation and deduce certain preliminary results associated with them.

Abstract:
Combinatorial interpretation of the fibonomial coefficients recently proposed by the present author results here in combinatorial interpretation of the recurrence relation for fibonomial coefficients . The presentation is provided with quite an exhaustive context where apart from plane grid coordinate system used several figures illustrate the exposition of statements and the derivation of the recurrence itself.