Space-efficient knot mosaics for prime knots with mosaic number 6. (English) Zbl 1426.57008

Heap, Aaron; Knowles, Douglas

Involve 12, No. 5, 767-789 (2019).

Summary: In [Quantum Inf. Process. 7, No. 2–3, 85–115 (2008; Zbl 1144.81013)] S. J. Lomonaco and L. H. Kauffman 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 [J. Knot Theory Ramifications 27, No. 6, Article ID 1850041, 18 p. (2018; Zbl 1392.57005)], 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.

MSC:

57K10

Knot theory

Keywords:

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

Citations:

Zbl 1144.81013; Zbl 1392.57005

References:

[1]	10.1142/S0218216518500414 · Zbl 1392.57005
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[5]	10.1007/s11128-008-0076-7 · Zbl 1144.81013
[6]	; Rolfsen, Knots and links. Math. Lecture Series, 7 (1976) · Zbl 0339.55004