Narrow Results

In this paper we discuss the relation between the combinatorial properties of cell decompositions of 3-spheres and the bridge index of knots contained in their 1-skeletons. The main result is to solve the conjecture of Ehrenborg and Hachimori which states that for a knot K in the 1-skeleton of a constructible 3-sphere satisfies e(K)≥2b(K), where e(K) is the number of edges K consists of, and b(K) is the bridge index of K. The key tool is a sharp inequality of the bridge index of tangles in relation with "tangle sum" operation, which improves the primitive rough inequality used by Eherenborg and Hachimori. We also present an application of our new tangle sum inequality to improve Armentrout's result on the relation between shellability of cell decompositions of 3-spheres and the bridge index of knots in a general position to their 2-skeletons.

Let L = K₁ ∪ K₂ be a 2-component link in S³ such that K₁ is a trivial knot. In this paper, we introduce for each n(∈ ℕ) a new bridge index b_K1=n([L]) of L called a constrained bridge index with respect to n-bridge K₁. Roughly speaking, b_K1=n([L]) is the minimum of the bridge numbers of the links ambient isotopic to L under the constraint that all of the bridge numbers of the components corresponding to K₁ are n. We give exact values b_K1=n([L]) (n = 1, 2, …) for particular constructed as follows: There exist unknotted solid tori V₁ ⊂ V₂ ⊂ ⋯ ⊂ V_m in S³, where the core of each V_j(j = 1, 2, …, m - 1) is a (1, α_j)-torus knot (α_j ∈ ℤ) in V_j+1, such that K₁ is the core of V₁, and is the core of the exterior of V_m.

We prove that the knots $13n_{592}$ and $15n_{41,127}$ both have stick number 10. These are the first non-torus prime knots with more than 9 crossings for which the exact stick number is known.

We improve the upper bound on superbridge index $\mathrm{\text{sb}}[K]$ in terms of bridge index $b[K]$ from $\mathrm{\text{sb}}[K]\le 5b[K]-3$ to $\mathrm{\text{sb}}[K]\le 3b[K]-1$.

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 $n$.

For a knot, the ascending number is the minimum number of crossing changes which are needed to obtain an descending diagram. We study relations between the ascending number and geometrical invariants; the crossing number, the genus and the bridge index. The main aim of this paper is to show that there exists a knot K such that $a(K)=2$ and $g(K)=n$, and that there exists a knot K’ such that $a(K^{\prime })\ge n$ and $g(K^{\prime })=1$ for any positive integer n. We also give an upper bound of the ascending number for a 2-bridge knot.

Some composite knots are known to be trivialized by twisting. However, the bridge index of the prime factors in known examples is two. In this note, for any integer n ≥ 1 we will construct composite knots which can be trivialized by twisting and which consist of two prime factors with bridge index greater than n.

We prove that genus one, three-bridge knots are pretzel knots.