Stick presentations of knots and their invariants

A stick knot is a knot which consists of finite line segments. This presentation of knots can be considered to be a reasonable mathematical model of cyclic molecules or molecular chains. The stick number $s(K)$ is defined to be the minimal number of sticks required to construct a knot $K$. A lattice knot is a stick knot in the cubic lattice. The lattice stick number $s_L(K)$ is defined to be the minimal number of sticks required to construct a knot $K$. The minimum lattice length $Len(K)$ is defined to be the minimum length to realize a knot $K$ as a lattice knot.
In this talk, we introduce upper bounds of $s(K)$, $s_L(K)$ and $Len(K)$ and show how to construct the stick knots. Furthermore, we introduce another upper bounds for some special knot types.