Vol. 12, No. 5, 2019

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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

In 2008, Kauffman and Lomonaco introduced the concepts of a knot mosaic and the mosaic number of a knot or link K, the smallest integer n such that K can be represented on an n-mosaic. In 2018, the authors of this paper introduced and explored space-efficient knot mosaics and the tile number of K, the smallest number of nonblank tiles necessary to depict K 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.

knots, knot mosaic, mosaic number, tile number, crossing number, space-efficient
Mathematical Subject Classification 2010
Primary: 57M25
Secondary: 57M27
Supplementary material

Table of knots

Received: 28 March 2018
Revised: 4 October 2018
Accepted: 27 December 2018
Published: 22 May 2019

Communicated by Kenneth S. Berenhaut
Aaron Heap
Department of Mathematics
State University of New York at Geneseo
Geneseo, NY
United States
Douglas Knowles
Department of Mathematics
Dartmouth College
Hanover, NH
United States