Abstract On the basis of the joint tree model introduced by Liu in 2003, the genus distributions of the orientable embeddings for further types of graphs can be obtained. These are apparently not easily obtained using overlap matrices, the formula of Jackson, etc. In this paper, however, by classifying the associated surfaces, we calculate the genus distributions of the orientable embeddings fo...

We show several geometric and algebraic aspects of a necklace: a link composed with a core circle and a series of circles linked to this core. We first prove that the fundamental group of the configuration space of necklaces (that we will call braid group of a necklace) is isomorphic to the braid group over an annulus quotiented by the square of the center. We then define braid groups of neckla...

Abstract We consider the problem of geometrically locally approximating a general complex analytic curve in plane at point by image polynomial parametrization t ? ( x 1 ) , 2 bidegree d . show number such curves is primitive necklaces on white beads and black beads. that this odd when = squarefree use to give partial solution conjecture Rababah. Our results naturally extend generalization regar...

Fredricksen, Kessler and Maiorana discovered a simple but elegant construction of a universal cycle for binary strings of length n: Concatenate the aperiodic prefixes of length n binary necklaces in lexicographic order. We generalize their construction to binary strings of length n whose weights are in the range c, c + 1, . . . , n by simply omitting the necklaces with weight less than c. We al...

Based on an exact trace formula for a one-dimensional ray-splitting system, we derive novel combinatorial identities for cyclic binary sequences (Pólya necklaces).

We perform numerical simulations of cosmic necklaces (systems of monopoles connected to two strings each) and investigate the conditions under which monopoles annihilate. When the total monopole energy is large compared to the string energy, we find that the string motion is no longer periodic, and thus the strings will be chopped up by self intersection. When the total monopole energy is much ...

Cosmic necklaces are hybrid topological defects consisting of monopoles and strings. We argue that primordial black holes(PBH) may have formed from loops of the necklaces, if there exist stable winding states, such as coils and cycloops. Unlike the standard scenario of PBH formation from string loops, in which it has been suggested that the production ratio f becomes a tiny fraction f ∼ 10−20, ...