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NEW KNOT TABLES
NEW KNOT TABLES
Slavik V. Jablan
jablans@mi.sanu.ac.yu
Abstract:
New knot tables based on the notion of knot families are given.
Using the methods of experimental mathematics, particular results obtained
for knots with
n£ 19 crossings
belonging to the families
p, pq,
p1q,
p11q,
p111q,
pqr,
pq1r
are extended, extrapolated and generalized to whole families. As the result,
general formulas for Alexander polynomials, signatures, unknotting numbers,
and data about symmetry properties of all knots belonging to the families
mentioned, are derived and estimated.
1. Introduction
A possibility to study knots from the
mathematical point of view was for the first time proposed by C.F. Gauss.
Gauss formulated the "crossing problem", by assigning letters to the crossing
points of a self-intersecting curve and trying to determine "words" defining
a closed curve. J.B. Listing represented knots by their projections (diagrams)
and made an attempt to derive and classify all projections having fewer
than seven crossings using so-called Complexions-Symbols. Almost complete
derivation of alternating knots having fewer than 11 crossings and non-alternating
knots with n £ 10 crossings was
given by P.G. Tait, T.P. Kirkman, and C.N. Little till the end of 19th
century [1,2]. Kirkman's geometrical
system for the systematic derivation of knot projections, closely connected
with the enumeration of polyhedra, represented at the same time the geometrical
method for the classification of knot projections [3].
In the 30-ties, after the appearance of
the first modern polynomial knot invariant, discovered by J.W. Alexander,
the knot theory was established as the part of topology, completely loosing
connection with its roots - geometry. In K. Redmeister's book "Knotentheorie"
(1932), each knot is represented by one projection, (randomly?) chosen
from several possible ones. After Redmeister [4], all
knot tables that can be found in knot theory books are simple copies of
the first: sometimes, some projection is slightly changed, or turned upside
down, and that's all. In order to compare them, the reader may consider
knot tables from the books [5,6,7,8,9,10].
All knot tables are followed by the corresponding
polynomial knot invariants: Alexander polynomials, Jones polynomials [8],
Laurent polynomials [7], and data about some other knot
invariants and properties - hyperbolic volumes [8], signatures
[6,9], unknotting numbers [9],
chirality and invertibility [6,9],
symmetry groups of knots [9], etc.). Usually,
knots are denoted in knot tables by their ordering numbers as 31,
41, 51, 52, 61, 62,
63, 71 - 77, 81 - 821,
91 - 949, 101 - 10166, without
any geometrical or topological "vertical" ordering principle connecting
knots with n and n+1 crossings. This classical notation, giving
no information about any knot or link (except its place in knot tables),
is preserved till now in the most of knot theory books. The most of knots
are alternating, and non-alternating knots will appear from n ³
8: 818 - 821, 942 - 949, 10124
- 10166, etc. The most complete printed knot tables contain
knots with n £ 10 crossings. The
only tables containing links are given by D. Rolfsen [5].
Today, with the development of computers,
the notation and enumeration of knots and links is very similar with the
situation occurring in different unordered structures: prime numbers, polyominoes
etc., giving no chance for any classification. This development
made possible to construct all possible permutations of n even numbers,
check their realizability as knot projections, and find the minimal Dowker
sequence for every knot [11].
In Dowker notation every knot is given
by its (minimal) Dowker sequence (e.g., 4 6 8 2 describing knot 41)
and the signs of crossings (necessary only in the case of non-alternating
knots), from which is possible to reconstruct the knot. Because Dowker
code is dependent from a minimal projection and from the choice of beginning
point, the mapping between knots and their Dowker sequences is one-to-many,
so it is necessary to find a minimal Dowker sequence for each knot. Hence,
Dowker codes are just minimal permutations representing certain knot projections,
without carrying any other geometrical or topological information about
knots, so they are absolutely non-useful in any attempt of knot classification.
Using computer enumeration and Dowker
algorithm, M.B. Thistlethwaite (by the program "Knotscape" [12])
and H. Doll & J. Hoste [13], obtained the tables
of knots with n £ 16 crossings
and non-isomorphic minimal link projections with n £
9 crossings. Similar program able to recognize all knot projections with
n£
10 crossings was developed by the author and V. Velickovic in 1995.
Continuing the "geometrical" line (Kirkman-Conway-Caudron)
[3,14,15] and the
classification of knots and links proposed in [16,17], in this paper we
will introduce new knot tables, based on the notion of knot families. Till
now, such new tables are completed only for prime knots with n £
8 crossings.
2. Notation
In the tables, knots are denoted by Conway
notation [5]. This unique symbolical notation for
knots and links was introduced by J.Conway in 1967 [14].
