相关论文: Minimum Number of Colors: the Turk's Head Knots Ca…
This article is about applications of linear algebra to knot theory. For example, for odd prime p, there is a rule (given in the article) for coloring the arcs of a knot or link diagram from the residues mod p. This is a knot invariant in…
In this paper we first investigate minimal sufficient sets of colors for p=11 and 13. For odd prime p and any p-colorable link L with non-zero determinant, we give alternative proofs of mincol_p L \geq 5 for p \geq 11 and mincol_p L \geq 6…
We show that the minimal number of colors for all effective $n$-colorings of a link with non-zero determinant is at least $1+\log_2 n$.
Every classical or virtual knot is equivalent to the unknot via a sequence of extended Reidemeister moves and the so-called forbidden moves. The minimum number of forbidden moves necessary to unknot a given knot is an invariant we call the…
The genus of knots is a one of the fundamental invariant and can be seen as a complexity of knots. In this paper, we give a lower bound of genus using Dehornoy floor, which is a measure of complexity of braids in terms of braid ordering.
The concordance genus of a knot K is the minimum three-genus among all knots concordant to K. For prime knots of 10 or fewer crossings there have been three knots for which the concordance genus was unknown. Those three cases are now…
We improve the lower bound for the minimum number of colors for linear Alexander quandle colorings of a knot given in Theorem 1.2 of Colorings beyond Fox: The other linear Alexander quandles (Linear Algebra and its Applications, Vol. 548,…
The Turaev genus and dealternating number of a link are two invariants that measure how far away a link is from alternating. We determine the Turaev genus of a torus knot with five or fewer strands either exactly or up to an error of at…
The lattice stick number of a knot type is defined to be the minimal number of straight line segments required to construct a polygon presentation of the knot type in the cubic lattice. In this paper, we mathematically prove that the…
Knots have been considered to be useful models for simulating molecular chains such as DNA and proteins. One quantity that we are interested on molecular knots is the minimum number of monomers necessary to realize a knot. In this paper we…
Simple closed curves in the plane can be mapped to nontrivial knots under the action of origami foldings that allow the paper to self-intersect. We show all tame knot types may be produced in this manner, motivating the development of a new…
This article concerns exact results on the minimum number of colors of a Fox coloring over the integers modulo r, of a link with non-null determinant. Specifically, we prove that whenever the least prime divisor of the determinant of such a…
This paper has two-fold goal: it provides gentle introduction to Knot Theory starting from 3-coloring, the concept introduced by R. Fox to allow undergraduate students to see that the trefoil knot is non-trivial, and ending with statistical…
It is known that the number of biquandle colorings of a long virtual knot diagram, with a fixed color of the initial arc, is a knot invariant. In this paper we describe a more subtle invariant: a family of biquandle endomorphisms obtained…
This paper starts a systematic description of colored knot polynomials, beginning from the first non-(anti)symmetric representation R=[2,1]. The project involves several steps: (i) parametrization of big families of knots a la…
We study knots whose $\mathrm{SL}_2(\mathbb{C})$-character varieties have a component of dimension greater than one. We call such knots $\mathcal{X}$-large and introduce two diagrammatic constructions that produce $\mathcal{X}$-large knots.…
K. Ichihara and E. Matsudo introduced the notions of $\mathbb{Z}$-colorable links and the minimal coloring number for $\mathbb{Z}$-colorable links, which is one of invariants for links. They proved that the lower bound of minimal coloring…
This survey article discusses three aspects of knot colorings. Fox colorings are assignments of labels to arcs, Dehn colorings are assignments of labels to regions, and Alexander-Briggs colorings assign labels to vertices. The labels are…
Knot colorings are one of the simplest ways to distinguish knots, dating back to Reidemeister, and popularized by Fox. In this mostly expository article, we discuss knot invariants like colorability, knot determinant and number of…
The transient number of a knot K, denoted tr(K), is the minimal number of simple arcs that have to be attached to K, in order that K can be homotoped to a trivial knot in a regular neighborhood of the union of K and the arcs. We give a…