celtic knot diagram obtained from honeycomb gridichihara/knots2019/slides/191221_2... · hexagon at...
TRANSCRIPT
![Page 1: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/1.jpg)
PreliminariesMain 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
![Page 2: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/2.jpg)
PreliminariesMain Results
1 Preliminaries
2 Main Results
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
![Page 3: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/3.jpg)
PreliminariesMain Results
1. Preliminaries
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
![Page 4: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/4.jpg)
PreliminariesMain Results
Preliminaries
Definition 1
Suppose that a plane is divided by polygons. A closed subset ofthis plane whose boundary is made up of some edges of thepolygons is called a grid.
Fact 1
There are only three types of regular polygons that can divide aplane by itself. regular triangle, Square, regular hexagon.
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
![Page 5: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/5.jpg)
PreliminariesMain Results
Preliminaries
Definition 2
p× q honeycomb grid
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
![Page 6: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/6.jpg)
PreliminariesMain 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
![Page 7: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/7.jpg)
PreliminariesMain 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
![Page 8: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/8.jpg)
PreliminariesMain 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
![Page 9: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/9.jpg)
PreliminariesMain 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
![Page 10: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/10.jpg)
PreliminariesMain 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
![Page 11: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/11.jpg)
PreliminariesMain 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
![Page 12: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/12.jpg)
PreliminariesMain 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
![Page 13: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/13.jpg)
PreliminariesMain Results
Preliminaries
Example 1
3× 4 4× 5
6× 10
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
![Page 14: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/14.jpg)
PreliminariesMain Results
Preliminaries
Example 1
3× 4 4× 5
6× 10
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
![Page 15: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/15.jpg)
PreliminariesMain Results
Preliminaries
Example 1
3× 4 4× 5
6× 10
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
![Page 16: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/16.jpg)
PreliminariesMain Results
Preliminaries
Example 1
3× 4 4× 5
6× 10
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
![Page 17: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/17.jpg)
PreliminariesMain Results
Preliminaries
Example 1
3× 4 4× 5
6× 10
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
![Page 18: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/18.jpg)
PreliminariesMain Results
Preliminaries
Example 1
3× 4 4× 5
6× 10
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
![Page 19: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/19.jpg)
PreliminariesMain Results
Preliminaries
Example 1
3× 4 4× 5
6× 10
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
![Page 20: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/20.jpg)
PreliminariesMain Results
Preliminaries
Example 1
3× 4 4× 5
6× 10
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
![Page 21: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/21.jpg)
PreliminariesMain Results
Preliminaries
Example 1
3× 4 4× 5
6× 10
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
![Page 22: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/22.jpg)
PreliminariesMain Results
Preliminaries
Example 1
3× 4 4× 5
6× 10
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
![Page 23: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/23.jpg)
PreliminariesMain Results
Preliminaries
Example 1
3× 4 4× 5
6× 10
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
![Page 24: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/24.jpg)
PreliminariesMain Results
Preliminaries
Example 2
r × r honeycomb grid
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
![Page 25: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/25.jpg)
PreliminariesMain Results
Preliminaries
Example 2
r × r honeycomb grid
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
![Page 26: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/26.jpg)
PreliminariesMain Results
Preliminaries
Example 2
r × r honeycomb grid
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
![Page 27: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/27.jpg)
PreliminariesMain Results
Preliminaries
Example 2
r × r honeycomb grid
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
![Page 28: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/28.jpg)
PreliminariesMain Results
Preliminaries
Example 2
r × r honeycomb grid
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
![Page 29: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/29.jpg)
PreliminariesMain Results
Preliminaries
Example 2
r × r honeycomb grid
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
![Page 30: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/30.jpg)
PreliminariesMain Results
Preliminaries
Example 2
r × r honeycomb grid
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
![Page 31: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/31.jpg)
PreliminariesMain Results
Preliminaries
Example 2
r × r honeycomb grid
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
![Page 32: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/32.jpg)
PreliminariesMain Results
Preliminaries
Example 3
3× 3
3× 7
3× 11
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
![Page 33: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/33.jpg)
PreliminariesMain Results
Preliminaries
Example 3
3× 3
3× 7
3× 11
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
![Page 34: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/34.jpg)
PreliminariesMain Results
Preliminaries
Example 3
3× 3
3× 7
3× 11
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
![Page 35: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/35.jpg)
PreliminariesMain Results
Preliminaries
Example 3
3× 3
3× 7
3× 11
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
![Page 36: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/36.jpg)
PreliminariesMain Results
Preliminaries
Example 3
3× 3
3× 7
3× 11
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
![Page 37: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/37.jpg)
PreliminariesMain Results
Preliminaries
Example 3
3× 3
3× 7
3× 11
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
![Page 38: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/38.jpg)
PreliminariesMain 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
![Page 39: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/39.jpg)
PreliminariesMain 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
![Page 40: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/40.jpg)
PreliminariesMain 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
![Page 41: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/41.jpg)
PreliminariesMain 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
![Page 42: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/42.jpg)
PreliminariesMain 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
![Page 43: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/43.jpg)
