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Polyac states that all directed versions of Leidemeister moves in knot and link diagrams can be generated by a set of four directed Leidemeister moves, and that four or more directed Leidemeister moves generate all of them. proved to do. As the minimal generating set, we refer to the set containing four directional Leidemeister movements that collectively generate all other directional Leidemeister movements. Polyak also found that a particular set containing two Leidemeister moves of type 1, one move of type 2, and one move of type 3 is the minimally generated set of all directional Leidemeister moves. proved to form We extend Polyak’s work by providing an additional 11 minimal four-element directional Leidemeister move generation sets, and that these 12 sets represent all possible minimal generations of directional Leidemeister moves. Prove that it represents a set. We also consider a Leidemeister-type move that associates a directed spatial trivalent graph diagram with the trivalent vertices that are sources and sinks, and assign a minimum generated set of directed Leidemeister-type moves for a spatial trivalent graph diagram of 10 Prove that movement is involved.

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