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We construct a group Γ 4 n corresponding to the motion of points in R 3 from the point of view of Delaunay triangulations. We study homomorphisms from pure braids on n strands to the product of copies of Γ 4n . We will also study the group of pure braids in R 3 , which is described by a fundamental group of the restricted configuration space of R 3 , and define the group homomorphism from the group of pure braids in R 3 to Γ 4 n . In the end of this paper we give some comments about relations between the restricted configuration space of R 3 and triangulations of the 3-dimensional ball and Pachner moves.1991 Mathematics Subject Classification. 57M25, 57M27.

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In [2] the author generalized the Conway algebra and constructed the invariant valued in the generalized Conway algebra defined by applying two skein relations to crossings, which is called a generalized Conway type invariant. The generalized Conway type invariant is a generalization of Homflypt polynomial.In this paper we show that an example of links, which have the same value of Homflypt polynomial, but have different values of the generalized Conway type invariant. We study a properties of Conway type invariant related to Vassiliev invariant. In section 3 we discuss about further researches.1991 Mathematics Subject Classification. 57M25.

A virtual knot, which is one of generalizations of knots in R 3 (or S 3 ), is, roughly speaking, an embedded circle in thickened surface Sg × I. In this talk we will discuss about knots in 3 dimensional Sg × S 1 . We introduce basic notions for knots in Sg × S 1 , for example, diagrams, moves for diagrams and so on. For knots in Sg × S 1 technically we lose over/under information, but we will have information how many times the knot rotates along S 1 . We will discuss the geometric meaning of the rotating information and how to construct invariants by using the "rotating" information.