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

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.


57K10 Knot theory
Full Text: DOI arXiv


[1] 10.1142/S0218216518500414 · Zbl 1392.57005
[2] 10.1142/S0218216518500566 · Zbl 1402.57009
[3] 10.1142/S0218216514500035 · Zbl 1287.57012
[4] 10.2140/involve.2018.11.13 · Zbl 1372.57018
[5] 10.1007/s11128-008-0076-7 · Zbl 1144.81013
[6] ; Rolfsen, Knots and links. Math. Lecture Series, 7 (1976) · Zbl 0339.55004
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