Related papers: Compositions of Knots Using Alexander Polynomial

Generalizing unknotting number, $n$-adjacent knots have $n$ crossings such that changing any non-empty subset of them results in the unknot. In this paper, we determine the 2-adjacent knots through 12 crossings. Using Heegaard Floer…

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 a homology for ternary groups using both associativity and skew elements. We describe the odd-even construction which yields many examples of ternary groups. We define the ternary knot group, consider its homomorphisms into…

We exhibit an encoding of knots into processes in the {\pi}-calculus such that knots are ambient isotopic if and only their encodings are weakly bisimilar.

In this article we shall give an account of certain developments in knot theory which followed upon the discovery of the Jones polynomial in 1984. The focus of our account will be recent glimmerings of understanding of the topological…

We take a close look at a classical magic trick performed with a string, where a trivial knot is seemingly isotoped into a trefoil, and generalize it to a family of magic tricks for transforming the unknot into other knots. We encode such a…

We examine computer experiments that can be performed to understand the dynamics of knots under self-repulsion. In the course of specific computer exploration we use the knot theory of rational knots and rational tangles to produce classes…

Knot diagrams are among the most common visual tools in topology. Computer programs now make it possible to draw, manipulate and render them digitally, which proves to be useful in knot theory teaching and research. Still, an openly…

By using double branched covers, we prove that there is a 1-1 correspondence between the set of knotoids in the 2-sphere, up to orientation reversion and rotation, and knots with a strong inversion, up to conjugacy. This correspondence…

We discuss the polynomial representation for long knots and elaborate on how to obtain them with a bound on degrees of the defining polynomials, for any knot-type.

Knots have a twisted history in quantum physics. They were abandoned as failed models of atoms. Only much later was the connection between knot invariants and Wilson loops in topological quantum field theory discovered. Here we show that…

Besides offering a friendly introduction to knot homologies and quantum curves, the goal of these lectures is to review some of the concrete predictions that follow from the physical interpretation of knot homologies. In particular, this…

The mock Alexander polynomial is an extension of the classical Alexander polynomial, defined and studied for (virtual) knots and knotoids by the second and third authors. In this paper we consider the mock Alexander polynomial for…

This is an expository article of our work on analogies between knot theory and algebraic number theory. We shall discuss foundational analogies between knots and primes, 3-manifolds and number rings mainly from the group-theoretic point of…

We give a combinatorial treatment of transverse homology, a new invariant of transverse knots that is an extension of knot contact homology. The theory comes in several flavors, including one that is an invariant of topological knots and…

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…

Two natural generalizations of knot theory are the study of spatial graphs and virtual knots. Our goal is to unify these two approaches into the study of virtual spatial graphs. This paper is a survey, and does not contain any new results.…

Families of alternating knots (links) and tangles are studied using as building block the conway defined as the twisting of two strands. The regular representation of knots assumes the projection has the minimal number of overpassings, and…

Computational topology is a vibrant contemporary subfield and this article integrates knot theory and mathematical visualization. Previous work on computer graphics developed a sequence of smooth knots that were shown to converge point wise…

We study the properties of glued knots, a sub-class of real rational knots, that can be constructed by gluing ellipses. We define an invariant called the gluing degree and relate it to various classical properties of knots and classify all…