The Milnor degree of a -manifold
The Milnor degree of a 3-manifold is an invariant that records the maximum simplicity, in terms of higher order linking, of any link in the 3-sphere that can be surgered to give the manifold. This invariant is investigated in the context of torsion linking forms, nilpotent quotients of the fundamental group, Massey products and quantum invariants, and the existence of 3-manifolds with any prescribed Milnor degree and first Betti number is established.
Along the way, it is shown that the number of linearly independent Milnor invariants of degree , for -component links in the -sphere whose lower degree invariants vanish, is positive except in the classically known cases (when , and when with , or ).
This paper initiates a study of the Milnor degree, a 3-manifold invariant introduced by the authors in [CM1]. The definition is recalled and motivated below.
All -manifolds considered here will be closed, connected and oriented. Any such manifold can be constructed by integral surgery on a framed link in the -sphere , written Indeed there are infinitely many choices for the link , and so in studying a given -manifold, it is natural to seek the simplest ones. But in what sense simplest? One measure of the complexity of a link is its linking matrix, or more generally its set of higher order linking numbers, a.k.a. Milnor’s -invariants [M2].
So first define the Milnor degree of a link in the -sphere to be the degree of its first nonvanishing -invariant. Here degree means length minus one, so the pairwise linking numbers are degree one. If all of the -invariants vanish, as for knots or more generally boundary links, then the link is said to have infinite Milnor degree. Thus the link invariant takes values in . Note that higher Milnor degree for a link indicates greater similarity with the unlink, and so in some sense greater simplicity. For example the Hopf link has Milnor degree one, since the components have nonzero linking number, whereas the link obtained by Whitehead doubling both components of the Hopf link is a boundary link, and so has infinite Milnor degree. Links with arbitrary finite degree can be constructed by repeatedly Bing doubling the Hopf link; see Figure 7 in [M1], and §8 in [C4].
Now define the Milnor degree of a -manifold to be the supremum of the degrees of all possible links that can be surgered to give ,
This complexity measure for 3-manifolds first arose in the authors’ study of cyclotomic orders of quantum -invariants at prime levels [CM1], and also appears as a measure of computational complexity for the quantum -invariants at the fourth root of unity [KM1]. As it turns out, this connection with quantum topology provides a powerful tool for analyzing the Milnor degree.
As with other topological invariants defined in a similar fashion – such as Heegaard genus or the surgery number – the Milnor degree is hard to compute. Indeed its value is unknown for many 3-manifolds, including even some lens spaces. It will be seen however that in some situations, especially in the absence of homological torsion, classical techniques from algebraic topology can be brought to bear on this computation.
In section 1 the manifolds of Milnor degree one are completely characterized in terms of their torsion linking forms. In particular, the case when the first homology of the manifold is cyclic (homology lens spaces) is discussed in some detail. In some circumstances one can also identify the manifolds of infinite degree in terms of their linking forms. For example, it will be seen that if the manifold has prime power order first homology and is not of Milnor degree one, then it must have infinite Milnor degree.
In section 2, the Milnor degree of a 3-manifold with torsion free homology is related to the lower central series of its fundamental group, and consequently to its cohomological Massey degree. The Massey degree is known to be algorithmically computable, but it is difficult to calculate in practice. Nevertheless, 3-manifolds with any given Massey degree are easily constructed and it follows that the Milnor degree assumes all values in . Building on the result mentioned in the abstract on the number of independent Milnor invariants of links (Lemma 2.5, proved in Appendix B), this section includes some realization results for the Milnor degree of manifolds with prescribed first homology.
It is a more difficult task to compute the Milnor degree in the presence of torsion, although Massey products can still be of some help. This problem is tackled in section 3 using quantum topology techniques. This leads to a construction of rational homology spheres of arbitrary Milnor degree, and more generally, -manifolds of arbitrary Milnor degree with any prescribed first Betti number.
Before embarking on a detailed analysis of the Milnor degree of -manifolds, we make a few basic observations.
The Milnor degree is invariant under change in orientation, that is where is with the opposite orientation: If is surgery on , then is surgery on the mirror image with negated framings, and clearly .
The Milnor degree of a connected sums satisfies the inequality For if and are surgery on and , then is surgery on the split union , and clearly .
Note that this inequality need not be an equality. For example the connected sum of lens spaces is surgery on the -torus knot [Mo], and so of infinite degree, whereas both lens spaces are of degree 1, as will be seen in the next section. As a consequence, realization results for the Milnor degree are subtler than one might first suspect; see Corollary 2.12 and (the proof of) Theorem LABEL:thm:realization2 for upper bounds on .
There exist -manifolds of infinite Milnor degree with any prescribed first homology
for example obtained by surgery on an unlink with framings .
All integral homology spheres have infinite Milnor degree. This follows from the well known fact that they are all constructible by surgery on boundary links in the -sphere, or it can be deduced from the following more general statement.
The Milnor degree is homological in the sense that it can be defined using any integral homology sphere in place of . In other words does not depend on which homology sphere is used as the base manifold, that is
where denotes the result of surgery on the framed link , and is the degree of the first non-vanishing -invariant of in (see section for a discussion of Milnor invariants in arbitrary integral homology spheres). This is proved in Appendix A using the work of Habegger and Orr; cf. the proof of 6.1 in [HO].
1. Manifolds of degree one
In this section, classical results from the theory of quadratic forms are used to characterize all 3-manifolds of Milnor degree one, and some of infinite degree, in terms of their torsion linking forms.
The linking form of a -manifold is the non-degenerate form
on the torsion subgroup of defined by , where is any -cycle representing , and is any -chain bounded by a positive integral multiple of a 1-cycle representing .
If is surgery on a framed link , then is computed from the (integer) linking matrix of , with framings on the diagonal, as follows. First change basis (pre and post multiply by a unimodular matrix and its transpose) to transform into a block sum , where is a zero matrix and is nonsingular (meaning invertible over , i.e. having nonzero determinant).*** To see how this is done when is singular, start with a primitive vector in with , and complete this to a basis for . Using these basis vectors as the columns of a matrix , we have where is the zero matrix and null( null(. The argument is completed by induction on the nullity of . This corresponds to a sequence of handleslides in the Kirby calculus [K], transforming into . Now following Seifert [Se], the linking form is presented by the (rational) matrix with respect to the generators of given by the meridians of the components of .
The purely algebraic procedure just described associates with any symmetric integer matrix a non-degenerate linking form on the torsion subgroup of , presented by where with nonsingular as above.
It is a classical fact that if is nonsingular, then it can be recovered up to stable equivalence from the isomorphism class of its linking form. (Stable equivalence allows change of basis and block sums with diagonal matrices of ’s; the former correspond to handleslides and the latter to blow ups in the Kirby calculus.) This was proved by Kneser and Puppe [kp] for the case when is odd, and in general by Durfee and Wilkens in their 1971 theses, later simplified by Wall [Wa3] and Kneser (see Durfee [Du, Theorem 4.1]).
For a singular matrix, one needs both its linking form and its nullity to recover its stable equivalence class. This can be seen by changing basis to transform the matrix into the form with nonsingular, as above, and then appealing to the nonsingular case. Although this fact is presumably well-known, we have not been able to find a proof in the literature, and so credit it to “folklore”:
(Folklore) Two symmetric integer matrices are stably equivalent if and only if they have the same nullity and isomorphic linking forms.
Needless to say, this theorem has a number of consequences regarding surgery presentations of -manifolds, many of which are presumably known in some form to experts in the field:
Let be a -manifold, and be a symmetric integer matrix with nullity equal to the first Betti number of and with linking form . Then can be constructed by surgery on a link with linking matrix in an integral homology sphere.
Suppose that is surgery on a framed link in with linking matrix . Then has nullity and , and so by the theorem, and are stably equivalent. Thus a basis change (as above) will transform into for suitable unimodular diagonal matrices and . Letting be an unlink far away from with linking matrix (so ), this means that can be transformed by handleslides to a link of the form with linking matrix . Note that need not be an unlink, but it does have a unimodular linking matrix , and so is a homology sphere containing the link with linking matrix , and , as desired.
