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Abstract:
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A main focus of the paper is construction of new methods for
defining diverse knot distance types - the distance of knots made by crossing
changes (Gordian distance) and the distance among knots made by crossing
smoothing (smoothing distance).
Different ways of knots presentation are introduced, with objective to a
mirror curve model. It is presented a purpose of the model, coding of knots, by
using the model preferences, as well as introduction of a method to determinate
a knots presented by the model and derived all the knots that could be placed to
a nets dimensions p×q (p ≤ 4, q ≤ 4). Diverse knot notations are described into
details, with a focus to Conway’s notation and its topological characteristics.
As it is known, a present algorithms are based on an algebra of chain fractions,
that are in close relation with a presentation of rational knots, which
results in an absence of a huge number of non-rational knots, in an existing
Gordian’s distance tables. The subject of the paper is an implementation of
methods with bases on determination of new distances equal 1. The methods
are based on a non-minimal presentation of rational and non-rational knots,
generation of algorithms established on geometrical characteristics of Conway’s
notation and a weighted graph search. The results are organized into Gordian’s
distance knots tables up to 9 crossings, and have been enclosed with the paper.
In order to append the table with knots having a bigger number of crossings, it
has been suggested a method for extension of results for knot families.
Using facts of relation among Gordian’s numbers and smoothing numbers,
a new method for smoothing number determination is presented, and results in
a form of lists for knots not having more then 11 crossings. In conjunction with
Conway’s notation concept and the method, algorithms for a smoothing distance
are generated. New results are organized in knot tables, up to 9 crossings,
combined with previous results, and enclosed with the paper.
A changes and smoothing to a knot crossing could be applied for modeling
topoisomerase and recombinase actions of DNA chains. It is presented the
method for studying changes introduced by the enzymes.
A main contribution to the paper is the concept of Conways notation, used
for all relevant results and methods, which led to introduction of a method for
derivation a new knots in Conways notation by extending C-links. In a lack of
an adequat pattern for an existing knot tables in DT-notation, there is usage of
a structure based on topological knot concepts. It is proposed a method for knot
classification based on Conways notation, tables of all knots with 13 crossings
and alternated knots with 14 crossings has been generated and enclosed.
The subject of the paper takes into consideration Bernhard-Jablan’s
hypothesis for a determination of unknotting number using minimal knot diagrams.
The determination is crucial in computation of diverse knot distances.
The paper covers one of main problems in knot theory and contains a new
method of knot minimization. The method is based on relevance of local and
global minimization.
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There are defined new terms such as a maximum and a mixed unknotting
number. The knots that do not change a minimum crossing number, after only
one crossing change are taken into consideration for the analyzes. Three classes
of the knots are recognized, and called by authors . Kauffman’s knots, Zekovic
knots and Taniyama’s knots. The most interesting conclusion correlated with
Zekovic knots is that all derived Perko’s knots (for n ≤ 13 crossings) are actually
Zekovic knots. Defining this class of knots provides opportunity to emphasize
new definitions of specifis featured for well-known Perko’s knots. |