Supercoiled tangles and stick numbers of 2-Bridge links

Utilizing both twisting and writhing, we construct integral tangles with few sticks, leading to an efficient method for constructing polygonal 2-bridge links. Let L be a two-bridge link with crossing number c, stick number s, and n tangles. It is shown that $s \le \frac{2}{3}c + 2n + 3$. We also show that if c > 12n + 3, then minimal stick representatives do not admit minimal crossing projections.