Slavik V. Jablan


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].


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.


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