ccerf@ulb.ac.be)
The first authors to address this problem were Doll and Hoste [2]. They introduced a nomenclature for oriented links by choosing for each of them a reference ordering of the components and a reference orientation of the components. For example, for a 2-component link L, the corresponding reference oriented link will be denoted by L++. Reversing the first component will give rise to the oriented link L-+, and so on. Doll and Hoste list all projections1 of oriented links up to nine crossings. There are, however, two practical problems when using their table. The first one is that it is published on microfiche, which makes it not so easily accessible. The second problem is that, as Doll and Hoste used a from scratch approach, the resulting link projections are not always the same as Rolfsen's, which hinders the comparisons. In some cases, a different point at infinity is chosen for the projection (e.g., 421). In some other cases, a projection of the mirror image link is chosen (e.g., 621). Actually, as Doll and Hoste were not addressing the question of chirality, they didn't care if two projections of the same oriented link were related not only by a flype but also by a mirror reflection.
The atlas presented here constitutes a new reference table for oriented knots and links, reconciling Rolfsen's and Doll/Hoste's points of view, while adding some data that will be described below. The first part of the atlas (Figure 1) transposes Doll/Hoste's nomenclature on Rolfsen's projections. That is, for alternating knots up to ten crossings, we draw Rolfsen's projection of the unoriented knot and add an arrow that will define a positive orientation of the knot. For alternating links up to nine crossings, we draw Rolfsen's projection of the unoriented link, then number and orient the components according to Doll/Hoste's choices in order to ensure that we get to the same nomenclature as theirs. Moreover, for each knot or link, we make sure that we use the link type as in Rolfsen's table and not the mirror image.
The second part of this atlas, the main one, consists in a big table (Table 1) where each entry is a different oriented link. From now on, by link we mean knot or link since knots are just links with one component. Corresponding to each unoriented link, the number of entries will be the number of nonequivalent possible orientations. For instance, the link 631 exists as two nonequivalent oriented links:
631+++ = 631-, and
631++- = 631-+ = 631+-+ = 631-+- = 631-++ = 631+-.To determine this, we used the program of Henry and Weeks, computing the symmetry group of hyperbolic links via a canonical decomposition of the link complement [4] (http://thames.northnet.org/weeks/index/SnapPea.html). This program was not yet available at the time Doll and Hoste made their table, thus the numbers of nonequivalent orientations corresponding to each unoriented link found by this method provide an independent check on the numbers indicated in Doll and Hoste's table. We are happy to report that those numbers are all consistent.
For each entry of Table 1, columns 2, 3, 4 list well-known numerical invariants of alternating oriented links, namely the writhe w, twice the linking number 2l and the self-writhe s, with w = 2l + s. Let us remind the following definitions:
Definition 1 The writhe w of an oriented link is the sum of the signs of the crossings of a minimal projection of the link.
Definition 2 The linking number l of an oriented link is half the sum of the signs of the intercomponent crossings of a minimal projection of the link.
Definition 3 The self-writhe s of an oriented link is the sum of the signs of the intracomponent crossings of a minimal projection of the link.
The sign convention commonly used is the following:
Column 5 of Table 1 lists the symmetry properties of the oriented links, with the following notations: a for achiral, c for chiral, i for invertible, n for noninvertible, -A for (-)amphicheiral. Proper definitions of these terms are given below:
Definition 4 An oriented link L is achiral if it is equivalent to its mirror image, denoted by L*. Otherwise, it is chiral.
For instance, 9234++ not = 9234++* hence 9234++ is chiral. The word chiral comes from the Greek ceir (\chi\epsilon\iota\rho) meaning hand. Indeed, a left hand is nonequivalent to its mirror image right hand.
Definition 5 An oriented link L is invertible if it is equivalent to the same link with all components reverted, denoted by -L. Otherwise it is noninvertible.
For instance, -9234++ means 9234-. We have 9234++ not = -9234++ hence 9234++ is noninvertible. Since most links are invertible, only the orientations beginning with a + are listed in the table. The corresponding orientations with + <-> - are implicitly equivalent. In case of noninvertible links as 9234++, the opposite orientation (here, 9234-) is written between parentheses. The invariants w, l, and s are insensitive to invertibility (they are the same for both orientations, be the link invertible or noninvertible), but they are sensitive to chirality: a non-zero w, l, or s implies that the link is chiral. Some oriented links L are chiral (L not = L*) and noninvertible (L not = -L) but have the symmetry property that L = -L*. Such links are called (-)amphicheiral (e.g., 817+).
The chirality of oriented links can be detected by several methods. A code (number from 1 to 5) is used in column 6 of Table 1 for the simplest method allowing to determine the chirality of each oriented link. By increasing complexity, these methods are:
Finally, last column of Table 1 displays for each chiral alternating oriented link the corresponding chirality specifier as determined by the method of Liang, Cerf and Mislow [7]. Every chiral link possesses two nonequivalent mirror images, denoted by L (from the Latin: laevus, left) and D (from the Latin: dexter, right), a new analogy with left and right hands. The above-mentioned authors developed an empirical method aimed at labelling the two mirror images unequivocally. For instance, the oriented trefoil (31+, with w = -3) represented in Figure 1 is L. Its mirror image, with all crossings changed (31+*), will be D. This corresponds to the implicit rule accepted by all authors. By contrast, there is no rule concerning links as 84+ (w = 0). Using the method of [7], we find that this link is D. Its mirror image, 84+*, will be L.
To allow an easy comparison of the present work with works using the tabulation of Doll and Hoste, we end this atlas with a third part (Table 2). All alternating oriented links of two or more components up to nine crossings are listed, using for each of them the same number of entries as the number of different projections in Doll and Hoste's tabulation. For each projection, the symbol = means that Doll and Hoste used the same link type as Rolfsen, and the symbol * means that they used the mirror image type compared to Rolfsen. Let us remind that we always used the same link type as Rolfsen.
Although every effort has been made to check the correctness of the tables, the author would be grateful that any remaining error be communicated to her.