Space-efficient knot mosaics for prime knots with mosaic number 6

Heap, Aaron; Knowles, Douglas

doi:10.2140/involve.2019.12.767

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1944-4184 (online)

ISSN 1944-4176 (print)

Author index

To appear

Other MSP journals

Space-efficient knot mosaics for prime knots with mosaic number 6

Aaron Heap and Douglas Knowles

Vol. 12 (2019), No. 5, 767–789

DOI: 10.2140/involve.2019.12.767

Abstract

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.

Keywords

knots, knot mosaic, mosaic number, tile number, crossing number, space-efficient

Mathematical Subject Classification 2010

Primary: 57M25

Secondary: 57M27

Supplementary material

Table of knots

Milestones

Received: 28 March 2018

Revised: 4 October 2018

Accepted: 27 December 2018

Published: 22 May 2019

Communicated by Kenneth S. Berenhaut

Authors

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

Vol. 12, No. 5, 2019