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The crossing numbers of knots and links via 4D topology
Akio Kawauchi
The diagrams of links (including knots) are characterized in terms of circular rigid disk-chord diagrams of their spun ribbon torus-links in the 4-sphere. As a result, the crossing…
Ribbonness on classical link
Akio Kawauchi
It is shown that if a link in 3-space bounds a proper oriented surface (without closed component) in the upper half 4-space, then the link bounds a proper oriented ribbon surface i…
Whitehead aspherical conjecture via ribbon sphere-link
Akio Kawauchi
Whitehead aspherical conjecture says that every connected subcomplex of every aspherical 2-complex is aspherical. By an argument on ribbon sphere-links, it is confirmed that the co…
Ribbonness of Kervaire's sphere-link in homotopy 4-sphere and its consequences to 2-complexes
Akio Kawauchi
M. A. Kervaire showed that every group of deficiency and weight is the fundamental group of a smooth sphere-link of components in a smooth homotopy 4-sphere. In the use…
Spatial graph as connected sum of a planar graph and a braid
Valeriy G. Bardakov, Akio Kawauchi
In this paper we show that every finite spatial graph is a connected sum of a planar graph, which is a forest, i.e. disjoint union of finite number of trees and a tangle. As a cons…
Unique diagram of a spatial arc and the knotting probability
Akio Kawauchi
It is shown that the projection image of an oriented spatial arc to any oriented plane is approximated by a unique arc diagram (up to isomorphic arc diagrams) determined from the s…