This document (this page with its associated tables) is intended as a resource for anyone interested in knots and links. Every knot up to 11 crossings and every link up to 10 crossings is listed here. The main feature of the listings is that every knot and link is shown with both its numeric notation and a braid notation. The numeric notation allows cross-reference with other works, while the braid notation is by far the easiest form to manipulate by computer.
An n-crossing knot or link has a braid notation on at most 5/9.(n+2) strings [A].
The tables are divided into sections to keep them to a sensible size. The largest is about 9K.
As far as possible, the braid notation chosen uses the fewest possible number of strings, using the techniques in [A]. For all the alternating knots and links, the minimum possible has been achieved.
The tables consist of 4 columns, read as follows.
The non-alternating 11-crossing knots have no standard numeric notation. They are first enumerated in [C], but they are not given indices there. I have therefore taken the liberty of using the order in that paper to define the indices as C1 to C182.
For links there is no standard numeric notation. In this document, 2-component links are given as Ln(x) and k-component links (for k > 2) as L(k)n(x), where n is the crossing number and x is an index. The index x comes from their order in [C].
There are some missing indices in the 2-component 10-crossing links. This is because there are duplicates in [A], and, rather than renumbering for this document, I have decided to retain the same numbering as [A] and omit the duplicates. The missing links are L10(146), L10(158) and L10(159).
There are three special cases to watch out for: infinity, [10] and [11]. They should be self-explanatory (the brackets mean that 10 is one number, not two), but they might trip up an automatic script.
For k-component links, the final column contains 2k-1 entries, one for each possible set of orientations of the components of the link, except that reversal of the orientation of all the components together does not need to be shown, so only one representative of each of these pairs appears. This is necessary because the Conway notation is less sensitive to the orientation than the braid notation.
Since this document will be compared with [C], it is worth pointing out some errors in [C] which came to light while preparing these tables.
This section is not intended as an introduction to knot theory; it just defines the terms used in this document. A good mathematical introduction is [B]. A more readable book is [R].
A knot is a 1-component link. In these definitions, the term link includes knots. In the less formal sections above, I have chosen to use the term "knot or link" to avoid confusion.
A link is an n-crossing link when it has a diagram with n crossings, and no diagram with fewer than n crossings.
A link is alternating when it has a diagram in which the crossings alternate over and under along every component. A link is alternating if and only if its Conway notation does not contain a - sign.
One might worry about the possibility of an alterating n-crossing link, all of whose alternating diagrams have at least m crossings, where n < m. In fact this cannot happen, a fact easily proved using the Kauffman polynomial [K].
The braid group on m strings, Bm, has generators g1 ... gm-1, where gi represents taking string i over string i+1 to swap them over. Multiplication of words corresponds to stacking the second word above the first. In B4, for example, g2-1 g1 g3 looks like:
The relations are that gi commutes with gj whenever |i-j| > 1, and gi gi+1 gi = gi+1 gi gi+1.
A braid is an element of a braid group. A braid can be closed to form a link by joining the m strings top to bottom without twisting. The closure of a braid in B3, for example, looks like:
A braid representation of a link is a braid whose closure is the link. The fewest number of strings required to represent the link by a braid is called the braid index of the link.
If you find this document useful, or if you have any suggestions, additions, complaints or corrections, please send me an e-mail.
The tables come from [A], and that work was supported by the Science, Engineering and Research Council of Great Britain.
Hugh Morton of Liverpool University, England, has helped by doing braid index calculations for the 31 knots and links added last to the table.
[A] D.A. Chalcraft, Low-Dimensional Topology, PhD Thesis at Cambridge University, England (25 September 1991)
[B] G. Burde and H. Zieschang, Knots, De Gruyter studies in mathematics 63 (Springer-Verlag, 1979)
[C] J.H. Conway, An enumeration of knots and links, and some of their algebraic properties, Comp. Probs. in Abstract Algebra 329-358 (Pergamon Press, 1970)
[F] P. Freyd, D. Yetter, J. Hoste, W.B.R. Lickorish, K.C. Millett and A. Ocneanu, A new polynomial invariant of knots and links, Bull. Amer. Math. Soc. 12 239-246 (1985)
[K] L.H. Kauffman, State models and the Jones polynomial, Topology 26 395-407 (1987)
[R] D. Rolfsen, Knots and Links, Mathematics Lecture Series 7 (Publish or Perish, Inc., 1976)
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