相关论文: On the Coloring of Pseudoknots
A virtual doodle is an equivalence class of virtual diagrams under an equivalence relation generated by flat version of classical Reidemesiter moves and virtual Reidemsiter moves such that Reidemeister moves of type 3 are forbidden. In this…
A table of the families of alternating knots formed by conways is presented. The Conway's function is shown with the use of linear algebra in terms of natural numbers, called conways, that represent the number of crossings along a…
We define strict confluent drawing, a form of confluent drawing in which the existence of an edge is indicated by the presence of a smooth path through a system of arcs and junctions (without crossings), and in which such a path, if it…
A holonomic knot is a knot in 3-space which arises as the 2-jet extension of a smooth function on the circle. A holonomic knot associated to a generic function is naturally framed by the blackboard framing of the knot diagram associated to…
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…
Coloring numbers are one of the simplest combinatorial invariants of knots and links to describe. And with Joyce's introduction of quandles, we can understand them more algebraically. But can we extend these invariants to tangles -- knots…
The alternating knots, links and twists projected on the $S_2$ sphere were identified with the phase space of a Hamiltonian dynamic system of one degree of freedom. The saddles of the system correspond to the crossings, the edges correspond…
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…
A string figure is topologically a trivial knot lying on an imaginary plane orthogonal to the fingers with some crossings. The fingers prevent cancellation of these crossings. As a mathematical model of string figure we consider a knot…
Defect characterizes the depth of factorization of terms in differential (cyclotomic) expansions of knot polynomials, i.e. of the non-perturbative Wilson averages in the Chern-Simons theory. We prove the conjecture that the defect can be…
Bankwitz characterized an alternating diagram representing the trivial knot. A non-alternating diagram is called almost alternating if one crossing change makes the diagram alternating. We characterize an almost alternaing diagram…
To supplement the already known classification of traces on classical pseudodifferential operators, we present a classification of traces on the algebras of odd-class pseudodifferential operators of non-positive order acting on smooth…
There exists many complicated $k$-noncrossing pseudoknot RNA structures in nature based on some special conditions. The special characteristic of RNA structures gives us great challenges in researching the enumeration, prediction and the…
Knot theory is actively studied both by physicists and mathematicians as it provides a connecting centerpiece for many physical and mathematical theories. One of the challenging problems in knot theory is distinguishing mutant knots. Mutant…
We introduce and study knotoids. Knotoids are represented by diagrams in a surface which differ from the usual knot diagrams in that the underlying curve is a segment rather than a circle. Knotoid diagrams are considered up to Reidemeister…
A knot is an an embedding of a circle into three-dimensional space. We say that a knot is unknotted if there is an ambient isotopy of the embedding to a standard circle. By representing knots via planar diagrams, we discuss the problem of…
A graph is pseudo-outerplanar if each of its blocks has an embedding in the plane so that the vertices lie on a fixed circle and the edges lie inside the disk of this circle with each of them crossing at most one another. It is proved that…
We study implications of unitarity for pseudo-orbit expansions of the spectral determinants of quantum maps and quantum graphs. In particular, we advocate to group pseudo-orbits into sub-determinants. We show explicitly that the…
This paper is a very brief introduction to knot theory. It describes knot coloring by quandles, the fundamental group of a knot complement, and handle-decompositions of knot complements.
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…