Knot Notation and Braiding (original) (raw)
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The Journal of Visualization and Computer Animation, 1992
We describe a new method for modelling braids and certain classes of knots and links, and how they may be visualized. The method uses generalized cylinders built around a bicubic spline centre line. We also show how multi-stranded, recursive, hawser laid ropes can be modelled using related techniques KEY WORDS Modelling Knots Generalized cylinders
Trivial or knot: A software tool and algorithms for knot simplification
Preprint, 1996
A special type of representation for knots and for local knot manipulations is described and used in a software tool called TOK to implement a number of algorithms on knots. Two algorithms for knot simplification are described: simulated annealing applied to the knot representation, and a “divide-simplify-join” algorithm. Both of these algorithms make use of the compact knot representation and of the basic mechanism TOK provides for carrying out a predefined knot manipulation on the knot representation. The simplification ...
We introduce braids via their historical roots and uses, make connections with knot theory and present the mathematical theory of braids through the braid group. Several basic mathematical properties of braids are explored and equivalence problems under several conditions defined and partly solved. The connection with knots is spelled out in detail and translation methods are presented. Finally a number of applications of braid theory are given. The presentation is pedagogical and principally aimed at interested readers from different fields of mathematics and natural science. The discussions are as self-contained as can be expected within the space limits and require very little previous mathematical knowledge. Literature references are given throughout to the original papers and to overview sources where more can be learned.
Minimum Braids: A Complete Invariant of Knots and Links
arXiv (Cornell University), 2004
Minimum braids are a complete invariant of knots and links. This paper defines minimum braids, describes how they can be generated, presents tables for knots up to ten crossings and oriented links up to nine crossings, and uses minimum braids to study graph trees, amphicheirality, unknotting numbers, and periodic tables.
An Algorithm for Generating a Family of Alternating Knots
ISRN Algebra, 2012
An algorithm for generating a family of alternating knots which are described by means of a chain code is presented. The family of alternating knots is represented on the cubic lattice, that is, each alternating knot is composed of constant orthogonal straight-line segments and is described by means of a chain code. This chain code is represented by a numerical string of finite length over a finite alphabet, allowing the usage of formal-language techniques for alternatingknot representation. When an alternating knot is described by a chain, it is possible to obtain its mirroring image in an easy way. Also, we have a compression efficiency for representing alternating knots, because chain codes preserve information and allow a considerable data reduction.
An algorithm for computing the Seifert matrix of a link from a braid representation
Ensaios Matemáticos, 2016
A Seifert surface of a knot or link in S 3 is an oriented surface in S 3 whose boundary coincides with that of the link. A corresponding Seifert matrix has as its entries the linking numbers of a set of homology generators of the surface. Thus a Seifert matrix encodes essential information about the structure of a link and, unsurprisingly, can be used to define powerful invariants, such as the Alexander polynomial. The program SeifertView has been designed to visualise Seifert surfaces given by braid representations, but it does not give the user any technical information about the knot or link. This article describes an algorithm which could work alongside SeifertView and compute a Seifert matrix from the same braids and surfaces. It also calculates the genus of the surface, the Alexander polynomial of the knot and the signature of the knot.
Journal of Knot Theory and Its Ramifications
We study collections of planar curves that yield diagrams for all knots. In particular, we show that a very special class called potholder curves carries all knots. This has implications for realizing all knots and links as special types of meanders and braids. We also introduce and apply a method to compare the efficiency of various classes of curves that represent all knots.
Fundamenta Mathematicae, 2004
This paper gives a new method for converting virtual knots and links to virtual braids. Indeed, the braiding method given here is quite general and applies to all the categories in which braiding can be accomplished. This includes the braiding of classical, virtual, flat, welded, unrestricted, and singular knots and links. We also give reduced presentations for the virtual braid group and for the flat virtual braid group (as well as for other categories). These reduced presentations are based on the fact that these virtual braid groups for n strands are generated by a single braiding element plus the generators of the symmetric group on n letters.
Knot the Usual Suspects: Finding the Diagrammatic Representations of Physical Knots
UF Journal of Undergraduate Research
In the last few hundred years, mathematicians have been attempting to describe the topological and algebraic properties of mathematical knots. Regarding the study of knots, there exists a disconnect between examining a knot’s mathematical and physical definitions. This is due to the inherent difference in the topology of an open-ended physical knot and a closed mathematical knot. By closing the ends of a physical knot, this paper presents a method to break this discontinuity by establishing a clear relation between physical and mathematical knots. By joining the ends and applying Reidemeister moves, this paper will calculate the equivalent mathematical prime or composite knots for several commonly used physical knots. In the future, it will be possible to study the physical properties of these knots and their potential to expand the field of mathematical knot theory.
Table of Families of Alternating Knots with their Conway's Function
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 direction, as it was used by J. Conway for the classification of knots. Colored figures and tangles show the parts of the knots or tangles with a definite handedness: all the colored parts of the knot family are associated to a particular orientation. For example all the colored conways have a right hand screw thread, and all the white conways have the opposite handedness. Figures for six conways were colored with two different colors for forty two families in order to show the dissection of the knot in two tangles corresponding to a particular factorization of the Conway's function. The Conway's function of each family is expressed as the internal product of two vectors corresponding to each of two colored family 2-tangles, and with a full fac...