# Space-Efficient Prime Knot 7-Mosaics

^{*}

## Abstract

**:**

## 1. Introduction

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

## 2. Space-Efficient 7-Mosaics

**Theorem**

**1.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

- (a)
- ${9}_{6}$, ${9}_{15}$, ${9}_{18}$,
- (b)
- ${10}_{5}$, ${10}_{6}$, ${10}_{7}$, ${10}_{8}$, ${10}_{9}$, ${10}_{10}$, ${10}_{13}$, ${10}_{14}$, ${10}_{15}$, ${10}_{16}$, ${10}_{17}$, ${10}_{18}$, ${10}_{19}$, ${10}_{24}$, ${10}_{25}$, ${10}_{26}$, ${10}_{29}$, ${10}_{30}$, ${10}_{31}$, ${10}_{32}$, ${10}_{33}$, ${10}_{35}$, ${10}_{36}$, ${10}_{38}$, ${10}_{39}$,
- (c)
- $11{a}_{90}$, $11{a}_{93}$, $11{a}_{119}$, $11{a}_{145}$, $11{a}_{180}$, $11{a}_{184}$, $11{a}_{185}$, $11{a}_{192}$, $11{a}_{203}$, $11{a}_{205}$, $11{a}_{210}$, $11{a}_{226}$, $11{a}_{306}$, $11{a}_{307}$, $11{a}_{308}$, $11{a}_{309}$, $11{a}_{311}$, $11{a}_{333}$, $11{a}_{336}$, $11{a}_{337}$, $11{a}_{363}$,
- (d)
- $12{a}_{541}$, $12{a}_{601}$, $12{a}_{1024}$, $12{a}_{1034}$, $12{a}_{1126}$, and
- (e)
- $13{a}_{4304}$.

**Proof.**

**Conjecture.**

## 3. Useful Observations and the Proof of Theorem 1

**Observation**

**1**

**Observation**

**2**

**Observation**

**3**

**Observation**

**4.**

**Observation**

**5.**

**Lemma**

**1.**

**Proof.**

**Case 1:**First, we consider the case where there are two consecutive top caps. The first five tile positions in the second row are determined by Observation 2, which also prevents the sixth tile position from being a horizontal segment tile. Otherwise, there would be a cap in the seventh row, which would necessitate a tile with four connection points in the sixth position. Thus the sixth tile must be a single arc tile ${T}_{1}$, and the final tile position must be blank.

**Case 2:**The next case has two top caps with a blank tile in between them. In this case, the second row is completely determined and must have a horizontal segment tile as seen in Figure 13. Our claim is that any completion of this mosaic will not be space-efficient or will not represent a prime knot. To see this, we will examine the remaining rows. There is actually only one possibility for the third row as well, which can be seen in the first mosaic of Figure 14. The first and last tiles in the third row must complete a left and right cap, respectively, and the second and sixth tiles must have four connection points. The third and fifth tile positions in the third row must also have four connection points. Otherwise, they would be single arc tiles, and according to Observation 5, any resulting space-efficient mosaic would not represent a prime knot.

**Case 3:**Now let us consider the case where there is only one top cap in the first occupied row, and it is located in the first two tile positions after the corner tile, as in Figure 17. Then the first tile in the second occupied row must be a single arc tile ${T}_{2}$, followed by two tiles with four connection points. There must also be a single arc tile ${T}_{1}$ in this row, but this ${T}_{1}$ tile cannot be part of a right cap (that is, the tile below it is not a ${T}_{4}$ tile). To see this, assume the ${T}_{1}$ tile is part of a right cap. If the ${T}_{1}$ tile is in the fourth tile position of the second row, then, using Observations 2 and 5, it is easy to see that the knot mosaic is either not prime or not space-efficient. If the ${T}_{1}$ tile is in the fifth, sixth, or seventh position, then Observation 2 says the preceding tile position must have four connection points, which contradicts the fact that the first row only has a single top cap.

**Case 4:**Now we consider the final case, shown in Figure 21, where there is a single top cap, and it is located in the third and fourth tile positions of the first occupied row. We will assume that the first column is occupied, otherwise a shift of the mosaic to the left would reduce this to Case 3.

**Proof of Theorem 1.**

## 4. Mosaics for Theorem 3

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 9.**Knots with mosaic number 6 whose minimal mosaic tile number is greater than their tile number.

**Figure 14.**The first three rows are determined and there must be a single arc tile in the fourth or fifth position of the middle column.

**Figure 15.**If the tile in the fourth row, fourth column is a ${T}_{1}$ tile, then the mosaic is not space-efficient.

**Figure 16.**If the tile in the fourth row, fourth column is blank or a horizontal segment, then the mosaic is not space-efficient.

**Figure 18.**Possible configurations with a horizontal segment in the fourth position of the second row.

**Figure 19.**Possible configurations of the first three rows when the second row has two horizontal line segments.

**Figure 22.**Possible configurations when there is a single top cap occupying the third and fourth position of the first row and a horizontal segment tile in the fifth position of the second row.

**Figure 26.**The second outer shell completed is not space-efficient, and the fifth outer shell can be completed in three ways.

**Figure 27.**The sixth, seventh, eleventh, twelfth, fifteenth, eighteenth, and twentieth outer shells completed in ways other than those given in Theorem 1.

**Figure 28.**The alternative completions of the fourteenth and seventeenth outer shells can be simplified to not use the vertical segment tiles.

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**MDPI and ACS Style**

Heap, A.; LaCourt, N.
Space-Efficient Prime Knot 7-Mosaics. *Symmetry* **2020**, *12*, 576.
https://doi.org/10.3390/sym12040576

**AMA Style**

Heap A, LaCourt N.
Space-Efficient Prime Knot 7-Mosaics. *Symmetry*. 2020; 12(4):576.
https://doi.org/10.3390/sym12040576

**Chicago/Turabian Style**

Heap, Aaron, and Natalie LaCourt.
2020. "Space-Efficient Prime Knot 7-Mosaics" *Symmetry* 12, no. 4: 576.
https://doi.org/10.3390/sym12040576