Determination of the Homotopy Type of a Morse – Smale Diffeomorphism on a 2-torus by Heteroclinic Intersection

According to the Nielsen – Thurston classification, the set of homotopy classes of orientation-preserving homeomorphisms of orientable surfaces is split into four disjoint subsets. Each subset consists of homotopy classes of homeomorphisms of one of the following types: $T_{1}$) periodic homeomorphism; $T_{2}$) reducible non-periodic homeomorphism of algebraically finite order; $T_{3}$) a reducible homeomorphism that is not a homeomorphism of algebraically finite order; $T_{4}$) pseudo-Anosov homeomorphism. It is known that the homotopic types of homeomorphisms of torus are $T_{1}$, $T_{2}$, $T_{4}$ only. Moreover, all representatives of the class $T_{4}$ have chaotic dynamics, while in each homotopy class of types $T_{1}$ and $T_{2}$ there are regular diffeomorphisms, in particular, Morse – Smale diffeomorphisms with a finite number of heteroclinic orbits. The author has found a criterion that allows one to uniquely determine the homotopy type of a Morse – Smale diffeomorphism with a finite number of heteroclinic orbits on a two-dimensional torus. For this, all heteroclinic domains of such a diffeomorphism are divided into trivial (contained in the disk) and non-trivial. It is proved that if the heteroclinic points of a Morse – Smale diffeomorphism are contained only in the trivial domains then such diffeomorphism has the homotopic type $T_{1}$. The orbit space of non-trivial heteroclinic domains consists of a finite number of two-dimensional tori, where the saddle separatrices participating in heteroclinic intersections are projected as transversally intersecting knots. That whether the Morse – Smale diffeomorphisms belong to types $T_{1}$ or $T_{2}$ is uniquely determined by the total intersection index of such knots.

Exciton scattering via algebraic topology

This paper introduces an intersection theory problem for maps into a smooth manifold equipped with a stratification. We investigate the problem in the special case when the target is the unitary group [Formula: see text] and the domain is a circle. The first main result is an index theorem that equates a global intersection index with a finite sum of locally defined intersection indices. The local indices are integers arising from the geometry of the stratification. The result is used to study a well-known problem in chemical physics, namely, the problem of enumerating the electronic excitations (excitons) of a molecule equipped with scattering data.

INDEX POLYNOMIAL INVARIANTS OF TWISTED LINKS

Bourgoin defined the notion of a twisted link. In a sense, it is a non-orientable version of a virtual link. Im, Lee and Lee defined a polynomial invariant of a virtual link by using the virtual intersection index. In this paper, we give an alternative definition of index polynomial by using indices of real crossings and extend it to a twisted links.

ON HOMOTOPIES WITH TRIPLE POINTS OF CLASSICAL KNOTS

We consider a knot homotopy as a cylinder in 4-space. An ordinary triple point p of the cylinder is called coherent if all three branches intersect at p pairwise with the same intersection index. A triple unknotting of a classical knot K is a homotopy which connects K with the trivial knot and which has as singularities only coherent triple points. We give a new formula for the first Vassiliev invariant v2(K) by using triple unknottings. As a corollary we obtain a very simple proof of the fact that passing a coherent triple point always changes the knot type. As another corollary we show that there are triple unknottings which are not homotopic as triple unknottings even if we allow more complicated singularities to appear in the homotopy of the triple homotopy.

Scaffold Library for Tissue Engineering: A Geometric Evaluation

Tissue engineering scaffold is a biological substitute that aims to restore, to maintain, or to improve tissue functions. Currently available manufacturing technology, that is, additive manufacturing is essentially applied to fabricate the scaffold according to the predefined computer aided design (CAD) model. To develop scaffold CAD libraries, the polyhedrons could be used in the scaffold libraries development. In this present study, one hundred and nineteen polyhedron models were evaluated according to the established criteria. The proposed criteria included considerations on geometry, manufacturing feasibility, and mechanical strength of these polyhedrons. CAD and finite element (FE) method were employed as tools in evaluation. The result of evaluation revealed that the close-cellular scaffold included truncated octahedron, rhombicuboctahedron, and rhombitruncated cuboctahedron. In addition, the suitable polyhedrons for using as open-cellular scaffold libraries included hexahedron, truncated octahedron, truncated hexahedron, cuboctahedron, rhombicuboctahedron, and rhombitruncated cuboctahedron. However, not all pore size to beam thickness ratios (PO : BT) were good for making the open-cellular scaffold. The PO : BT ratio of each library, generating the enclosed pore inside the scaffold, was excluded to avoid the impossibility of material removal after the fabrication. The close-cellular libraries presented the constant porosity which is irrespective to the different pore sizes. The relationship between PO : BT ratio and porosity of open-cellular scaffold libraries was displayed in the form of Logistic Power function. The possibility of merging two different types of libraries to produce the composite structure was geometrically evaluated in terms of the intersection index and was mechanically evaluated by means of FE analysis to observe the stress level. The couples of polyhedrons presenting low intersection index and high stress level were excluded. Good couples for producing the reinforced scaffold were hexahedron-truncated hexahedron and cuboctahedron-rhombitruncated cuboctahedron.

GAUSS' LINKING NUMBER REVISITED

In this paper we provide a mathematical reconstruction of what might have been Gauss' own derivation of the linking number of 1833, providing also an alternative, explicit proof of its modern interpretation in terms of degree, signed crossings and intersection number. The reconstruction presented here is entirely based on an accurate study of Gauss' own work on terrestrial magnetism. A brief discussion of a possibly independent derivation made by Maxwell in 1867 completes this reconstruction. Since the linking number interpretations in terms of degree, signed crossings and intersection index play such an important role in modern mathematical physics, we offer a direct proof of their equivalence. Explicit examples of its interpretation in terms of oriented area are also provided.