BIRS Workshop Lecture Videos

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BIRS Workshop Lecture Videos

3-dimensional topology and polycontinuous pattern Shimokawa, Koya


Block copolymers produce spherical, cylindrical, lamellar and bicontinuous patterns as microphase-separated structures. Typical examples of bicontinuous patterns are Gyroid, D-surface and P-surface. A mathematical model of such a structure is a triply periodic non-compact surface embedded in the 3-dimensional space which divides it into two possibly disconnected submanifolds. We will consider the case where submanifolds are open neighborhood of networks. Here a network means an infinite graph embedded in the 3-dimensional space. In this case the bicontinuous pattern is uniquely determined by networks up to isotopy. We say such a bicontinuous pattern is associated to a network. On the other hand, for example triblock-arm star-shaped molecules yields a tricontinuous pattern. One mathematical model of such a tricontinuous (resp. poly-continuous) pattern is a triply periodic non-compact multibranched surface (or more generally polyhedron) dividing it into 3 (resp. several) possibly disconnected non-compact submanifolds. We assume that each submanifold is the open neighborhood of three (resp. several) networks. We call such a multibranched surface a tricontinuous pattern(resp. poly-continuous pattern). The relation between poly-continuous patterns and networks is not obvious in this case. Two different poly-continuous patterns are associated to one network and vice versa. In this talk we will give a condition for poly-continuous patterns to give the same network. We will also show that two poly-continuous patterns can be related by a finite sequence of moves and discuss further applications.

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