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
PreliminariesMain Results
1 Preliminaries
2 Main Results
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
PreliminariesMain Results
1. Preliminaries
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
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
PreliminariesMain Results
Preliminaries
Definition 2
p× q honeycomb grid
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
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
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
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
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
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
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
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
PreliminariesMain Results
Preliminaries
Example 1
3× 4 4× 5
6× 10
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
PreliminariesMain Results
Preliminaries
Example 1
3× 4 4× 5
6× 10
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
PreliminariesMain Results
Preliminaries
Example 1
3× 4 4× 5
6× 10
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
PreliminariesMain Results
Preliminaries
Example 1
3× 4 4× 5
6× 10
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
PreliminariesMain Results
Preliminaries
Example 1
3× 4 4× 5
6× 10
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
PreliminariesMain Results
Preliminaries
Example 1
3× 4 4× 5
6× 10
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
PreliminariesMain Results
Preliminaries
Example 1
3× 4 4× 5
6× 10
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
PreliminariesMain Results
Preliminaries
Example 1
3× 4 4× 5
6× 10
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
PreliminariesMain Results
Preliminaries
Example 1
3× 4 4× 5
6× 10
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
PreliminariesMain Results
Preliminaries
Example 1
3× 4 4× 5
6× 10
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
PreliminariesMain Results
Preliminaries
Example 1
3× 4 4× 5
6× 10
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
PreliminariesMain Results
Preliminaries
Example 2
r × r honeycomb grid
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
PreliminariesMain Results
Preliminaries
Example 2
r × r honeycomb grid
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
PreliminariesMain Results
Preliminaries
Example 2
r × r honeycomb grid
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
PreliminariesMain Results
Preliminaries
Example 2
r × r honeycomb grid
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
PreliminariesMain Results
Preliminaries
Example 2
r × r honeycomb grid
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
PreliminariesMain Results
Preliminaries
Example 2
r × r honeycomb grid
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
PreliminariesMain Results
Preliminaries
Example 2
r × r honeycomb grid
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
PreliminariesMain Results
Preliminaries
Example 2
r × r honeycomb grid
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
PreliminariesMain Results
Preliminaries
Example 3
3× 3
3× 7
3× 11
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
PreliminariesMain Results
Preliminaries
Example 3
3× 3
3× 7
3× 11
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
PreliminariesMain Results
Preliminaries
Example 3
3× 3
3× 7
3× 11
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
PreliminariesMain Results
Preliminaries
Example 3
3× 3
3× 7
3× 11
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
PreliminariesMain Results
Preliminaries
Example 3
3× 3
3× 7
3× 11
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
PreliminariesMain Results
Preliminaries
Example 3
3× 3
3× 7
3× 11
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
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
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
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
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
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
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
PreliminariesMain Results
2. Main Results
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
PreliminariesMain Results
Main Result
Notation 1
Focus on the red component
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
PreliminariesMain Results
Main Result
Notation 2
Focus on the red component
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
PreliminariesMain Results
Main Result
Notation 3
Focus on the red component
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
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
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
PreliminariesMain Results
Main Result
recall
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
PreliminariesMain Results
14× 24
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
PreliminariesMain Results
Main Result3
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
PreliminariesMain Results
Main Result3
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
PreliminariesMain Results
Main Result3
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
PreliminariesMain Results
Main Result3
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
PreliminariesMain Results
Main Result3
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
PreliminariesMain Results
Main Result3
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
PreliminariesMain Results
Main Result3
Yukari Funakoshi Celtic Knot Diagram obtained from honeycomb grid
