Circuit presentation and lattice stick number with exactly 4 $z$-sticks

Abstract

The lattice stick number $s_L(L)$ of a link $L$ is defined to be the minimal number of straight line segments required to construct a stick presentation of $L$ in the cubic lattice. Hong, No and Oh found a general upper bound $s_L(K) \leq 3 c(K) +2$. A rational link can be represented by a lattice presentation with exactly 4 $z$-sticks. An $n$-circuit is the disjoint union of $n$ arcs in the lattice plane $\mathbb{Z}^2$. An $n$-circuit presentation is an embedding obtained from the $n$-circuit by connecting each $n$ pair of vertices with one line segment above the circuit. By using a 2-circuit presentation, we can easily find the lattice presentation with exactly 4 $z$-sticks. In this paper, we show that an upper bound for the lattice stick number of rational $\dfrac{p}{q}$-links realized with exactly 4 $z$-sticks is $2p+6$. Furthermore it is $2p+5$ if $L$ is a 2-component link.