Related papers: Immersed flat ribbon knots
In this paper we give a lower bound for the least distortion embedding of a distance regular graph into Euclidean space. We use the lower bound for finding the least distortion for Hamming graphs, Johnson graphs, and all strongly regular…
Manturov recently introduced the idea of a free knot, i.e. an equivalence class of virtual knots where equivalence is generated by crossing change and virtualization moves. He showed that if a free knot diagram is associated to a graph that…
This paper gives mathematical models for flat knotted ribbons, and makes specific conjectures for the least length of ribbon (for a given width) needed to tie the trefoil knot and the figure eight knot. The first conjecture states that (for…
Twisted torus knots and links are given by twisting adjacent strands of a torus link. They are geometrically simple and contain many examples of the smallest volume hyperbolic knots. Many are also Lorenz links. We study the geometry of…
We classify graphs that are 0, 1, or 2 edges short of being complete partite graphs with respect to intrinsic linking and intrinsic knotting. In addition, we classify intrinsic knotting of graphs on 8 vertices. For graphs in these families,…
In this paper we present a systematic method to generate prime knot and prime link minimal triple-point projections, and then classify all classical prime knots and prime links with triple-crossing number at most four. We also extend the…
In "Width complexes for knots and 3-manifolds," Jennifer Schultens defines the width complex for a knot in order to understand the different positions a knot can occupy in the 3-sphere and the isotopies between these positions. She poses…
A {\it stuck knot} is a knot diagram containing designated crossings, called {\it stuck crossings}, whose incident strands are required to remain locally non-separable. These rigidity constraints restrict the allowable ambient isotopies and…
This paper focuses on the graphs in the Petersen family, the set of minor minimal intrinsically linked graphs. We prove there is a relationship between algebraic linking of an embedding and knotting in an embedding. We also present a more…
One of the important features of an interconnection network is its ability to efficiently simulate programs or parallel algorithms written for other architectures. Such a simulation problem can be mathematically formulated as a graph…
It is shown that given any link-manifold, there is an algorithm to decide if the manifold contains an embedded, essential planar surface; if it does, the algorithm will construct one. If a slope on the boundary of the link-manifold is…
Every link is shown to be presentable as a boundary of an unknotted flat banded surface. A (flat) banded link is defined as a boundary of an unknotted (flat) banded surface. A link's (flat) band index is defined as the minimum number of…
In this note, we attempt to find counterexamples to the conjecture that the ideal form of a knot, that which minimizes its contour length while respecting a no-overlap constraint, also minimizes the volume of the knot, as determined by its…
A ribbon is, intuitively, a smooth mapping of an annulus $S^1 \times I$ in 3-space having constant width $\varepsilon$. This can be formalized as a triple $(x,\varepsilon, \mathbf{u})$ where $x$ is smooth curve in 3-space and $\mathbf{u}$…
A knot is a circle piecewise-linearly embedded into the 3-sphere. The topology of a knot is intimately related to that of its exterior, which is the complement of an open regular neighborhood of the knot. Knots are typically encoded by…
The warping degree of an oriented knot diagram is the minimal number of crossings which we meet as an under-crossing first when we travel along the diagram from a fixed point. The warping degree of a knot projection is the minimal value of…
We classify all the maximal linklessly embeddable graphs of order 12 and show that their complements are all intrinsically knotted. We derive results about the connected domination numbers of a graph and its complement. We provide an answer…
The Bandwidth Theorem of B\"ottcher, Schacht and Taraz [Mathematische Annalen 343 (1), 175-205] gives minimum degree conditions for the containment of spanning graphs H with small bandwidth and bounded maximum degree. We generalise this…
We introduce a notion of intrinsic linking and knotting for virtual spatial graphs. Our theory gives two filtrations of the set of all graphs, allowing us to measure, in a sense, how intrinsically linked or knotted a graph is; we show that…
This is the third paper in a series devoted to enumerating the prime alternating knots and links. This paper establishes a method for enumerating the prime alternating links. It is shown that one may choose any prime alternating link…