Celtic Knot Diagram obtained from honeycomb grid

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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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1 Preliminaries

2 Main Results

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1. Preliminaries

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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 agrid.

Fact 1

There are only three types of regular polygons that can divide a plane by itself. regular triangle, Square, regular hexagon.

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Preliminaries

Definition 2

p×q honeycomb grid

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Definition 3

Celtic Knot Projection (CKP) induced by p×q honeycomb grid

Celtic Knot Design (CKD) induced by p×q honeycomb grid

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Preliminaries

Definition 3

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Definition 3

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Preliminaries

Definition 3

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Definition 3

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Preliminaries

Definition 3

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Remark 1

CKP^：

←→

mirror

CKD D1 CKDD2

Today,







・We will use D₁.

・We will express it in CDP, if we don’t need the information above and below of intersections.

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Preliminaries Example 1

3×4 4×5

6×10

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Example 1

3×4 4×5

6×10

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3×4 4×5

6×10

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Example 1

3×4 4×5

6×10

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3×4 4×5

6×10

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Example 1

3×4 4×5

6×10

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3×4 4×5

6×10

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Example 1

3×4 4×5

6×10

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3×4 4×5

6×10

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Example 1

3×4 4×5

6×10

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3×4 4×5

6×10

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Example 2

r×r honeycomb grid

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Preliminaries

Example 2

r×r honeycomb grid

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Example 2

r×r honeycomb grid

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Preliminaries

Example 2

r×r honeycomb grid

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Example 2

r×r honeycomb grid

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Preliminaries

Example 2

r×r honeycomb grid

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Example 2

r×r honeycomb grid

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Preliminaries

Example 2

r×r honeycomb grid

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Example 3

3×3

3×7

3×11

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Preliminaries

Example 3

3×3

3×7

3×11

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Example 3

3×3

3×7

3×11

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Preliminaries

Example 3

3×3

3×7

3×11

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Example 3

3×3

3×7

3×11

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Preliminaries

Example 3

3×3

3×7

3×11

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Example 4

Examples of other types of links^： 5×8

7×11

The number of components? The characteristics of each component?

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Preliminaries

Example 4

7×11

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Example 4

7×11

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Preliminaries

Example 4

7×11

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Example 4

7×11

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Preliminaries

Example 4

7×11

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2. Main Results

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Main Result

Notation 1

Focus on the red component

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Notation 2

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Main Result

Notation 3

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G^：p×q honeycomb grid D：a CKD induced from G

Main Result 1

The number of components of a link obtained fromDis r.

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Main Result

G^：p×q honeycomb grid , D^：a CKD induced fromG Proposition 1

Ifq=m(p+ 1)−1 (m= 1,2,3, . . .), then the number of components of a links obtained fromDis 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”.

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recall

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Main Result

3×3 3×7

3×11

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3×3 3×7

3×11

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Main Result

3×3 3×7

3×11

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3×3 3×7

3×11

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Main Result

G^：p×q honeycomb grid , D^：a CKD induced fromG Main Result 2

The link component obtained fromDpassing through the regular hexagon at the upper left ofGand the component passing through the regular hexagon at the lower right ofGare coincide.

3×7

7×11

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Definition 4

The component of the link obtained fromDthat passing through the regular hexagon at the top left ofG is called bySpur.

7×11

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Main Result

Definition 4

3×7 4×5

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Definition 4

3×7 4×5

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Main Result

Definition 4

3×7 4×5

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Definition 4

3×7 4×5

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Main Result

Definition 4

3×7 4×5

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Definition 4

3×7 4×5

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Main Result

Definition 4

3×7 4×5

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Definition 4

3×7 4×5

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Main Result

Definition 4

3×7 4×5

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Definition 4

3×7 4×5

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Main Result

7×11

Main Result 3

For link obtained fromD, each component except Spur are same knot.

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7×11

Main Result 3

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Main Result

7×11

Main Result 3

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14× 24

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Main Result3

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Main Result3

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Main Result3

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Main Result3