Narrow Results

In this paper, we study the parabolic representations of 2-bridge links by finiding arc coloring vectors on the Conway diagram. The method we use is to convert the system of conjugation quandle equations to that of symplectic quandle equations. In this approach, we have an integer coefficient monic polynomial $P_K(u)$ for each 2-bridge link $K$, and each zero of this polynomial gives a set of arc coloring vectors on the diagram of $K$ satisfying the system of symplectic quandle equations, which gives an explicit formula for a parabolic representation of $K$. We then explain how these arc coloring vectors give us the closed form formulas of the complex volume and the cusp shape of the representation. As other applications of this method, we show some interesting arithmetic properties of the Riley polynomial and of the trace field, and also describe a necessary and sufficient condition for the existence of epimorphisms between 2-bridge link groups in terms of divisibility of the corresponding Riley polynomials.

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.

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 . We also show that if c > 12n + 3, then minimal stick representatives do not admit minimal crossing projections.

Kuperberg introduced web spaces for some Lie algebras which are generalizations of the Kauffman bracket skein module on a disk. We derive some formulas for $A_1$ and $A_2$ clasped web spaces by graphical calculus using skein theory. These formulas are colored version of skein relations, twist formulas and bubble skein expansion formulas. We calculate the ${𝔰𝔩}_2$ and ${𝔰𝔩}_3$ colored Jones polynomials of $2$-bridge knots and links explicitly using twist formulas.

In the class of links with one unknotting tunnel the 2-bridge links are characterized by the property of consisting of two trivial components. The tunnels of 2-bridge links are classified.

We show that minimal genus Seifert surfaces for 2-bridge links can be isotoped to braided Seifert surfaces on minimal string braids. In particular, we can obtain a minimal string braid for a given 2-bridge link, where the word length of the braid thus obtained is shortest, among all braid presentations, in terms of the band generators of braids.