From Conway symbols it is possible to read directly many of important knot
or link properties: their symmetry, to recognize the world [15]
to which they belong, to prove the equality of rational knots or links
using very simple calculation, and even to derive some general conclusions:
for example, all rational knots with symmetrical Conway symbol are amphicheiral
[15].
A prime knot or link with singular digons,
expressed by a Conway symbol, is called generating, and a knot or link
without digons is called a basic polyhedron [14,16,17].
Any other knot or link can be derived from some generating knot or link,
by replacing singular digons by chains of digons. All knots and links that
can be derived from a generating knot or link by such replacement make
a family [16,17]. All knots
and links are distributed into disjoint sets, called by A.Caudron worlds
[15].
It is interesting that the term "family"
is very rarely mentioned and used in knot theory: its description can be
found only in [18], where a family of knots is introduced
as an ïnformal term used to describe a list of knots where each successive
knot is obtained from the previous one by a simple process. The twist knots
are an example, as are the knots 31, 51, 71,...".
The other recent use of knot and link families can be found in the CD-R
"Raising Public Awareness of Mathematics" by R. Brown [19].
3. New knot tables
In the new knot tables based on knot families,
every family is given by its general Conway symbol and existential conditions
(i.e. conditions necessary that a given Conway symbol represents a knot,
and not a link). In each "Notation" subsection it is given a comparative
classical notation of knots with n £
10 crossings belonging to certain family and their corresponding Conway
symbols. For each knot with n £
19 crossings, its Dowker sequence is given. After the list of particular
Alexander polynomials for the knots with n £
19 crossings, it is given a general formula for the Alexander polynomial
of a family considered, and the list of particular Jones polynomials for
knots with
n£ 19 crossings. For
every family they are determined in the general form the symmetry group,
symmetry type, signatures, and unknotting numbers. All that data are calculated
for the knots with
n £ 19 crossings
first by using program "Knotplot" [20] and its tangle
calculator for calculating Dowker sequences. After that, Alexander polynomials
given in the form proposed by Rolfsen [5], Jones polynomials,
symmetry groups, symmetry types and signatures are calculated by putting
Dowker sequences mentioned in the program "Knotscape" [12].
Unknotting numbers are calculated for the knots with n £
19 crossings mentioned according to Bernhard-Jablan Conjecture [17,22,23]
by using a program developed by the author and V. Velickovic. Finally,
particular results obtained for the knots with n £
19 crossings are extrapolated to whole families in order to derive general
formulas for the Alexander polynomials, symmetry groups, symmetry types,
signatures, and unknotting numbers.
This way, all the general formulas in
this paper belong to the experimental mathematics: they represent
the results that are estimated, extrapolated and conjectured,
and need to be proved (or disproved!). The general Alexander polynomials
derived that way coincide with the general Alexander polynomials for the
family
p (p = 2k+1) and
subfamilies
p2, p12 (p = 2k+1)
proved in [24].
Tables:
p, pq,
p1q,
p11q,
p111q,
pqr,
pq1r
4. Conclusions
The concept of new knot tables based on
knot families can be naturally extended to links,
in the spirit of [15,16,17].
For that, it is necessary to develop programs able to work with links and
calculate polynomial invariants, and other data already calculated for
knots.
Because the complete concept of new knot
tables is based on the notion of generating knots and links and families
originating from them, one of the possible future aims can be a search
for new knot and link invariants that will be the invariants of families.
If we will be able for a given knot to recognize a family to which it belongs,
even Alexander polynomial maybe can be sufficient for the recognition of
particular knots.
From the results obtained, it looks that
all properties of knots or links belonging to some family are well-ordered,
so it is possible to extend them to some general form. It works for Alexander
polynomials, Jones polynomials, symmetry properties, unknotting numbers,
and even for Dowker sequences.
Next interesting question is a possibility
to try to establish connections between coefficients of polynomial invariants
and other knot or link invariants and understand the topological meaning
of certain coefficients.
One of main open questions is Bernhard-Jablan
Conjecture on unknotting number. According to that Conjecture,
a number u(k) is defined in the following way:
(a) u(1) = 0, where 1 is the unknot;
(b) u(k) = min(u(k¢)+1),
where the minimum is taken over all the knots k¢
obtained from a minimal projection of k by a crossing change.
It is conjectured that the number u(k)
will be the unknotting number of the knot k.
