To a link (obtained by closing a string link) we associate a certain diagram of groups. If two links are concordant, we show that there exists a certain type of isomorphism between the group diagrams.
One of the challenges of Volume Modelling is the definition of theoretical frameworks to support object manipulation and representation. Opposite to what is seen in the field of surface modelling, few satisfactory tools have been presented that ensure topological robustness to volumetric models. In this paper we offer one such mathematical framework for volumetric model definition based on tetrahedral meshes. From a complete tetrahedron topological characterization, a set of Morse Operators is given, which enable global tolopogical control during tetrahedron addition or removal. A number of applications can be envisaged, from geometrical modelling to volume reconstruction. We show the effectiveness of the tetrahedron characterization framework for volume reconstruction from images, showing that the method is capable of handling certain types of noise topologically without the need for a time consuming preprocessing step, or a post-processing step to detect cavities and holes.
The group BSL(k) of boundary cobordism classes of boundary k-string links is defined. An epimorphism from BSL(k) to a group of cobordism classes of matrices is defined. An action of a certain group of pure braids on BSL(k) provides all possible splittings for a given boundary k-link. A necessary and sufficient condition is given for two elements of BSL(k) to have the same closure as an F(k)-link (i.e., a boundary k-link with one of its splittings), up to F(k)-cobordism.
Let H (k) be the group of all homotopy classes of k-string links. It has been proved that f, g ∈ H (k) have the same closure if and only if there is β ∈ S k (1) such that β • f = g, where S k ( 1) is the stabilizer of 1 for a certain action of the group H (2k) on the set H (k). If β ∈ S k (1), Artin's automorphism β , induced by β on RF (2k), the reduced free group in 2k generators, induces an automorphism β ∈ A(RF (k)), the group of all automorphisms of RF (k) that send each generator to one of its conjugates. This can be used to compare the homotopy classes of links obtained by closing f and g . The association β → β is a homomorphism from S k (1) to A(RF (k)). In this paper we determine its kernel.
We give a direct proof that if two string links have isotopic closures, then there is a braid-special isomorphism between their n-level group diagrams, for every n 2. In the case of link-homotopy, we give an alternative proof to our previous result that there is a braid-special isomorphism between the group diagrams for the homotopy classes of two string links if and only if they have link-homotopic closures.
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