Reidemeister Moves On Graphs With Dowker Notation And Coloring

The main content of this paper discusses the influence of R-moves on Dowker Notationof knots, and the influence of R-moves on coloring universal graph of knots. It analyzes theinfluence of R-moves on a projection of a knot and R-moves on universal graph of a knot, sothat we have the transformation rule of Dowker Notation sequence of a knot and the changeof the color number of coloring a universal graph of a knot.The purpose of this review is to define an equivalence sequence of Dowker Notationthrough Dowker Notation on knots to show three kinds of R-moves, and three discussiongroups of the influence of R-moves on a projection’s Dowker Notation sequence of a knot.The first discusses the influence of a projection’s Dowker Notation sequence of a knot afterR-1move. The second discusses the influence of R-2move. The third discusses the influenceof R-3move. Then it gets a result that graph equivalence and dual graph equivalence. Theuniversal graph can get a corresponding graph G through putting a vertex at the center ofeach closed region and connecting the vertex of adjacent region. Then coloring the universalgraph can translate into calculating the chromatic polynomial of G. It can be obtained thatthe influence of colored number of R-moves on universal graph coloring from calculatingchromatic polynomial of G after R-moves. It has three discussion groups. The firstdiscusses the influence of the least coloring number after R-1move on a universal graph. Thesecond discusses the influence of the least coloring number of R-2move on a universal graph.The third discusses the influence of the least coloring number of R-3move on a universalgraph.