Celtic Knot Diagram obtained from honeycomb grid Yukari Funakoshi - PowerPoint PPT Presentation

Preliminaries Main Results Celtic Knot Diagram obtained from honeycomb grid Yukari Funakoshi Collaborative Organization for Research in women’s Education of Science, Technology, Engineering, and Mathematics (Nara Women’s University) December 21, 2019 Joint work with M. Hashizume. Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

Preliminaries Main Results 1 Preliminaries 2 Main Results Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

Preliminaries Main Results 1. Preliminaries Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

Preliminaries Main Results Preliminaries Definition 1 Suppose that a plane is divided by polygons. A closed subset of this plane whose boundary is made up of some edges of the polygons is called a grid . Fact 1 There are only three types of regular polygons that can divide a plane by itself. regular triangle, Square, regular hexagon. Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

Preliminaries Main Results Preliminaries Definition 2 p × q honeycomb grid Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

Preliminaries Main Results Preliminaries Definition 3 Celtic Knot Projection (CKP) induced by p × q honeycomb grid Celtic Knot Design (CKD) induced by p × q honeycomb grid Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

Preliminaries Main Results Preliminaries Definition 3 Celtic Knot Projection (CKP) induced by p × q honeycomb grid Celtic Knot Design (CKD) induced by p × q honeycomb grid Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

Preliminaries Main Results Preliminaries Definition 3 Celtic Knot Projection (CKP) induced by p × q honeycomb grid Celtic Knot Design (CKD) induced by p × q honeycomb grid Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

Preliminaries Main Results Preliminaries Definition 3 Celtic Knot Projection (CKP) induced by p × q honeycomb grid Celtic Knot Design (CKD) induced by p × q honeycomb grid Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

Preliminaries Main Results Preliminaries Definition 3 Celtic Knot Projection (CKP) induced by p × q honeycomb grid Celtic Knot Design (CKD) induced by p × q honeycomb grid Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

Preliminaries Main Results Preliminaries Definition 3 Celtic Knot Projection (CKP) induced by p × q honeycomb grid Celtic Knot Design (CKD) induced by p × q honeycomb grid Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

Preliminaries Main Results Preliminaries Remark 1 CKP ： ← → mirror CKD D 1 CKD D 2  ・ We will use D 1 .   ・ We will express it in CDP, if we don’t need the information Today,  above and below of intersections.  Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

Preliminaries Main Results Preliminaries Example 1 3 × 4 4 × 5 6 × 10 Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

Preliminaries Main Results Preliminaries Example 1 3 × 4 4 × 5 6 × 10 Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

Preliminaries Main Results Preliminaries Example 1 3 × 4 4 × 5 6 × 10 Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

Preliminaries Main Results Preliminaries Example 1 3 × 4 4 × 5 6 × 10 Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

Preliminaries Main Results Preliminaries Example 1 3 × 4 4 × 5 6 × 10 Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

Preliminaries Main Results Preliminaries Example 1 3 × 4 4 × 5 6 × 10 Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

Preliminaries Main Results Preliminaries Example 1 3 × 4 4 × 5 6 × 10 Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

Preliminaries Main Results Preliminaries Example 1 3 × 4 4 × 5 6 × 10 Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

Preliminaries Main Results Preliminaries Example 1 3 × 4 4 × 5 6 × 10 Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

Preliminaries Main Results Preliminaries Example 1 3 × 4 4 × 5 6 × 10 Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

Preliminaries Main Results Preliminaries Example 1 3 × 4 4 × 5 6 × 10 Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

Preliminaries Main Results Preliminaries Example 2 r × r honeycomb grid Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

Preliminaries Main Results Preliminaries Example 2 r × r honeycomb grid Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

Preliminaries Main Results Preliminaries Example 2 r × r honeycomb grid Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

Preliminaries Main Results Preliminaries Example 2 r × r honeycomb grid Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

Preliminaries Main Results Preliminaries Example 2 r × r honeycomb grid Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

Preliminaries Main Results Preliminaries Example 2 r × r honeycomb grid Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

Preliminaries Main Results Preliminaries Example 2 r × r honeycomb grid Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

Preliminaries Main Results Preliminaries Example 2 r × r honeycomb grid Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

Preliminaries Main Results Preliminaries Example 3 3 × 3 3 × 7 3 × 11 Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

Preliminaries Main Results Preliminaries Example 3 3 × 3 3 × 7 3 × 11 Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

Preliminaries Main Results Preliminaries Example 3 3 × 3 3 × 7 3 × 11 Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

Preliminaries Main Results Preliminaries Example 3 3 × 3 3 × 7 3 × 11 Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

Preliminaries Main Results Preliminaries Example 3 3 × 3 3 × 7 3 × 11 Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

Preliminaries Main Results Preliminaries Example 3 3 × 3 3 × 7 3 × 11 Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

Preliminaries Main Results Preliminaries Example 4 Examples of other types of links ： 5 × 8 7 × 11 The number of components? The characteristics of each component? Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

Preliminaries Main Results Preliminaries Example 4 Examples of other types of links ： 5 × 8 7 × 11 The number of components? The characteristics of each component? Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

Preliminaries Main Results Preliminaries Example 4 Examples of other types of links ： 5 × 8 7 × 11 The number of components? The characteristics of each component? Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

Preliminaries Main Results Preliminaries Example 4 Examples of other types of links ： 5 × 8 7 × 11 The number of components? The characteristics of each component? Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

Preliminaries Main Results Preliminaries Example 4 Examples of other types of links ： 5 × 8 7 × 11 The number of components? The characteristics of each component? Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

Preliminaries Main Results Preliminaries Example 4 Examples of other types of links ： 5 × 8 7 × 11 The number of components? The characteristics of each component? Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

Preliminaries Main Results 2. Main Results Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

Preliminaries Main Results Main Result Notation 1 Focus on the red component Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

Preliminaries Main Results Main Result Notation 2 Focus on the red component Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

Preliminaries Main Results Main Result Notation 3 Focus on the red component Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

Preliminaries Main Results Main Result G ： p × q honeycomb grid D ： a CKD induced from G Main Result 1 The number of components of a link obtained from D is r . Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

Preliminaries Main Results Main Result G ： p × q honeycomb grid , D ： a CKD induced from G Proposition 1 If q = m ( p + 1) − 1 ( m = 1 , 2 , 3 , . . . ) , then the number of components of a links obtained from D is p and each component is a trivial knot. 3 × 3 3 × 7 3 × 11 There are other types of links, depending on “size of grid”. Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid