Intrinsic symmetry groups of links with 8 and fewer crossings

We present an elementary derivation of the"intrinsic"symmetry groups for knots and links of 8 or fewer crossings. The standard symmetry group for a link is the mapping class group $\MCG(S^3,L)$ or $\Sym(L)$ of the pair $(S^3,L)$. Elements in this symmetry group can (and often do) fix the link and act nontrivially only on its complement. We ignore such elements and focus on the"intrinsic"symmetry group of a link, defined to be the image $\Sigma(L)$ of the natural homomorphism $\MCG(S^3,L) \rightarrow \MCG(S^3) \cross \MCG(L)$. This different symmetry group, first defined by Whitten in 1969, records directly whether $L$ is isotopic to a link $L'$ obtained from $L$ by permuting components or reversing orientations. For hyperbolic links both $\Sym(L)$ and $\Sigma(L)$ can be obtained using the output of \texttt{SnapPea}, but this proof does not give any hints about how to actually construct isotopies realizing $\Sigma(L)$. We show that standard invariants are enough to rule out all the isotopies outside $\Sigma(L)$ for all links except $7^2_6$, $8^2_{13}$ and $8^3_5$ where an additional construction is needed to use the Jones polynomial to rule out"component exchange"symmetries. On the other hand, we present explicit isotopies starting with the positions in Cerf's table of oriented links which generate $\Sigma(L)$ for each link in our table. Our approach gives a constructive proof of the $\Sigma(L)$ groups.


INTRODUCTION
The symmetry group of a link L is defined to be the mapping class group MCG(L) (or Sym(L)) of the pair (S 3 , L). The study of this symmetry group is a classical topic in knot theory, and these groups have now been computed for prime knots and links in several ways. Kodama and Sakuma [KS92] used a method in Bonahon and Siebenmann [BS09] to compute these groups for all but three of the knots of 10 and fewer crossings in 1992. In the same year, Weeks and Henry used the program SnapPea to compute the symmetry groups for hyperbolic knots and links of 9 and fewer crossings [HW92]. These efforts followed earlier tabulations of symmetry groups by Boileau and Zimmermann [BZ87], who found symmetry groups for nonelliptic Montesinos links with 11 or fewer crossings.
We consider a different group of symmetries of a link L given by the image of the natural homomorphism π : Sym(L) = MCG(S 3 , L) → MCG(S 3 ) × MCG(L).
Since these symmetries record an action on L itself (and only record the orientation of the ambient S 3 ), we will call them "intrinsic" symmetries of L to distinguish them from the standard symmetry group. Unlike the elements in the Sym group, which may be somewhat difficult to describe explicitly, each of the elements in Σ(L) corresponds to an isotopy of L which may exchange the position of some components, which may mirror crossings, and which may reverse orientations of some components. Neither the Boileau-Zimmerman or the Henry-Weeks-SnapPea method gives much insight into what those isotopies might look like. In addition, it is worth noting that SnapPea is a large and complicated computer program, and while its results are accurate for the links in our table, it is always worthwhile to have alternate proofs for results that depend essentially on nontrivial computer calculations.
In this spirit, the present paper presents an elementary and explicit derivation of the Σ(L) groups for all links of 8 and fewer crossings. We rule out certain isotopies using elementary and polynomial invariants to

Number of link types by components and crossings
Crossings 1-U 1-OS 2-U 2-OS 3-U 3-OS 4-U 4-OS All Links-U All Links-OS   0 1  1  1  1  1  1  2  1  2  1  2  3 1  2  4 1  1  1  4  1  4  5 2  4  1  2  1  2  6 3  5  3  10  3  18  6  44  7 7  14  8  38  1  8  9  40  8 21  38  16  78  10 200  3  120  29  398   TABLE 1. This table shows the number of distinct isotopy classes of links by crossing number and number of components in the link. The columns labeled "n-U" count link types for n-component links in the usual way, where the components are unlabeled and unoriented, and the mirror image of a given link is considered to have the same link type, regardless of whether the two are ambient isotopic. The columns labeled "n-OS" give finer information, considering two n-component links to have the same type if and only if they are oriented, labeled ambient isotopic. As we can see, this stricter definition leads to a much larger collection of link types.
provide an upper bound on the size of Σ(L) for each link in our table and then present explicit isotopies generating Σ(L) starting with the configurations of the link in Cerf's table of alternating oriented links [Cer98] or (for nonalternating links) Doll and Hoste's table [DH91]. For three links in our table, 7 2 6 , 8 2 13 and 8 3 5 , an additional construction is needed to rule out certain "component exchange" symmetries using satellites and the Jones polynomial. This shows that the polynomial invariants are powerful enough to compute Σ(L) for these links. We give the first comprehensive list of Σ(L) groups that we know of, though Hillman [Hil86] provides examples of various two-component links (including some split links) with symmetry groups equal to 12 different subgroups of Γ 2 .
Why are the Σ(L) groups interesting? First, it is often more natural to consider the restricted group Σ(L) than the generally larger Sym(L). Sakuma [Sak89] has shown that a knot K has a finite symmetry group if and only if K is a hyperbolic knot, a torus knot, or a cable of a torus knot. Thus for many 1 knots, the group Sym(L) contains infinitely many elements which act nontrivially on the complement of L but fix the link itself. We ignore such elements, which lie in the kernel of the natural homomorphism π : MCG(S 3 , L) → MCG(S 3 ) × MCG(L). In fact, even when MCG(S 3 , L) is finite, we give various examples below where π has nontrivial kernel. It is often difficult to describe an element of Sym(L) in ker π explicitly, but it is always simple to understand the exact meaning of the statement γ ∈ Σ(L).
As an application, if one is interested in classifying knots and links up to oriented, labeled ambient isotopy, it is important to know the symmetry group Σ(L) for each prime link type L, since links related by an element in Γ(L) outside Σ(L) are not (oriented, labeled) ambient isotopic. The number of different links related by an element of Γ(L) to a given link of prime link type L is given by the number of cosets of Σ(L) in Γ(L). If we count these cosets instead of prime link types, the number of actual knots and link types of a given crossing number is actually quite a bit larger than the usual table of prime knot and link types suggests. (See Table 1.) Second, Σ(L) seems likely to be eventually relevant in applications. For instance, when studying DNA links, each loop of the link has generally has a unique sequence of base pairs which provide an orientation and an unambiguous labeling of each component of the link. In such a case, the question of whether two components in a link can be interchanged may prove to be of real significance.
Last, we are interested in the topic of tabulating composite knots and links. Since the connect sum of different symmetry versions of the same knot type can produce different knots (such as the square knot, which is the connect sum of a trefoil and its mirror image, and the granny knot, which is the connect sum of two trefoils with the same handedness), keeping track of the action of Σ(L) is a crucial element in this calculation. We treat this topic in a forthcoming manuscript [CMP11].

THE SYMMETRY AND INTRINSIC SYMMETRY GROUPS
As we will describe below, the group MCG(S 3 )×MCG(L) was first studied by Whitten in 1969 [Whi69], following ideas of Fox. They denoted this group Γ(L) or Γ µ , where µ is the number of components of L. We can write this group as a semidirect product of Z 2 groups encoding the orientation of each component of the link L with the permutation group S µ exchanging components of L (cf. Definition 4.1), finally crossed with another Z 2 recording the orientation of S 3 : It is clear that an element γ = ( 0 , 1 , . . . , µ , p) ∈ Γ(L) acts on L to produce a new link L γ . If 0 = +1, then L γ and L are the same as sets (but the components of L have been renumbered and reoriented), while if 0 = −1 the new link L γ is the mirror image of L (again with renumbering and reorientation). We can then define the symmetry subgroup Σ(L) by γ ∈ Σ(L) ⇐⇒ there is an isotopy from L to L γ preserving component numbering and orientation.
For knots, Σ(L) < Γ 1 = Z 2 × Z 2 . Here the five subgroups of Z 2 × Z 2 correspond to the standard descriptions of the possible symmetries for a knot, as shown in Table 2  For links, the situation is more interesting, as the group Γ(L) is more complicated. In the case of twocomponent links, the group Γ 2 = Z 2 × (Z 2 × Z 2 S 2 ) is a nonabelian 16 element group isomorphic to Z 2 × D 4 . The various subgroups of Γ 2 do not all have standard names, but we will call a link purely invertible if (1, −1, . . . , −1, e) ∈ Σ(L), and say that components (i, j) have a pure exchange symmetry if (1, 1, . . . , 1, (ij)) ∈ Σ(L). For two-component links, we will say that L has pure exchange symmetry if its two components have that symmetry. The question of which links have this symmetry goes back at least to Fox's 1962 problem list in knot theory [Fox62,Problem 11] 4 . For example, 2 2 1 (the Hopf link) has pure exchange symmetry while we will show that 7 2 4 does not.
We similarly focus on the pure invertibility symmetry: we show that 45 out of the 47 prime links with 8 crossings or less are purely invertible. The two exceptions are 6 3 2 (the Borromean rings) and 8 3 5 ; we show that both of these are invertible using some nontrivial permutation, i.e., (1, −1, −1, −1, p) ∈ Σ(L) for some p = e. (Whitten found examples of more complicated links which are not invertible even when a nontrivial permutation is allowed [Whi71].) Increasing the number of components in a link greatly increases the number of possible types of symmetry. Table 3 lists the number of subgroups of Γ µ ; each different subgroup represents a different intrinsic symmetry group that a µ-component link might have. We note that if Σ(L) is the symmetry subgroup of link L, then the symmetry subgroup of L γ is the conjugate subgroup Σ(L γ ) = γΣ(L)γ −1 . Therefore, it suffices to only examine the number of mutually nonconjugate subgroups of Γ µ in order to specify all of the different intrinsic symmetry groups.

METHODS AND NOTATION
We started from the excellent table of oriented alternating links provided by Cerf [Cer98]. Cerf considers the effect of reversing orientations of components of her links, but does not compute the effect of permutations. Our more detailed calculation does not depend on this information, so our paper also provides a check on Cerf's symmetry calculations. For alternating links, we have kept component numbers and orientations consistent with her table. For nonalternating links, we use component numbers and orientations consistent with those of Doll and Hoste [DH91]. Like Cerf, Doll and Hoste considered the effect of reversing the orientation of individual components of these links, but not the effect of permutations of components. We note that these component numbers and orientations are not consistent with those used in SnapPea. The component numbers and orientations in SnapPea seem to have been chosen arbitrarily sometime in the 1980s by Joe Christy when he digitized the Rolfsen table [CD10].
To check our results against the SnapPea calculations, we used the Python interface provided by SnapPy to compute the image of π : MCG(S 3 , L) → MCG(S 3 ) × MCG(L). The results agreed 5 with the tables we give below, meaning that our results serve as an independent verification of SnapPea's accuracy for these links. 5 We had to redraw several links where the component numbering and orientation given by default in SnapPea differed from our choice of numbering and orientation to make the groups agree. The default link data in SnapPea results in several cases in subgroups conjugate to those we give below.
Finally, some notational comments: we henceforth use the term 'link' to refer to a prime, multicomponent link unless otherwise indicated. We use the Rolfsen notation for links but provide the Thistlethwaite notation as well in our summary tables (Tables 9, 11, and 12).

THE WHITTEN GROUP
We begin by giving the details of our construction of the Whitten group Γ(L) and the symmetry group Σ(L). Consider operations on an oriented, labeled link L with µ components. We may reverse the orientation of any of the components of L or permute the components of L by any element of the permutation group S µ . However, these operations must interact with each as well: if we reverse component 3 and exchange components 3 and 5, we must decide whether the orientation is reversed before or after the permutation. Further, we can reverse the orientation on the ambient S 3 as well, a process which is clearly unaffected by the permutation. To formalize our choices, we follow [Whi69] to introduce the Whitten group of a µ-component link.
Consider the following operations on L: (1) Permuting the K i .
(2) Reversing the orientation of any set of K i 's (3) Reversing the orientation on S 3 (mirroring L). Let γ be a combination of any of the moves (1), (2), or (3). We think of γ = ( 0 , 1 , ... µ , p) as an element of the set Z µ+1 2 × S µ in the following way. Let Lastly, let p ∈ S µ be the permutation of the K i associated to γ. To be explicitly clear, permutation p permutes the labels of the components; the component originally labeled i will be labeled p(i) after the action of γ.
For each element, γ in Z µ+1 where −K i is K i with orientation reversed, K * i is the mirror image of K i and the ( * ) appears above if and only if 0 = −1. Note that the ith component of γ(L) is i K ( * ) p(i) the possibly reversed or mirrored p(i)th component of L. Since we are applying i instead of p(i) to K p(i) we are taking the convention of first permuting and then reversing the appropriate components.
We now confirm that this operation defines a group action of the Whitten group Γ(L) on the set of links obtained from L by such transformations. Proof. We know that L is a disjoint union of µ copies of S 1 denoted L = K 1 · · · K µ . Further, the mapping class groups of S 1 and S 3 are both Z 2 , where the elements ±1 correspond to orientation preserving and reversing diffeomorphisms of S 1 and S 3 . In general, the mapping class group of µ disjoint copies of a space is the semidirect product of the individual mapping class groups with the permutation group S µ . This means that MCG(L) = (Z 2 ) µ S µ and the Whitten group Γ(L) has a bijective map to MCG(S 3 ) × MCG(L).
It remains to show that the group operation * in the Whitten group maps to the group operation (composition of maps) in MCG(S 3 ) × MCG(L). To do so, we introduce some notation. Let γ = ( 0 , 1 , ... µ , p) and γ = ( 0 , 1 , ... µ , q). We must show where γ * γ is the operation of the Whitten Group, Γ µ . Then, Note that X p(i) = p(i) K q(p(i)) , which implies Now, γ * γ = ( 0 0 , 1 p(1) , 2 p(2) , ..., µ p(µ) , qp) and acts on L as We have dropped the notation for mirroring throughout the proof, because the two links clearly agree in this regard. The element γ * γ preserves the orientation of S 3 if and only if 0 0 = 1, i.e., if either both or neither of γ and γ mirror L.
We can now define the subgroup of Γ(L) which corresponds to the symmetries of the link L.
Definition 4.5. Given a link, L and γ ∈ Γ(L), we say that L admits γ when there exists an isotopy taking each component of L to the corresponding component of L γ which respects the orientations of the components. We define as the Whitten symmetry group of L, The Whitten symmetry group Σ(L) is a subgroup of Γ µ , and its left cosets represent the different isotopy classes of links L γ among all symmetries γ. By counting the number of cosets, we determine the number of (labeled, oriented) isotopy classes of a particular prime link.
Next, we provide a few examples of symmetry subgroups. Recall that the first Whitten group Γ 1 = Z 2 × Z 2 has order four and that Γ 2 = Z 2 × (Z 2 × Z 2 S 2 ) is a nonabelian 16 element group.
Example 4.6. Let L = 4 1 , the figure eight knot. Since L ∼ −L ∼ L * ∼ −L * , we have Σ(4 1 ) = Γ 1 , so the figure eight knot has full symmetry. There is only one coset of Σ(4 1 ) and hence only one isotopy class of 4 1 knots. Example 4.8. Let L = 7 2 5 , whose components are an unknot K 1 and a trefoil K 2 . In section 7.2, we determine all symmetry groups for two-component links, but we provide details for 7 2 5 here. Since the components K 1 and K 2 are of different knot types, we conclude that no symmetry in Σ(7 2 5 ) can contain the permutation (12). Since K 2 K * 2 , we cannot mirror L, i.e., the first entry of γ ∈ Σ(7 2 5 ) cannot equal −1. The linking number of L is nonzero, so we can rule out the symmetries (1, −1, 1, e) and (1, 1, −1, e) by Lemma 7.6. Last, L is purely invertible, meaning isotopic to −L = −K 1 ∪ −K 2 . Thus, Σ(L) is the two element group Σ 2,1 = {(1, 1, 1, e), (1, −1, −1, e)}. There are 8 cosets of this two element group in the 16 element group Γ 2 , so there are 8 isotopy classes of 7 2 5 links. We now prove Proof. Given a map f : S 3 → S 3 ∈ MCG(S 3 , L), we see that if f is orientation-preserving on S 3 , then it is homotopic to the identity on S 3 since MCG(S 3 ) = Z 2 . This homotopy yields an ambient isotopy between L and f (L), proving that f | L = π(f ) ∈ Σ(L). If f is orientation-reversing on S 3 , it is homotopic to a standard reflection r. Composing the homotopy with r provides an ambient isotopy between L and r(f (L)), proving that π(f ) ∈ Σ(L). This shows π(Sym(L)) ⊂ Σ(L). Now suppose g ∈ Σ(L). The isotopy from L to g(L) generates an orientation-preserving (since it is homotopic to the identity) diffeomorphism f : S 3 → S 3 which either fixes L or takes L to rL. In the first case, f ∈ Sym(L). In the second, the map rf ∈ Sym(L).

THE LINKING MATRIX
For each link L, our overall strategy will be to explicitly give isotopies for certain elements of the symmetry subgroup Σ(L), generate the subgroup containing those elements, and then rule out the remainder of Γ(L) using invariants. For three-and four-component links, a great deal of information about Σ(L) can usually be obtained by considering the collection of pairwise linking numbers of the components of the link.
We recall a few definitions: Definition 5.1. Given a n-component link, with components L 1 , L 2 , . . . , L n , we let the n×n linking matrix of L be the matrix Lk(L) so that Lk(L) ij = Lk(L) ji = Lk(L i , L j ) and Lk(L) ii = 0 where Lk(L i , L j ) is the linking number of L i and L j . We let Lk(n) denote the set of n × n symmetric, integer-valued matrices with zeros on the diagonal.
The linking number can also be computed by counting the signed crossings of one knot over another. Among minimal crossing number diagrams of alternating links, the following three numbers are also useful link invariants: Definition 5.2. The (overall) linking number k(L) of L is half the sum of the entries of the linking matrix. (This is half of the intercomponent signed crossings of L.) The writhe wr(L) is the sum of all signed crossings of L. The self-writhe s(L) is the sum of the intracomponent signed crossings of L. Clearly, wr = 2 k + s.
Murasugi and Thistlethwaite separately showed that writhe was an invariant of reduced alternating link diagrams [Mur87,Thi88]. Since linking number is an invariant for all links, self-writhe is also an invariant of reduced alternating diagrams. We will utilize this invariance to rule out certain symmetries of links, cf. Lemma 7.6.
The Whitten group Γ n acts on the set of linking matrices Lk(n). Further, for a given link L, the symmetry subgroup Σ(L) must be a subgroup of the stabilizer of Lk(L) under this action. This means that it is worthwhile for us to understand this action and make a classification of linking matrices according to their orbit types. We start by writing down the action: Proposition 5.3. The action of the Whitten group Γ n on n-component links gives rise to the following Γ n action on the set of n × n linking matrices Lk(n): Proof. Equation (1) reminds us that Recall that linking number is reversed by changing the orientation of either curve or the ambient S 3 , which proves that we should multiply by 0 i j as claimed.
In principle, this description of the action provides all the information one needs to compute orbits and stabilizers for any given matrix (for instance, by computer). However, that brute force approach doesn't yield much insight into the structure of the problem. We now develop enough theory to understand the situation without computer assistance in the case of three-component links.

5.1.
Linking matrix for three-component links. We first observe that there is a bijection between Lk(3) and Z 3 given by We would like to understand the action of Γ 3 on Lk(3) by reducing it to the natural action of the simpler group (Z 2 ) 3 S 3 on Z 3 .
Since Γ 3 is a group of order 96 and the kernel of f has order 2, the image of f has order 48. Since the target group (Z 2 ) 3 S 3 also has order 48, we conclude that f is surjective, as claimed.
By Corollary 5.4, the Γ 3 action on Lk(3) maps each entry z k = Lk(L) ij in the linking matrix to By the definition of f , the natural action of (Z 2 ) 3 S 3 on Z 3 is This triple corresponds precisely to the new linking matrix (3) obtained from the Γ 3 action, so we have shown the two actions correspond.
We are now in a position to classify 3 × 3 linking matrices according to their orbit types, and compute their stabilizers in Γ 3 . The stabilizer of a linking matrix A ∈ Lk(3) as a subgroup of Γ(3) under the group action of Proposition 5.3 is the preimage under the homomorphism f of Proposition 5.5 of the stabilizer of the corresponding triple (z 1 , z 2 , z 3 ) ∈ Z 3 under the natural action of (Z 2 ) 3 S 3 on Z 3 . Since the kernel of f has order 2, stabilizers in Γ 3 are twice the size of the corresponding stabilizers in (Z 2 ) 3 S 3 .
There are 10 orbit types of triples (z 1 , z 2 , z 3 ) under this action. To list the orbit types, we write a representative triple in terms of variables a, b, and c which are assumed to be integers with distinct nonzero magnitudes. To list the stabilizers, we either give the group explicitly as a subgroup of (Z 2 ) 3 S 3 or provide a list of generators in the form g 1 , g 2 , . . . , g n . One of these groups, S(a, a, −a), is more complicated and is described below.
Using the preimage formula in the proof of Proposition 5.5, it is easy to compute the stabilizer of a given linking matrix in Γ 3 directly from the table above; we simply conjugate by a permutation to bring the corresponding triple into one of the forms above and then apply the preimage formula.
We can now draw some amusing conclusions which might not be obvious otherwise, such as: Lemma 5.6. If L is a three-component link and any element of Σ(L) reverses orientation on S 3 then at least one pair of components of L has linking number zero.
Proof. The stabilizer of Lk(L) includes an element of the form (−1, 1 , 2 , 3 , p) if and only if some element (δ 1 , δ 2 , δ 3 , p) in the stabilizer of the corresponding triple (z 1 , z 2 , z 3 ) has δ 1 δ 2 δ 3 = −1 since we showed in the proof of Proposition 5.5 that 0 equaled δ 1 δ 2 δ 3 . A negative δ i will switch the sign of the linking number z p(i) ; to stabilize the triple (z 1 , z 2 , z 3 ) there must be an even number of sign changes unless some z i = 0. Hence, if L has some mirror symmetry, i.e., one with 0 = −1, then δ 1 δ 2 δ 3 = −1 which produces an odd number of sign changes, so some linking number z i = 0.

5.2.
Linking matrix for four-component links. For four-component links, we will need to develop a different observation. There are only three prime four-component links with 8 or fewer crossings, and fortunately they all possess a particular type of linking matrix. While the situation seems too complicated to make a full analysis of the S 4 action on the 6 nonzero elements of a general 4 × 4 linking matrix, it is relatively simple to come up with a theory which covers our cases. We first give a correspondence between certain 4 × 4 linking matrices and elements of Z 4 : If we let Γ 4 act on the matrix A ij as usual, matrices in this form are fixed by the subgroup with permutations in an 8 element subgroup of S 4 isomorphic to D 4 which we will call G 0 . and the image of f in (Z 2 ) 4 S 4 is {(δ 1 , δ 2 , δ 3 , δ 4 , f 0 (p)) : δ 4 = δ 1 δ 2 δ 3 , p ∈ G 0 }. The preimage of such an element is the 4 element set {( 1 2 δ 1 , 1 , 2 , 2 δ 1 δ 3 , 1 δ 1 δ 2 , p)}, where 1 and 2 are arbitrary.
To show that the action of G on Lk(4) descends to the natural action of (Z 2 ) 4 S 4 on Z 4 , i.e.
we need only check that for all p ∈ G 0 . This is another straightforward, if lengthy, computation.
We now need to find stabilizers for a few carefully chosen linking matrix types.

THE SATELLITE LEMMA
We begin with two definitions.
Definition 6.1. A link L is invertible if reversing the orientation of all of its components produces a link isotopic to L, i.e., if (1, −1, . . . , −1, p) ∈ Σ(L) for some permutation p. If the trivial permutation suffices, we call L purely invertible.
We note that if components K i and K j are not of the same knot type, then it is impossible for them to have a pure exchange symmetry. We also note that if a link admits all pure exchange symmetries, the terms invertible and purely invertible are equivalent.
The most difficult part of our work below will be in ruling out pure exchange symmetries. So far, we have two (crude) tools; we can rule out pure exchange when the two components have different knot types or when the pure exchange does not preserve the linking matrix. In a few cases, these tools will not be enough and we will need the following: Proof. Such a pure exchange would carry an oriented solid tube around L i to a corresponding oriented solid tube around L j . If we imagine K embedded in this tube, this generates an isotopy between L(K, i) and L(K, j).
The point of this lemma is that we can often distinguish L(K, i) and L(K, j) using classical invariants which are insensitive to the original labeling of the link. This seems like a general technique, and it would be interesting to explore this topic further.

TWO-COMPONENT LINKS
This section records the symmetry group Σ(L) for all prime two-component links with eight or fewer crossings; there are 30 such links to consider. Our results are summarized in section 7.1, which names and lists the symmetry groups which appear (see Table 6). We count how frequently each group appears by crossing number in Table 7. The symmetry group for each link is listed in Tables 8 and 9, by group and by link, respectively. Proofs of these assertions appear in section 7.2 7.1. Symmetry names and results. The Whitten group Γ 2 = Z 2 × (Z 2 × Z 2 S 2 ) of all possible symmetries for two-component links is a nonabelian 16 element group isomorphic to Z 2 × D 4 . The symmetry group Σ(L) of a given link must form a subgroup of Γ 2 . There are 27 mutually nonconjugate subgroups of Γ 2 ; of these possibilities, only seven are realized as the symmetry subgroup of a prime link with 9 or fewer crossings (see Table 6). An eighth appears as the symmetry subgroup of a 10-crossing link.
Question 7.1. Do all 27 nonconjugate subgroups of Γ 2 appear as the symmetry group of some (possibly composite, split) link? Of some prime, non-split link?
Hillmann [Hil86] provided examples for a few of these symmetry subgroups, but some of his examples were split links. Here are the groups we found among links with 8 or fewer crossings.

Symmetry name
Notation Subgroup of Isomorphic to Even number of operations Even ops & pure exchange Σ 8,2 {( 0 , 1 , 2 , p) : 0 1 2 = 1} D 4 No exchanges The first seven nontrivial groups in Table 6 are realized as the symmetry group of a link with nine or fewer crossings, while Σ 8,3 appears to be the symmetry group of a 10-crossing link. We know of no nontrivial links with full symmetry but speculate that they exist.
The next table records the frequency of each group.     Table 9.
The proof of this theorem is divided into five cases, based on the five symmetry groups that appear in Table 9; these proofs are found in section 7.2. Many of our arguments generalize to various families of links. As this paper focuses on these first examples, we ask for the reader's understanding when we eschew the most general argument in favor of a simpler, more expedient one. 7.2. Proofs for two-component links. Below, we attempt to provide a general framework for determining symmetry groups for two-component links. Since Σ(L) is a subgroup of Γ 2 , the order of the symmetry group must divide |Γ 2 | = 16. Our strategy begins by exhibiting certain symmetries via explicit isotopies. With these in hand, we next use various techniques to rule out some symmetries until we can finally determine the symmetry group Σ(L). These techniques generally involve using some link invariant to show L γ L. Among link invariants, the linking number and self-writhe (for alternating links) are easily applied since they count signed crossings; we also use polynomials and other methods.
We focus on the 30 prime links with eight or fewer crossings. Our first results indicate which of these 30 links have either a pure invertibility or a pure exchange symmetries, which we prove explicitly by exhibiting isotopies. Recall that a link is purely invertible if reversing all components' orientations produces an isotopic link; a link has pure exchange symmetry if swapping its two components is an isotopy. Lemma 7.3. Via the isotopies exhibited in Figures 6, 7, and 20-26 in Appendices B.1 and C.1, the following 17 links have pure exchange symmetry: (6) 2 2 1 , 4 2 1 , 5 2 1 , 6 2 1 , 6 2 2 , 6 2 3 , 7 2 1 , 7 2 2 , 7 2 3 , i.e., (1, 1, 1, (12)) belongs to the symmetry group of each of these links.
As we determine symmetry groups, we will establish that the remaining 13 links in consideration do not have pure exchange symmetry.
Cerf [Cer98] states that all prime, alternating two-component links of 8 or fewer crossings are invertible, though this may involve exchanging components. Via the isotopies exhibited in Figures 8 and 27-34 of Appendices B.2 and C.2, we extend Cerf's result to non-alternating links, and we show that the invertibility is pure (i.e., without exchanging components). To obtain invertibility for 7 2 8 , combine the results of Figures 9 and 10, which show that each of its components can be individually inverted.
Lemma 7.4. All 30 prime two-component links with eight or fewer crossings are purely invertible.
Lemma 7.5. Any two-component link, such as those listed in (6), that has both pure exchange symmetry and (pure) invertibility, must have Σ 4,1 as a subgroup of its symmetry group Σ(L).
By examining signed crossings of a link, we calculate its linking number and self-writhe; if one of these is nonzero, we may rule out some symmetries. Proof. Consider the effect of each symmetry operation upon linking number (see Proposition 5.3): mirroring a link or inverting one of its components will swap the sign of the linking number, while exchanging its components fixes the linking number. As for the self-writhe of a link, it is fixed by inverting any component or exchanging the two components; however, mirroring a link swaps the sign of s(L).
Thus, the elements of Γ 2 that will swap the sign of a linking number are precisely those of the form γ = ( 0 , 1 , 2 , p) with := 0 1 2 = −1. If the linking number Lk(L) is nonzero, these cannot possibly produce a link L γ isotopic to the original link L, so these eight elements are not part of Σ(L). The remaining eight symmetry elements form Σ 8,2 = {( 0 , 1 , 2 , p) : 0 1 2 = 1}, which proves the first assertion.
Self-writhe is an invariant of reduced diagrams of alternating links, and any symmetry operation which mirrors the link will swap the sign of s(L). If the self-writhe s(L) is nonzero, then no element which mirrors, i.e., (−1, 1 , 2 , p), can lie in Σ(L). The remaining elements form Σ 8,1 = {1} × (Z 2 × Z 2 S 2 ).
Lemma 7.7. Let L be a two-component link.
(2) If the components of L are different knot types, then Σ(L) < Σ 8,3 . Proof. The first assertion is immediate, since the purely invertible symmetry generates Σ 2,1 . If the components of L have different knot types, then no exchange symmetries are permissible; the permutation p = (12) never appears in Σ(L). Hence the symmetry group Σ(L) is contained in the 'No exchanges' group Σ 8,3 .
With these five lemmas in hand, we are now prepared to begin proving Theorem 7.2, which we treat by each symmetry group. 7.2.1. Links with symmetry group Σ 2,1 .
Three of the alternating links (7 2 5 , 8 2 11 , 8 2 14 ) have nonzero self-writhe, so we apply Lemma 7.7 again. We conclude that they have only the purely invertible symmetry, and Σ 2,1 is their symmetry group.

THREE-COMPONENT LINKS
There are 14 three-component links with 8 or fewer crossings. In this section, we determine the symmetry group for each one; Table 11 summarizes the results. We obtain 11 different symmetry subgroups inside Γ 3 , which represent 7 different conjugacy classes of subgroups (out of the 131 possible).
For each link, our first task is to calculate the linking matrix. Then, we utilize Table 4 to determine the stabilizer of this matrix within Γ 3 ; we know that the symmetry group Σ(L) must be a subgroup of this stabilizer. From there, we proceed by ruling out other elements using polynomial invariants and by exhibiting isotopies to show that certain symmetries do lie in Σ(L) until we can discern the symmetry group.
We note that all but two of these links are purely invertible, even though PI might not be part of a minimal set of generators. Neither the Borromean rings 6 3 2 or the link 8 3 5 are purely invertible. Both of these links are, however, invertible. The Borromean rings can be inverted using any odd permutation p, i.e., they admit the symmetry γ = (1, −1, −1, −1, p); the link 8 3 5 is invertible using p = (23). Claim 8.1. The subgroup Σ(6 3 1 ) < Γ 3 is the 12 element group isomorphic to D 6 generated by pure exchanges and pure invertibility.
Proof. The linking matrix for 6 3 1 is 1 2 3 1 0 -1 -1 2 -1 0 -1 3 -1 -1 0 which is in the standard form (a, a, a). We know that Σ(6 3 1 ) is a subgroup of the stabilizer of this matrix under the action of Γ 3 on linking matrices. Consulting Table 4, we see that this stabilizer is the group in the claim. We must now show that all these elements are in the group. Figures 7 and 36 show that   (1, 1, 1, 1, (123)), (1, 1, 1, 1, (23)) ∈ Σ(6 3 1 ) Since any 3-cycle and 2-cycle generate S 3 , we have the rest of the pure exchanges as well. Figure 35 shows that this link is purely invertible, completing the proof.
These three elements generate the 12 element group of the claim. Now the linking matrix for 6 3 3 is 1 2 3 1 0 1 -1 2 1 0 -1 3 -1 -1 0 which is in the standard form (a, a, −a). Consulting Table 4, we see that this means |Σ(6 3 3 )| divides 12, the order of the stabilizer. Since we already have 12 elements in the symmetry group, it must equal the stabilizer, which completes the proof.
This rules out all but the four element subgroup above, completing the proof.

ISOTOPIES FOR FOUR-COMPONENT LINKS
There are three prime four-component links with 8 crossings. They are quite similar in appearance, with only some crossing changes distinguishing them. Their symmetry computations are made somewhat more difficult by the fact that we are working in the 768 element group Γ 4 . All three of these links are composed of four unknots linked together so that component 1 is linked to 2 and 3, and 2 and 3 are linked to 4.
Here are the symmetry groups for these links, listed in terms of generators for each group. Again, we denote the purely invertible symmetry, i.e., element (1, −1, −1, −1, −1, e), by PI; all three links admit this symmetry.
Claim 9.1. The symmetry subgroup for 8 4 1 is the 16 element group isomorphic to Z 2 × D 4 given by the S 3 -orientation-preserving elements of the inverse image f −1 (S(a, a, a, a)).
Claim 9.2. The symmetry subgroup for 8 4 2 is the 16 element group isomorphic to D 8 given by the S 3orientation-preserving elements of f −1 (S(a, a, −a, a)).
This rules out these 16 remaining elements, so the claim is proven.

LINKS
We now compare our results on intrinsic symmetry groups to the existing literature on symmetry groups for links. Henry and Weeks [HW92,Wee93] report Sym(L) groups for hyperbolic links up to 9 crossings, while Boileau and Zimmerman [BZ87] computed Sym(L) groups for nonelliptic Montesinos links with up to 11 crossings, and Bonahon and Siebenmann computed Sym(L) for the Borromean rings link (6 3 2 ) as an example of their methods in [BS09,Theorem 16.18].
Comparing all this data with ours, we see that Lemma 10.1. Among all links of 8 and fewer crossings with known Sym(L) groups, the Whitten symmetry group Σ(L) is not isomorphic to Sym(L) only for the links in Table 13.  Our results provide some data on symmetry groups of torus links as well.
Proof. Goldsmith computed [Gol82] the "motion groups" M (S 3 , L (np,nq) ) for torus knots and links in S 3 . Combining her Corollary 1.13 and Theorem 3.7, we see that for the (np, nq) torus link L (np,nq) , the subgroup Sym + of orientation preserving symmetries is homeomorphic to M (S 3 , L (np,nq) ) if p + q is odd, and an index two quotient group of M (S 3 , L (np,nq) ) if p + q is even. The motion group itself is either D 2 or an 8-element quaternion group. But in either case Sym + D 2 . Now the motion group by itself does not provide any information about the orientation reversing elements in Sym. However, any such element in Sym \ Sym + would map to an element in Σ(L (np,nq) ) which reversed orientation on S 3 . Since we have already shown that there are no such elements in Σ(L (np,nq) ) for n = 1, p = 1, q = 2, 3, 4, we see that for these links Sym = Sym + = Σ. For the Hopf link such an orientation reversing element does exist in Sym(L). So, Sym(L) is a Z 2 extension of D 2 and thus is isomorphic to (Z 2 ) 3 By Proposition 4.9, the group Σ(L) is the image of Sym(L) under a homomorphism, and hence a quotient group of Sym(L). Further, if Σ(L) has only orientation-preserving elements (on S 3 ) then Sym(L) does as well. Thus we know something about the Sym(L) groups of all our links; Table 14 summarizes the new information provided by our approach.
Link | Sym(L)| divisible by Sym(L) has quotient Our results imply that the unknown Sym(L) groups listed above must have a certain quotient.

FUTURE DIRECTIONS
We have now presented explicit computations of the Whitten symmetry groups for all links with 8 and fewer crossings. In all the cases we studied, the most difficult part of the computation was obtaining explicit isotopies to generate the symmetry group; ruling out the remaining elements of Γ n was generally done by the application of classical invariants. The most difficult of these cases required us to use the "satellite lemma" and study the classical invariants of satellites of our original link.
While we have presented conventional proofs of all our results, we used computer methods extensively in determining the right line of attack for each link -our method was to use the Mathematica package KnotTheory to systematically apply all possible Whitten group elements to each link and then check the knot types of the components, the linking matrix, the Jones polynomial, and the HOMFLYPT polynomial in an attempt to distinguish the new link from the original one. We then checked the computer calculations by hand. This automated method clearly cannot compute Σ(L), but it does provide a subgroup Σ (L) of Γ(L) which is known to contain Σ(L). While we do not currently intend to generate isotopies for links with higher crossing number, we intend to present our computationally-obtained Σ (L) groups for 9, 10, and 11 crossing links in a future publication.
Some of the most natural questions about the Whitten symmetry groups remain unanswered by this type of explicit enumeration: which groups can arise as Whitten groups? Does every subgroup of a given Γ n arise as a Σ(L)? We have observed 6 different subgroups of Γ 2 , 11 different subgroups of Γ 3 , and 3 different subgroups of Γ 4 so far. This subject is certainly worth further exploration: can one generate a carefully chosen link with a given symmetry group?
computations of subgroups of Γ µ . We are also grateful to the anonymous referee, who made a number of helpful suggestions as we revised the paper. Our work was supported by the UGA VIGRE grants DMS-07-38586 and DMS-00-89927 and by the UGA REU site grant DMS-06-49242.

APPENDIX A. GUIDE TO LINK ISOTOPY FIGURES
The Appendices contain figures representing the 101 isotopies presented in our paper; they are organized as follows.
• Appendix A explains the moves depicted in these isotopies.
• Appendix B exhibits 41 isotopies which can be found by simply rotating about one axis.
• Appendix C contains the remaining isotopies for 2-component links • Appendix D contains the remaining isotopies for 3-component links • Appendix E contains the remaining isotopies for 4-component links For coordinates on our figures, we assume the diagrams appear in the xy-plane with the z-axis coming out of the page. With this orientation, Figure 4 shows symbols for various rotations around lines. We will also denote moves during an isotopy with arrows, as shown in Figure 5.
x-axis y-axis z-axis line y = x line y = −x  In all of our diagrams, the component numbered 1 is drawn with a thin (1 pt) line, while other components are denoted by thicker (2, 3, or 4 pt lines). We also number the components explicitly to prevent any confusion on this point. As usual, arrows denote orientation on the link components.

APPENDIX B. ISOTOPY FIGURES FOUND BY ROTATIONS
In this section, we display the simplest type of isotopies: mere rotation of a link about a coordinate axis. Such a rotation suffices for 41 of the 101 isotopies we present in this paper.