Narrow Results

We have derived rigorously the generating function for the number of pyramid-like polyhedra g(a,b,c) on the cubic lattice with width a, depth b, and height c.

The main theorem in this article shows that the link cannot be realized with fewer than 28 steps on the cubical lattice. There are at least two very different 28-step realizations of on the cubical lattice in which the length of the components of the link differs. The two 28-step realizations of do not have minimal curvature. A realization with a minimal curvature of 6.5 π of the link with 30 steps is also shown.

A method for representing knots by means of a chain code is presented. Knots which are digitalized and represented by the orthogonal direction change chain code are called discrete knots. Discrete knots are composed of constant straight-line segments using only orthogonal directions. The chain elements represent the orthogonal direction changes of the constant straight-line segments of the discrete knot. There are only five possible orthogonal direction changes for representing any discrete knot. Thus, this chain code only considers relatives directions changes, which allows us to have a unique knot descriptor invariant under translation and rotation. Also, this knot descriptor may be starting point normalized. Finally, this unique knot descriptor produces a numerical string of finite length over a finite alphabet, allows us the usage of grammatical techniques for discrete-knot analysis. Thus, we present some prime discrete knot detection examples within composite discrete knots.

A method for generating families of particular 3D curves which differs from random self-avoiding walks is presented. This method is based on a chain code called knot numbers. The knot-number notation describes discrete knots. A discrete knot is the digitalized representation of a knot and is composed of constant orthogonal straight-line segments. The orthogonal direction changes of the contiguous straight-line segments of the knot define the chain elements. There are only five possible orthogonal direction changes for representing any discrete knot. Thus, the chain elements are considered as one base-five integer number (knot number). In this manner, we obtain a unique knot descriptor. This description is invariant under translation, rotation, starting point, and coding direction of the discrete knot.

By evaluating all possible combinations of chain elements of curves and considering some restrictions, we obtain interesting families of curves which allow us to find the smallest nontrivial knot. Finally, we present a modest attempt at building the table of minimal discrete knots.

Lattice stick number s_L(K) is defined to be the minimal number of sticks required to construct a polygonal representation of the knot K in the cubic lattice. In this paper, we give lattice stick numbers of small knots such as 3₁ and 4₁. More precisely we prove that s_L(3₁) = 12 and s_L(K) ≥ 14 for any other non-trivial knot K.

An easy and fast algorithm for obtaining minimal discrete knots is presented. A minimal discrete knot is the digitalized representation of a knot, which is composed of the minimum number of constant orthogonal straight-line segments and is represented by the knot-number notation. The proposed algorithm for obtaining minimal discrete knots tries to reduce the number of straight-line segments of a given discrete knot preserving its shape. This algorithm is based on two fundamental transformations: U and L.

In order to prove the efficiency and rapidity of the algorithm, a great variety of examples of discrete knots are presented: complete families of discrete knots at different orders; random discrete knots; examples of non-trivial and unsolved discrete knots.

The polygons on the cubic-lattice have played an important role in simulating various circular molecules, especially the ones with relatively big volumes. There have been a lot of theoretical studies and computer simulations devoted to this subject. The questions are mostly around the knottedness of such a polygon, such as what kind of knots can appear in a polygon of given length, how often it can occur, etc. A very often asked and long standing question is about the minimal length of a knotted polygon. It is well-known that there are knotted polygons on the lattice with 24 steps yet it is unproved up to this date that 24 is the minimal number of steps needed. In this paper, we prove that in order to obtain a knotted polygon on the cubic lattice, at least 24 steps are needed and we can only have trefoils with 24 steps.

How many edges are necessary and sufficient to construct a knot of type K in the cubic lattice? Define the minimal edge number of a knot to be this number of edges. To what extend does the minimal edge number measure the complexity of a knot? What is the behaviour of the minimal edge number under the connected sum of knots, and what is its limiting behaviour? We consider these questions and show that the minimal edge number may be computed using simulated annealing.

This chapter reveals the depth of the transformational nature inherent in the Mereon Matrix showing how clusters and singles of polyhedra, 33 unified forms, scale to infinity, internally and externally. This chapter suggests significant physical, psychological and sociological implications inherent in the structure and dynamics of the Matrix’s outermost polyhedron. Referred to as the Context, this geosphere iterates between 120 and 180 faces as it opens and closes, expands and contracts. Image intense, it is revealed how localised environmental energy is required, received and processed to eventually produce new systems and/or product that expands the global environment.