Narrow Results

We prove that the character variety of a family of one-relator groups has only one defining polynomial and we provide the means to compute it. Consequently, we give a basis for the Kauffman bracket skein module of the exterior of the rational link L_p/q of two components modulo the (A + 1)-torsion.

For each odd prime p, and for each non-split link admitting non-trivial p-colorings, we prove that the maximum number of colors, mod p, is p. We also prove that we can assemble a non-trivial p-coloring with any number of colors, from the minimum to the maximum number of colors. Furthermore, for any rational link, we prove that there exists a non-trivial coloring of any Schubert Normal Form (SNF) of it, modulo its determinant, which uses all colors available. If this determinant is an odd prime, then any non-trivial coloring of this SNF, modulo the determinant, uses all available colors. We prove also that the number of crossings in the SNF equals twice the determinant of the link minus 2. Facts about torus links and their coloring abilities are also proved.

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 relate some terms on the boundary of the Newton polygon of the Alexander polynomial $\mathrm{\Delta }(x,y)$ of a rational link to the number and length of monochromatic twist sites in a particular diagram that we call the standard form. Normalize $\mathrm{\Delta }(x,y)$ to be a true polynomial (as opposed to a Laurent polynomial), in such a way that terms of even total degree have positive coefficients and terms of odd total degree have negative coefficients. If the rational link has a reduced alternating diagram with no self-crossings, then $\mathrm{\Delta }(-1,0)=1$. If the standard form of the rational link has $M$ monochromatic twist sites, and the $j$th monochromatic twist site has $m_j$ crossings, then $\mathrm{\Delta }(-1,0)={\prod }_{j=1}^M(m_j+1)$. Our proof employs Kauffman’s clock moves and a lattice for the terms of $\mathrm{\Delta }(x,y)$ in which the $y$-power cannot decrease.

Let ${{𝒰}}_n$ be the set of un-oriented and rational links with crossing number $n$, a precise formula for $\left|{{𝒰}}_n\right|$ was obtained by Ernst and Sumners in 1987. In this paper, we study the enumeration problem of oriented rational links. Let ${\mathrm{\Lambda }}_n$ be the set of oriented rational links with crossing number $n$ and let ${\mathrm{\Lambda }}_n(d)$ be the set of oriented rational links with crossing number $n$ ($n\ge 2$) and deficiency $d$. In this paper, we derive precise formulas for $\left|{\mathrm{\Lambda }}_n\right|$ and $|{\mathrm{\Lambda }}_n(d)|$ for any given $n$ and $d$ and show that

In this paper, we enumerate the number of oriented rational knots and the number of oriented rational links with any given crossing number and minimum genus. This allows us to obtain a precise formula for the average minimal genus of oriented rational knots and links with any given crossing number.