The concept of Smarandache Bryant Schneider Group of a Smarandache loop is introduced. Relationship(s) between the Bryant Schneider Group and the Smarandache Bryant Schneider Group of an S-loop are discovered and the later is found to be useful in finding Smarandache isotopy-isomorphy condition(s) in S-loops just like the formal is useful in finding isotopy-isomorphy condition(s) in loops. Some...

Let Hg be a genus g handlebody, and Tg = ∂Hg a closed connected orientable surface. In this paper we find a finite set of generators for Eg 2 , the subgroup of PMCG2(Tg) consisting of the isotopy classes of homeomorphisms of Tg which admit an extension to the handlebody keeping a properly embedded trivial arc fixed. This subgroup turns out to be important for the study of knots in closed 3-mani...

Let X be an oriented 4-manifold which does not have simple SW-type, for example a blow-up of a rational or ruled surface. We show that any two cohomologous and deformation equivalent symplectic forms on X are isotopic. This implies that blow-ups of these manifolds are unique, thus extending work of Biran. We also establish uniqueness of structure for certain fibered 4-manifolds.

We construct infinite families of topologically isotopic, but smoothly distinct knotted spheres, in many simply connected 4-manifolds that become smoothly isotopic after stabilizing by connected summing with S2×S2, and as a consequence, analogous families of diffeomorphisms and metrics of positive scalar curvature for such 4-manifolds. We also construct families of smoothly distinct links, all ...

Our main result demonstrates the existence of different Hamiltonian isotopy classes of symplectically embedded polydisks inside a 4-ball, and by the same argument also in the complex projective plane. Furthermore, we find exactly how large the ball can be before the embeddings become isotopic; the optimal isotopy is a version of symplectic folding. Before stating the result precisely we fix som...

Classical knot theory studies the position of a circle (knot) or of several circles (link) in R3 or S3 = R3 ∪∞. The fundamental problem of classical knot theory is the classification of links (including knots) up to the natural movement in space which is called an ambient isotopy. To distinguish knots or links we look for invariants of links, that is, properties of links which are unchanged und...