## NEW KNOT TABLES## Slavik V. Jablan
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 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], 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
In Dowker notation every knot is given
by its (minimal) Dowker sequence (e.g., 4 6 8 2 describing knot 4 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 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
In the tables, knots are denoted by 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 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 3
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 This way, all the general formulas in
this paper belong to the p = 2k+1) and
subfamilies
p2, (p12p = 2k+1)
proved in [24].
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 (a) (b) u(k) = min( It is conjectured that the number For all alternating knots with The present work was restricted to a very
small part of knots: only to several families of rational knots, because
for
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2. Thistlethwaite M.
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A tabulation of oriented links.
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17. Jablan S. V.: Ordering Knots. http://members.tripod.com/vismath/sl/ 18. Farmer D. F., Stanford
T. B.: 19. Brown R.: 20. 21. Bernhard J. A.:
Unknotting numbers and their minimal knot diagrams. 22. Jablan S. V.: Unknotting
number and ¥-unknotting number of a knot.
23. Stoimenow, A.: Vassiliev Invariants and Rational Knots of Unknotting Number One. http://www.informatik.hu-berlin.de/~stoimeno/ 24. Cavicchiolli A.,
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Address:
1991 |