For all alternating knots with n
£ 10 crossings, the obtained numbers u(k)
coincide with the unknotting numbers of those knots [21],
calculated (or even estimated by giving two or more expected values for
the unknotting number [9]) by other methods. Because
the unknotting number of every knot is greater or equal to the half of
signature s(k), for a lot of knots with
n £ 19 crossings used in the experimental
work we namely proved that the numbers
u(k) obtained from
their minimal projections by recursive unknotting process are really the
unknotting numbers of those knots, because they are equal to [(s(k))/
2]. This property holds as well for the general formulas for s(k)
and u(k), extrapolated for several families of knots, where
u(k)
= [(s(k))/ 2], so we strongly believe
that at least for such families we succeeded to find the correct unknotting
number, even in its general form. Certainly, the results obtained in experimenting
with all knots of the families discussed with n £
19 crossings are out of the question, so for those knots with u(k)
= [(s(k))/ 2] we are sure that the unknotting
numbers are determined correctly.
The present work was restricted to a very
small part of knots: only to several families of rational knots, because
for n£ 7 there are no other generating
knots except rational ones. Thanks to that, out of consideration remained
non-alternating knots, so it is possible that some of the conjectures or
estimations need to be restricted to alternating knots or even only to
rational knots. For example, till now it is proved that Bernhard-Jablan
Conjecture holds for rational knots with the unknotting number one
[23], so all the general formulas presented in this
work expect very serious proving (or disproving) procedure and verification.
References:
1. Turner J. C., Van
De Griend P. (Eds.):
History and Science of Knots. World Scientific,
Singapoore, New Jersey, London, Hong Kong, 1995.
2. Thistlethwaite M.
B.: Knots tabulations and related topics. In Aspects of Topology,
Eds. I.M.James and E.H.Kronheimer, Cambridge University Press, Cambridge,
1985, 1-76.
3. Kirkman T. P.: The
enumeration, description and construction of knots of fewer than ten crossings.
Trans. Roy. Soc. Edinburgh 32 (1885), 281-309.
4. Reidemeister K.: Knotentheorie.
Springer Verlag, Berlin, 1932.
5. Rolfsen D.: Knots
and Links. Publish & Perish Inc., Berkeley, 1976.
6. Burde G., Zieschang
H.: Knots. Walter de Greyter, Berlin, New York 1985.
7. Kauffman L. A.: On
Knots. Princeton University Press, Princeton, 1987.
8. Adams C. C.: The
Knot Book. Freeman, New York, 1994.
9. A Survey of Knot
Theory. (Ed. A. Kawauchi), Birkhäuser, Basel, Boston, Berlin,
1996.
10. Murasugi K.: Knot
theory and its applications. Birkhäuser, Boston, 1996.
11. Dowker C. H.; Thistlethwaite
M. B.: Classification of knot projections. Topology Appl. 16
(1983), 19-31.
12. Knotscape.
http://www.math.utk.edu/~morwen/
13. Doll H., Hoste J.:
A tabulation of oriented links.
Mathematics of Computation 57
196 (1991), 747-761.
14. Conway J.: An enumeration
of knots and links and some of their related properties. In Computational
Problems in Abstract Algebra, Proc. Conf. Oxford 1967 (Ed. J. Leech),
Pergamon Press, New York, 1970, 329-358.
15. Caudron A.: Classification
des noeuds et des enlancements. Public. Math. d'Orsay 82. Univ. Paris
Sud, Dept. Math., Orsay, 1982.
16. Geometry of Links.
Novi Sad J. Math., 29, 3 (1999), 121-139.
17. Jablan S. V.: Ordering
Knots. http://members.tripod.com/vismath/sl/
18. Farmer D. F., Stanford
T. B.: Knots and Surfaces. Mathematical World, Vol. 6, American
Mathematical Society, Providence, Rhode Island, 1996.
19. Brown R.: Raising
Public Awareness of Mathematics. CD-ROM, Centre for the Popularization
of Mathematics, Bangor, 2001.
20. KnotPlot.
http://www.cs.ubc.ca/nest/contributions/scharein/KnotPlot.html
21. Bernhard J. A.:
Unknotting numbers and their minimal knot diagrams. Journal of Knot
Theory and Its Ramifications
3, 1 (1994), 1-5.
22. Jablan S. V.: Unknotting
number and ¥-unknotting number of a knot.
Filomat (Nis), 12:1, (1998), 113-120.
23. Stoimenow, A.: Vassiliev
Invariants and Rational Knots of Unknotting Number One. http://www.informatik.hu-berlin.de/~stoimeno/
24. Cavicchiolli A.,
Ruini B., Spaggiari F.: On a conjecture of M.J.Dunwoody (to appear).
Address:
The Mathematical Institute,
Knez Mihailova 35,
P.O.Box 367,
11001 Belgrade,
Yugoslavia
1991 Mathematics Subject Classification: 57M25
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