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

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celtic knot diagram obtained from honeycomb grid

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

Dec 13, 2023 625 likes •1.41k views

Preliminaries Main Results Celtic Knot Diagram obtained from honeycomb grid Yukari Funakoshi Collaborative Organization for Research in womens Education of Science, Technology, Engineering, and Mathematics (Nara Womens University)

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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

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SLIDE 2

Preliminaries Main Results

1 Preliminaries 2 Main Results

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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Preliminaries Main Results

1. Preliminaries

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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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

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Preliminaries Main Results

Preliminaries

Definition 2 p × q honeycomb grid

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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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

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SLIDE 7

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

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SLIDE 8

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

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SLIDE 9

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

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SLIDE 10

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

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SLIDE 11

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

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Preliminaries Main Results

Preliminaries

Remark 1 CKP： ← → mirror CKD D1 CKD D2 Today,      ・We will use D1. ・We will express it in CDP, if we don’t need the information above and below of intersections.

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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Preliminaries Main Results

Preliminaries

Example 1 3 × 4 4 × 5 6 × 10

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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Preliminaries Main Results

Preliminaries

Example 1 3 × 4 4 × 5 6 × 10

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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SLIDE 15

Preliminaries Main Results

Preliminaries

Example 1 3 × 4 4 × 5 6 × 10

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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SLIDE 16

Preliminaries Main Results

Preliminaries

Example 1 3 × 4 4 × 5 6 × 10

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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SLIDE 17

Preliminaries Main Results

Preliminaries

Example 1 3 × 4 4 × 5 6 × 10

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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SLIDE 18

Preliminaries Main Results

Preliminaries

Example 1 3 × 4 4 × 5 6 × 10

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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SLIDE 19

Preliminaries Main Results

Preliminaries

Example 1 3 × 4 4 × 5 6 × 10

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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SLIDE 20

Preliminaries Main Results

Preliminaries

Example 1 3 × 4 4 × 5 6 × 10

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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SLIDE 21

Preliminaries Main Results

Preliminaries

Example 1 3 × 4 4 × 5 6 × 10

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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SLIDE 22

Preliminaries Main Results

Preliminaries

Example 1 3 × 4 4 × 5 6 × 10

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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SLIDE 23

Preliminaries Main Results

Preliminaries

Example 1 3 × 4 4 × 5 6 × 10

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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Preliminaries Main Results

Preliminaries

Example 2 r × r honeycomb grid

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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Preliminaries Main Results

Preliminaries

Example 2 r × r honeycomb grid

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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Preliminaries Main Results

Preliminaries

Example 2 r × r honeycomb grid

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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Preliminaries Main Results

Preliminaries

Example 2 r × r honeycomb grid

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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Preliminaries Main Results

Preliminaries

Example 2 r × r honeycomb grid

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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Preliminaries Main Results

Preliminaries

Example 2 r × r honeycomb grid

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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Preliminaries Main Results

Preliminaries

Example 2 r × r honeycomb grid

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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Preliminaries Main Results

Preliminaries

Example 2 r × r honeycomb grid

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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Preliminaries Main Results

Preliminaries

Example 3 3 × 3 3 × 7 3 × 11

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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Preliminaries Main Results

Preliminaries

Example 3 3 × 3 3 × 7 3 × 11

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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Preliminaries Main Results

Preliminaries

Example 3 3 × 3 3 × 7 3 × 11

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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SLIDE 35

Preliminaries Main Results

Preliminaries

Example 3 3 × 3 3 × 7 3 × 11

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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SLIDE 36

Preliminaries Main Results

Preliminaries

Example 3 3 × 3 3 × 7 3 × 11

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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SLIDE 37

Preliminaries Main Results

Preliminaries

Example 3 3 × 3 3 × 7 3 × 11

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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SLIDE 38

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

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SLIDE 39

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

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SLIDE 40

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

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SLIDE 41

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

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SLIDE 42

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

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SLIDE 43

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

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Preliminaries Main Results

2. Main Results

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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Preliminaries Main Results

Main Result

Notation 1 Focus on the red component

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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Preliminaries Main Results

Main Result

Notation 2 Focus on the red component

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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Preliminaries Main Results

Main Result

Notation 3 Focus on the red component

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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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

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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

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Preliminaries Main Results

Main Result

recall

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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SLIDE 51

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

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SLIDE 52

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

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SLIDE 53

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

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SLIDE 54

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

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Preliminaries Main Results

Main Result

G：p × q honeycomb grid , D：a CKD induced from G Main Result 2 The link component obtained from D passing through the regular hexagon at the upper left of G and the component passing through the regular hexagon at the lower right of G are coincide. 3 × 7 7 × 11

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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Preliminaries Main Results

Main Result

Definition 4 The component of the link obtained from D that passing through the regular hexagon at the top left of G is called by Spur. 7 × 11

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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Preliminaries Main Results

Main Result

Definition 4 The component of the link obtained from D that passing through the regular hexagon at the top left of G is called by Spur. 3 × 7 4 × 5

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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SLIDE 58

Preliminaries Main Results

Main Result

Definition 4 The component of the link obtained from D that passing through the regular hexagon at the top left of G is called by Spur. 3 × 7 4 × 5

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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SLIDE 59

Preliminaries Main Results

Main Result

Definition 4 The component of the link obtained from D that passing through the regular hexagon at the top left of G is called by Spur. 3 × 7 4 × 5

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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SLIDE 60

Preliminaries Main Results

Main Result

Definition 4 The component of the link obtained from D that passing through the regular hexagon at the top left of G is called by Spur. 3 × 7 4 × 5

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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SLIDE 61

Preliminaries Main Results

Main Result

Definition 4 The component of the link obtained from D that passing through the regular hexagon at the top left of G is called by Spur. 3 × 7 4 × 5

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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SLIDE 62

Preliminaries Main Results

Main Result

Definition 4 The component of the link obtained from D that passing through the regular hexagon at the top left of G is called by Spur. 3 × 7 4 × 5

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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SLIDE 63

Preliminaries Main Results

Main Result

Definition 4 The component of the link obtained from D that passing through the regular hexagon at the top left of G is called by Spur. 3 × 7 4 × 5

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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SLIDE 64

Preliminaries Main Results

Main Result

Definition 4 The component of the link obtained from D that passing through the regular hexagon at the top left of G is called by Spur. 3 × 7 4 × 5

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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SLIDE 65

Preliminaries Main Results

Main Result

Definition 4 The component of the link obtained from D that passing through the regular hexagon at the top left of G is called by Spur. 3 × 7 4 × 5

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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SLIDE 66

Preliminaries Main Results

Main Result

Definition 4 The component of the link obtained from D that passing through the regular hexagon at the top left of G is called by Spur. 3 × 7 4 × 5

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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Preliminaries Main Results

Main Result

7 × 11 Main Result 3 For link obtained from D, each component except Spur are same knot.

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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Preliminaries Main Results

Main Result

7 × 11 Main Result 3 For link obtained from D, each component except Spur are same knot.

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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Preliminaries Main Results

Main Result

7 × 11 Main Result 3 For link obtained from D, each component except Spur are same knot.

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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Preliminaries Main Results

14 × 24

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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Preliminaries Main Results

Main Result3

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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Preliminaries Main Results

Main Result3

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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Preliminaries Main Results

Main Result3

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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Preliminaries Main Results

Main Result3

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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Preliminaries Main Results

Main Result3

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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Preliminaries Main Results

Main Result3

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid

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Preliminaries Main Results

Main Result3

Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid