Narrow Results

Let K be a knot or link and let p = det(K). Using integral colorings of rational tangles, Kauffman and Lambropoulou showed that every rational K has a mod p coloring with distinct colors. If p is prime this holds for all mod p colorings. Harary and Kauffman conjectured that this should hold for prime, alternating knot diagrams without nugatory crossings for p prime. Asaeda, Przyticki and Sikora proved the conjecture for Montesinos knots. In this paper, we use an elementary combinatorial argument to prove the conjecture for prime alternating algebraic knots with prime determinant. We also give a procedure for coloring any prime alternating knot or link diagram and demonstrate the conjecture for non-algebraic examples.

Given a link $L$, a representation ${\pi }_1(S^3-L)\to \mathrm{\text{SL}}(2,ℂ)$ is trace-free if the image of each meridian has trace zero. We determine the conjugacy classes of trace-free representations when $L$ is a Montesinos link.

Negami found an upper bound on the stick number $s(K)$ of a nontrivial knot $K$ in terms of the minimal crossing number $c(K)$: $s(K)\le 2c(K)$. Huh and Oh found an improved upper bound: $s(K)\le \frac{3}{2}(c(K)+1)$. Huh, No and Oh proved that $s(K)\le c(K)+2$ for a $2$-bridge knot or link $K$ with at least six crossings. As a sequel to this study, we present an upper bound on the stick number of Montesinos knots and links. Let $K$ be a knot or link which admits a reduced Montesinos diagram with $c(K)$ crossings. If each rational tangle in the diagram has five or more index of the related Conway notation, then $s(K)\le c(K)+3$. Furthermore, if $K$ is alternating, then we can additionally reduce the upper bound by $2$.

In this paper, we are interested in BB knots, namely knots and links whose bridge index and braid index are equal. Supported by observations from experiments, it is conjectured that BB knots possess a special geometric/physical property (and might even be characterized by it): if the knot is realized by a (closed) springy metal wire, then the equilibrium state of the wire is in an almost planar configuration of multiple (overlapping) circles. In this paper, we provide a heuristic explanation to the conjecture and explore the plausibility of the conjecture numerically. We also identify BB knots among various knot families. For example, we are able to identify all BB knots in the family of alternating Montesinos knots, as well as some BB knots in the family of the non-alternating Montesinos knots, and more generally in the family of the Conway algebraic knots. The BB knots we identified in the knot families we considered include all of the 182 one component BB knots with crossing number up to 12. Furthermore, we show that the number of BB knots with a given crossing number $n$ grows exponentially with $n$.