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Space-efficient
knot mosaics for prime knots with mosaic number 6
Aaron Heap and Douglas Knowles
Vol. 12 (2019), No. 5, 767–789
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Abstract |
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In 2008, Kauffman and Lomonaco introduced the concepts of a knot mosaic and the mosaic number of a knot or link , the smallest integer such that can be represented on an -mosaic. In 2018, the authors of this paper introduced and explored space-efficient knot mosaics and the tile number of , the smallest number of nonblank tiles necessary to depict on a knot mosaic. They determine bounds for the tile number in terms of the mosaic number. In this paper, we focus specifically on prime knots with mosaic number 6. We determine a complete list of these knots, provide a minimal, space-efficient knot mosaic for each of them, and determine the tile number (or minimal mosaic tile number) of each of them. |
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Keywords
knots, knot mosaic, mosaic number, tile number, crossing
number, space-efficient
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Mathematical Subject Classification 2010
Primary: 57M25
Secondary: 57M27
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Supplementary material |
Milestones
Received: 28 March 2018
Revised: 4 October 2018
Accepted: 27 December 2018
Published: 22 May 2019
Communicated by Kenneth S. Berenhaut
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Authors |
| Aaron Heap | |
| Department of Mathematics State University of New York at Geneseo Geneseo, NY United States |
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| Douglas Knowles | |
| Department of Mathematics Dartmouth College Hanover, NH United States |
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