Supercoiled Tangles and Stick Numbers of 2-bridge Links

TL;DR

This paper introduces a method for constructing polygonal 2-bridge links with fewer sticks by utilizing twisting and writhing, and establishes bounds relating stick number and crossing number, revealing limitations of minimal crossing projections.

Contribution

It presents a new construction technique for 2-bridge links and derives bounds on stick numbers, highlighting when minimal crossing projections are not minimal in stick number.

Findings

Stick number s ≤ (2/3) * crossing number c + 2n + 3

Minimal stick representatives may not have minimal crossing projections when c > 12n + 3

Efficient construction of polygonal 2-bridge links using twisting and writhing

Abstract

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 is less than or equal to 2/3 c + 2n+3 . We also show that if c > 12n+3, then minimal stick representatives do not admit minimal crossing projections.