Related papers: Colored Unlinking
The number of colorings of a knot diagram by a quandle has been shown to be a knot invariant by CJKLS using quandle cohomology methods. In a previous paper by the second named author, the CJKLS invariant was refined and, in particular, it…
Consider the following two ways to colour the vertices of a graph where the requirement that adjacent vertices get distinct colours is relaxed. A colouring has "defect" $d$ if each monochromatic component has maximum degree at most $d$. A…
In this paper we compute the sharp lower bounds for the crossing number of $n$-string $k$-loop essential tangles. For essential tangles with only string components, we characterise the ones with the minimum crossing number for a given…
This paper gives the first examples of gordian unlinks. The components of these unlinks cannot be separated while maintaining constant length and thickness. We construct infinite families of 2-component gordian unlinks and also construct…
Noncrossing linked partitions arise in the study of certain transforms in free probability theory. We explore the connection between noncrossing linked partitions and colored Motzkin paths. A (3,2)-Motzkin path can be viewed as a colored…
We consider the maximum chromatic number of hypergraphs consisting of cliques that have pairwise small intersections. Designs of the appropriate parameters produce optimal constructions, but these are known to exist only when the number of…
Charts are oriented labeled graphs in a disk. Any simple surface braid (2-dimensional braid) can be described by using a chart. Also, a chart represents an oriented closed surface (called a surface-link) embedded in 4-space. In this paper,…
In this paper we give two new criteria of detecting the checkerboard colorability of virtual links by using odd writhe and arrow polynomial of virtual links, respectively. By applying new criteria, we prove that 6 virtual knots are not…
We introduce a framework to analyze knots and links in an unmarked solid torus. We discuss invariants that detect when such links are equivalent under an ambient homeomorphism, and show that the multivariable Alexander polynomial is such in…
In this paper, we define a lassoing on a link, a local addition of a trivial knot to a link. Let K be an s-component link with the Conway polynomial non-zero. Let L be a link which is obtained from K by r-iterated lassoings. The complete…
New presentations of a link and a virtual link are introduced and algebraic systems on links and virtual links are constructed respectively. Based on the algebraic systems, Reduction Crossing Algorithms for them are proposed which are used…
A link is almost alternating if it is non-alternating and has a diagram that can be transformed into an alternating diagram via one crossing change. We give formulas for the first two and last two potential coefficients of the Jones…
We present computational results about quasi-alternating knots and links and odd homology obtained by looking at link families in the Conway notation. More precisely, we list quasi-alternating links up to 12 crossings and the first examples…
Let $S$ be a set of four points chosen independently, uniformly at random from a square. Join every pair of points of $S$ with a straight line segment. Color these edges red if they have positive slope and blue, otherwise. We show that the…
It is known that there are 21 ribbon knots with 10 crossings or fewer. We show that for every ribbon knot, there exists a tangle that satisfies two properties associated with the knot. First, under a specific closure, the closed tangle is…
Symmetries of knots have been studied extensively, and strongly invertible knots are one of them. Lamm defined the equivariant crossing number $c_t(K)$, the minimum crossing number among all symmetric diagrams for a strongly invertible knot…
We construct a graph G such that any embedding of G into R^{3} contains a nonsplit link of two components, where at least one of the components is a nontrivial knot. Further, for any m < n we produce a graph H so that every embedding of H…
In this article, we give a list of minimal grid diagrams of the 12 crossing prime alternating knots. This is a continuation of the work in https://doi.org/10.1142/S0218216520500765
We introduce a variant of the vertex-distinguishing edge coloring problem, where each edge is assigned a subset of colors. The label of a vertex is the union of the sets of colors on edges incident to it. In this paper we investigate the…
The weak splitting number $wsp(L)$ of a link $L$ is the minimal number of crossing changes needed to turn $L$ into a split union of knots. We describe conditions under which certain $\mathbb{R}$-valued link invariants give lower bounds on…