# FIGURE-8 KNOT ON THE CUBIC LATTICE

Oh, Seung-Sang

• Accepted : 2008.03.03
• Published : 2008.03.25
• 25 2

#### Abstract

We will examine the stick number of knots on the cubic lattice which is called the lattice stick number. The lattice stick numbers of knots $3_1$ and $4_1$ are known as 12 and 14, respectively. In this paper, we will show that only $3_1$ and $4_1$ have representations of irreducible non-trivial polygons, both numbers of whose sticks parallel to the y-axis and the z-axis are exactly four.

#### Keywords

knot;stick number

#### References

1. C. Adams, The Knot Book, W.H. Freedman & Co., New York, 1994.
2. Colin C. Adams, Belvin M. Brennan, Deborah L. Greilsheimer and Alexander K. Woo, Stick numbers and composition of knots and links, J. Knot Theory Ramif. 6 (1997) 149-161. https://doi.org/10.1142/S0218216597000121
3. Y. Huh and S. Oh, Lattice stick numbers of small knots, J. Knot Theory Ramif. 14 (2005) 859-867. https://doi.org/10.1142/S0218216505004160
4. S. Negami, Ramsey theorems for knots, links, and spatial graphs, Trans. Amer. Math. Soc. 324 (1991) 527- 541. https://doi.org/10.2307/2001731
5. R. Randell, An elementary invariant of knots, J. Knot Theory Ramif. 3 (1994) 279-286. https://doi.org/10.1142/S0218216594000216