Circuit presentation and lattice stick number with exactly four z-sticks

Abstract

The lattice stick number sL(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 [5] found a general upper bound sL(K) ≤ 3c(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 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 p q -links realized with exactly 4 z-sticks is 2p + 6.Furthermore it is 2p + 5 if L is a 2-component link.