PreliminariesMain 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
![Page 44: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/44.jpg)
PreliminariesMain Results
2. Main Results
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
![Page 45: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/45.jpg)
PreliminariesMain Results
Main Result
Notation 1
Focus on the red component
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
![Page 46: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/46.jpg)
PreliminariesMain Results
Main Result
Notation 2
Focus on the red component
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
![Page 47: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/47.jpg)
PreliminariesMain Results
Main Result
Notation 3
Focus on the red component
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
![Page 48: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/48.jpg)
PreliminariesMain Results
Main Result
G:p× q honeycomb gridD: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
![Page 49: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/49.jpg)
PreliminariesMain 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 ofcomponents of a links obtained from D is p and each componentis 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
![Page 50: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/50.jpg)
PreliminariesMain Results
Main Result
recall
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
![Page 51: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/51.jpg)
PreliminariesMain 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 ofcomponents of a links obtained from D is p and each componentis 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
![Page 52: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/52.jpg)
PreliminariesMain 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 ofcomponents of a links obtained from D is p and each componentis 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
![Page 53: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/53.jpg)
PreliminariesMain 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 ofcomponents of a links obtained from D is p and each componentis 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
![Page 54: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/54.jpg)
PreliminariesMain 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 ofcomponents of a links obtained from D is p and each componentis 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
![Page 55: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/55.jpg)
PreliminariesMain 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 regularhexagon at the upper left of G and the component passingthrough the regular hexagon at the lower right of G are coincide.
3× 7
7× 11
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
![Page 56: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/56.jpg)
PreliminariesMain Results
Main Result
Definition 4
The component of the link obtained from D that passing throughthe regular hexagon at the top left of G is called by Spur.
7× 11
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
![Page 57: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/57.jpg)
PreliminariesMain Results
Main Result
Definition 4
The component of the link obtained from D that passing throughthe 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
![Page 58: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/58.jpg)
PreliminariesMain Results
Main Result
Definition 4
The component of the link obtained from D that passing throughthe 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
![Page 59: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/59.jpg)
PreliminariesMain Results
Main Result
Definition 4
The component of the link obtained from D that passing throughthe 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
![Page 60: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/60.jpg)
PreliminariesMain Results
Main Result
Definition 4
The component of the link obtained from D that passing throughthe 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
![Page 61: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/61.jpg)
PreliminariesMain Results
Main Result
Definition 4
The component of the link obtained from D that passing throughthe 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
![Page 62: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/62.jpg)
PreliminariesMain Results
Main Result
Definition 4
The component of the link obtained from D that passing throughthe 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
![Page 63: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/63.jpg)
PreliminariesMain Results
Main Result
Definition 4
The component of the link obtained from D that passing throughthe 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
![Page 64: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/64.jpg)
PreliminariesMain Results
Main Result
Definition 4
The component of the link obtained from D that passing throughthe 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
![Page 65: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/65.jpg)
PreliminariesMain Results
Main Result
Definition 4
The component of the link obtained from D that passing throughthe 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
![Page 66: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/66.jpg)
PreliminariesMain Results
Main Result
Definition 4
The component of the link obtained from D that passing throughthe 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
![Page 67: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/67.jpg)
PreliminariesMain Results
Main Result
7× 11
Main Result 3
For link obtained from D, each component except Spur are sameknot.
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
![Page 68: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/68.jpg)
PreliminariesMain Results
Main Result
7× 11
Main Result 3
For link obtained from D, each component except Spur are sameknot.
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
![Page 69: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/69.jpg)
PreliminariesMain Results
Main Result
7× 11
Main Result 3
For link obtained from D, each component except Spur are sameknot.
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
![Page 70: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/70.jpg)
PreliminariesMain Results
14× 24
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
![Page 71: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/71.jpg)
PreliminariesMain Results
Main Result3
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
![Page 72: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/72.jpg)
PreliminariesMain Results
Main Result3
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
![Page 73: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/73.jpg)
PreliminariesMain Results
Main Result3
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
![Page 74: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/74.jpg)
PreliminariesMain Results
Main Result3
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
![Page 75: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/75.jpg)
PreliminariesMain Results
Main Result3
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
![Page 76: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/76.jpg)
PreliminariesMain Results
Main Result3
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
![Page 77: Celtic Knot Diagram obtained from honeycomb gridichihara/Knots2019/Slides/191221_2... · hexagon at the upper left of G and the component passing through the regular hexagon at the](https://reader035.vdocuments.net/reader035/viewer/2022071117/60016be96ff2c23b1f5bdb45/html5/thumbnails/77.jpg)
PreliminariesMain Results
Main Result3
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid