Journal of Knot Theory and Its Ramifications

For an arbitrary 1-knot k¹, the spun 2-knot of k¹, denoted by spun(k¹), is a ribbon 2-knot in R⁴. Hence for a ribbon 2-knot K², we can also induce a notion corresponding to the crossing number on a 1-knot, and it is said to be the crossing number of K², denoted by cr(K²). In this note, we will show that the Alexander polynomial plays an important role in determining the crossing number of a ribbon 2-knot. Lastly, we will prove the following: If k¹ is a (p,q)-torus knot, then cr(spun(k¹)) is equal to (p - 1)(q - 1).

A generic immersion of a finite graph into the 2-space with p double points is said to be completely distinguishable if any two of the 2^p embeddings of the graph into the 3-space obtained from the immersion by giving over/under information to each double point are not ambient isotopic in the 3-space. We show that only non-trivializable graphs and non-planar graphs have a non-trivial completely distinguishable immersion. We give examples of non-trivial completely distinguishable immersions of several non-trivializable graphs, the complete graph on n vertices and the complete bipartite graph on m + n vertices.

A left orderable completely metrizable topological group is exhibited containing Artin's braid group on infinitely many strands. The group is the mapping class group (rel boundary) of the closed unit disk with a sequence of interior punctures converging to the boundary. This resolves an issue suggested by work of Dehornoy.

An embedded graph G in the 3-sphere S³ is called 2-irreducible if there are no separating spheres, cutting spheres, singular separating spheres, singular cutting spheres or 2-cutting spheres of G. Let D be a 2-disk in S³ that is very good for G. Let G′ be an embedded graph in S³ obtained from G by contracting D to a point. We show that if G′ is 2-irreducible then G is 2-irreducible. By this criterion certain graphs are easily shown to be 2-irreducible. As an application we show a pair of embedded graphs in the 3-sphere which is distinguished by 2-irreducibility.

Alexander group systems for virtual long knots are defined and used to show that any virtual knot is the closure of infinitely many long virtual knots. Manturov's result that there exists a pair of long virtual knots that do not commute is reproved.

We extend the notion of intersection graphs for chord diagrams in the theory of finite type knot invariants to chord diagrams for string links. We use our definition to develop weight systems for string links via the adjacency matrix of the intersection graphs, and show that these weight systems are related to the weight systems induced by the Conway and Homfly polynomials.

Hirasawa and Uchida defined the Gordian complex of knots which is a simplicial complex whose vertices consist of all knot types in S³ by using "a crossing change". In this paper, we define the C_k-Gordian complex of knots which is an extension of the Gordian complex of knots. Let k be a natural number more than 2 and we show that for any knot K₀ and any given natural number n, there exists a family of knots {K₀, K₁,…, K_n} such that for any pair (K_i, K_j) of distinct elements of the family, the C_k-distance d_Ck(K_i, K_j) = 1.

We define a linear algebraic extension of the Jones polynomial of classical knots, and prove that certain key properties of the classical Jones polynomial are properties of the extension. This shows that these properties are linear algebraic in nature, not topological. We identify a topological property of the classical Jones polynomial, that is, a property of the classical Jones polynomial that the extension does not possess. We discuss ortho-projection matrices, ortho-projection graphs, and their Jones polynomials. We classify, up to isomorphism, the connected ortho-projection graphs with at most eight vertices, and show that each such isomorphism class corresponds to a prime alternating classical knot diagram. We give an example of a connected ortho-projection graph with nine vertices that does not correspond to such a diagram.

For a hyperbolic knot in the 3-sphere, the distance between toroidal surgeries is at most 5, except the figure eight knot. In this paper, we determine all hyperbolic knots that admit two toroidal surgeries with distance 5.

For a knot or link K, let L(K) denote the rope length of K and let Cr(K) denote the crossing number of K. An important problem in geometric knot theory concerns the bound on L(K) in terms of Cr(K). It is well-known that there exist positive constants c₁, c₂ such that for any knot or link K, c₁ · (Cr(K))^3/4 ≤ L(K) ≤ c₂ · (Cr(K))^3/2. It is also known that for any real number p such that 3/4 ≤ p ≤ 1, there exists a family of knots {K_n} with the property that Cr(K_n) → ∞ (as n → ∞) such that L(K_n) = O(Cr(K_n)^p). However, it is still an open question whether there exists a family of knots {K_n} with the property that Cr(K_n) → ∞ (as n → ∞) such that L(K_n) = O(Cr(K_n)^p) for some 1 < p ≤ 3/2. In this paper, we show that there are many families of prime alternating Conway algebraic knots {K_n} with the property that Cr(K_n) → ∞ (as n → ∞) such that L(K_n) can grow no faster than linearly with respect to Cr(K_n).