Narrow Results

In this paper we give a simple proof of the equivalence between the rational link associated to the continued fraction [a₁, a₂,…,a_m], a_i ∈ ℕ, and the 2-bridge link of type p/q, where p/q is the rational number given by [a₁, a₂,…,a_m]. The known proof of this equivalence relies on the 2-fold cover of a link and the classification of the lens spaces. Our proof is elementary and combinatorial and follows the naïve approach of finding a set of movements to transform the rational link given by [a₁, a₂,…,a_m] into the 2-bridge link of type p/q.

We prove that for any odd $n\ge 3$, the $n$-palette number of any effectively $n$-colorable $2$-bridge knot is equal to $2+\lfloor {log}_{2}n\rfloor$. Namely, there is an effectively $n$-colored diagram of the $2$-bridge knot such that the number of distinct colors that appeared in the diagram is exactly equal to $2+\lfloor {log}_{2}n\rfloor$.

We introduce the Schubert form for a $3$-bridge link diagram, as a generalization of the Schubert normal form of a $2$-bridge link. It consists of a set of six positive integers, written as $(p/n,q/m,s/l)$, with some conditions and it is based on the concept of $3$-butterfly. Using the Schubert normal form of a $3$-bridge link diagram, we give two presentations of the 3-bridge link group. These presentations are given by concrete formulas that depend on the integers $\{p,n,q,m,s,l\}$. The construction is a generalization of the form the link group presentation of the $2$-bridge link $p/q$ depends on the integers $p$ and $q$.