We have found new types of transition metal chalcogenide crystals (NbSe_{3}, TaS_{3}, TaSe_{3}) which have the shapes of a ring, a disk, or a Mobius strip of 10-100 micrometers size.

Aside from their peculiarity in the outer shapes, it is found that the lattice spaces of these striking crystal forms have topologies of twisted loops themselves. In this sense the matters are distinctive compared to ordinary rectilinear crystals, whose topology is just trivial.

In fact, this new group of crystals is a physical realization of the "ribbon-knots" group in the field of topology, which deals with geometries of two-faced, closed loops. Hence the name "topological matters".

We researchers of solid state physics are interested in quantum mechanical behavior of electrons in such a twisted, closed loop system. However, approaches from various research fields, e.g. chemistry in their formation process, crystallographic view on their structural distortion, or mathematical determination of the optimal surfaces for a topology etc. will be granted.

2. Low dimensionality

3. Possible applications