Abstract
The concepts of tile number and space-efficiency for knot mosaics were first explored by Heap and Knowles in 2018, where they determined the possible tile numbers and space-efficient layouts for every prime knot with mosaic number 6 or less. In this paper, we extend those results to prime knots with mosaic number 7. Specifically, we find the possible values for the number of non-blank tiles used in a space-efficient 7 × 7 mosaic of a prime knot are 27, 29, 31, 32, 34, 36, 37, 39, and 41. We also provide the possible layouts for the mosaics that lead to these values. Finally, we determine which prime knots can be placed within the first of these layouts, resulting in a list of knots with mosaic number 7 and tile number 27.
Highlights
Knot mosaics were first introduced by Lomonaco and Kauffman in [1] as a basic building block of blueprints for constructing an actual physical quantum system, with a mosaic knot representing a quantum knot
We use KnotScape to determine what knots are depicted in the mosaic by choosing the crossings so that they are alternating, as well as all possible non-alternating combinations. Doing this for all three layouts with 27 non-blank tiles, we find all knots with mosaic number 7 and tile number
We find prime knots with mosaic number 6 and minimal mosaic tile number 32 but whose tile number 27 is only realized on mosaics of size 7 or larger
Summary
Knot mosaics were first introduced by Lomonaco and Kauffman in [1] as a basic building block of blueprints for constructing an actual physical quantum system, with a mosaic knot representing a quantum knot. It is of some interest to know how many non-blank tiles are necessary to depict a knot on a minimal mosaic, which is known as the minimal mosaic tile number of a knot, first introduced in [3]. We will make significant use of these moves throughout this paper, as we attempt to construct knot mosaics that use the least number of non-blank tiles. If we want to create knot mosaics efficiently, using the least number of non-blank tiles necessary, we will want to avoid these reducible crossings. A knot n-mosaic is space-efficient if it is reduced and the number of non-blank tiles is as small as possible on an n-mosaic without changing the knot type of the depicted knot. The number of non-blank tiles in a knot mosaic that is space-efficient cannot be decreased through a sequence of mosaic planar isotopy moves. Special thanks are due to James Canning, who was kind enough and brilliant enough to create for us a program that automated the process of creating the mosaics in the Table of Mosaics of Section 4
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