# Table of Knot Mosaics

Mosaic number 5 or less

Mosaic number 6 or less

Mosaic number 7

Crossing number:

### 10 or less

11

12

13

14

15

16

(Click mosaic for larger view.)

When listing prime knots with crossing number 10 or less, we will use the Alexander-Briggs notation, matching Rolfsen’s table of knots. [Rolfsen]

• m = mosaic number
• t = tile number
• tm = minimal mosaic tile number*
: m = 5, t = 17 : m = 6, t = 22
: m = 6, t = 24
: m = 6, t = 27
: m = 6, tm = 32*
: m = 7, t = 27
: m = 7, t = 29
: m = 7, t = 31
* Note: Every prime knot that requires 32 non-blank tiles to fit on a 6-mosaic (i.e. tm = 32) has tile number less than 32, and this tile number can only be achieved on a 7-mosaic.

*These knots are listed as 10162‑10166 in Rolfsen due to the Perko Pair.

## References:

Heap, A.; Knowles, D. Tile Number and Space-Efficient Knot Mosaics; J. Knot Theory Ramif. 2018, 27.
Heap, A.; Knowles, D. Space-Efficient Knot Mosaics for Prime Knots with Mosaic Number 6; Involve 2019, 12.
Heap, A.; LaCourt, N. Space-Efficient Prime Knot 7-Mosaics; Symmetry 2020, 12.
Heap, A.; Baldwin, D.; Canning, J.; Vinal, G. Knot Mosaics For Prime Knots with Crossing Number 10 or Less; in preparation.
Kuriya, T.; Shehab, O. The Lomonaco–Kauffman Conjecture; J. Knot Theory Ramif. 2014, 23.
Lee, H.; Ludwig, L.; Paat, J.; Peiffer, A. Knot Mosaic Tabulation; Involve 2018, 11.
Lomonaco, S.J.; Kauffman, L.H. Quantum Knots and Mosaics; Quantum Inf. Process. 2008, 7, 85–115.
Ludwig, L.; Evans, E. An Infinite Family of Knots Whose Mosaic Number Is Realized in Non-reduce Projections; J. Knot Theory Ramif. 2013, 22.
Rolfsen, D. Knots and Links; Publish or Perish Press: Berkeley, CA, USA, 1976.