Mutant knots, in the sense of Conway, are known to share the same Homfly polynomial. Their 2-string satellites also share the same Homfly polynomial, but in general their m-string satellites can have different Homfly polynomials for m > 2. We show that, under conditions of extra symmetry on the constituent 2-tangles, the directed m-string satellites of mutants share the same Homfly polynomial f...

We study the problem of factoring univariate polynomials over finite fields. Under the assumption of the Extended Riemann Hypothesis (ERH), Gao [Gao01] designed a polynomial time algorithm that fails to factor only if the input polynomial satisfies a strong symmetry property, namely square balance. In this paper, we propose an extension of Gao’s algorithm that fails only under an even stronger ...

We derive eight basic identities of symmetry in three variables related to Euler polynomials and alternating power sums. These and most of their corollaries are new, since there have been results only about identities of symmetry in two variables. These abundances of symmetries shed new light even on the existing identities so as to yield some further interesting ones. The derivations of identi...

We give a complete classification of all pairs of cyclotomic polynomials whose zeros interlace on the unit circle, making explicit a result essentially contained in work of Beukers and Heckman. We show that each such pair corresponds to a single polynomial from a certain special class of integer polynomials, the 2-reciprocal discbionic polynomials. We also show that each such pair also correspo...

Computational methods are given in [3] to determine the maximum possible absolute value of each of the coe cients of the polynomials in this class when satis es some restrictions. It is also shown that there exist polynomials P and Q in the class such that each of the nonzero coe cients of P and Q have the largest possible absolute value of any polynomial in the class. The extremal polynomials ...

We define two symmetric $q,t$-Catalan polynomials in terms of the area and depth statistic dinv statistics. prove symmetry using an involution on plane trees. The same proves Tutte polynomials. also provide a combinatorial proof remark by Garsia et al. regarding parking functions number connected graphs fixed vertices.

Taking a combinatorial point of view on cyclotomic polynomials leads to a larger class of polynomials we shall call the inclusion-exclusion polynomials. This gives a more appropriate setting for certain types of questions about the coefficients of these polynomials. After establishing some basic properties of inclusion-exclusion polynomials we turn to a detailed study of the structure of ternar...

The wild-relative gene pools of wheat are a rich source of genetic variation for wheat improvement.Karyotypes of eight genotypes of Triticum monococcum subsp. aegilopoides and eight genotypes of Aegilops cylindrica,which grown wild next to each other in different regions in the west of Iran were studied. Root tips were treated in α-bromonaphthalene solution, fixed in chromic acid-formaldehyde f...