Linking forms on finite abelian groups have been classified. They decompose, albeit non-uniquely, as orthogonal sums of forms on cyclic groups and on certain non-cyclic 2-groups [kk][Wa2]. The form on the cyclic group with self-linking on a generator will be denoted by . Note that must be relatively prime to , since the form is non-degenerate, and that
for some relatively prime to .
Any form isomorphic to for some will be called simple. This includes the trivial form on . A direct sum of simple forms will be called semisimple. But beware, such forms might also be sums of non-simple or even non-semisimple forms, for example
In terms of their associated stable equivalence classes of symmetric integer matrices, semisimple forms correspond to diagonal matrices, and the simple forms correspond to diagonal matrices with at most one nonzero entry.
Now observe that surgery on any diagonal framed link in (meaning its linking matrix is diagonal, i.e. pairwise linking numbers vanish) can be performed in two stages: first surger the sublink of -framed components, giving an integral homology sphere, and then surger the remaining components. With this perspective, Corollary 1.2 has the following immediate consequence.
The linking form of a -manifold is semisimple if and only if can be obtained by surgery on a diagonal link in an integral homology sphere, and is simple if and only there is such a link with at most one nonzero framing.
By definition, a -manifold is of degree greater than one if and only if it can be obtained by surgery on a diagonal link. Hence the proposition yields the following characterization of manifolds of degree one.
The -manifolds of Milnor degree one are exactly those with non-semisimple linking forms.
Noting that the linking form of any -manifold with torsion free homology is trivial, we deduce the following well-known result (see [Les, Lemma 5.1.1] for a direct proof):
If , then has Milnor degree greater than one. In fact can be obtained by zero surgery on an -component diagonal link in an integral homology sphere by the last statement in Proposition .
In particular if then is an integral homology sphere, and so as noted in the introduction is of infinite degree. If then is surgery on a knot in a homology sphere, and so again is of infinite degree:
If then has infinite Milnor degree.
If , then it will be shown in §2 that , and as consequence that the Milnor degree can assume any value greater than one (except , and when ).
The situation is more complex when has torsion. In the simplest case when is a finite cyclic group, i.e. is a homology lens space, the theorem gives the complete story in the extreme situations when is either simple or non-semisimple. But it tells us nothing when is semisimple but not simple. When can this happen? This and related questions will be addressed below.
Homology lens spaces
Assume that for some . Then for some relatively prime to . The application of Corollary 1.3 to this situation requires an understanding of when this form is (semi)simple.
The linking form on is simple if and only if is plus or minus a quadratic residue mod .
The linking form on is semisimple if and only if there is a factorization with the pairwise relatively prime such that each form is simple, where .
Proof. Criterion is just the definition of a simple form. To prove (b), note that for any factorization with the pairwise relatively prime, there exist integers for which which yields an orthogonal splitting of the form
In fact any such splitting arises in this way. Furthermore, the are uniquely determined mod ; indeed where is any mod inverse of . Therefore by definition is semisimple if and only if has a factorization as above for which each form is simple. But , and so . This completes the proof.
Elementary number theoretic considerations show that (prime to ) is plus or minus a quadratic residue mod , denoted to evoke the Legendre symbol , if and only if one of the following holds for all odd prime divisors of :
and mod , or
(see [IR, §5.1]). If , set .
(Prime powers) For prime, for all when either mod (since ) or and .
If mod , then (since ) and so is a quadratic residue mod .
And if then .
With this notation, the proposition says that is simple if and only if , and semisimple if and only if can be written as a product of “-quadratic” factors, defined as follows.
A divisor of is -quadratic if and are relatively prime and .
Let be a -manifold with first homology and linking form . Then has Milnor degree one if and only if is not a product of -quadratic factors. Furthermore, if is -quadratic which for lens spaces just means that is homotopy equivalent to then has infinite degree.
It is straightforward (e.g. using Mathematica [Wo]) to generate a complete list of non-semisimple forms for small , as in Table 1: For each , the smallest residue values of are listed, one for each pair of non-semisimple forms. These correspond exactly to the lens spaces of Milnor degree one.
|5||2||24||7||39||2, 5, 7|
|8||3||25||2, 3, 7||40||7, 11, 19|
|13||2, 5||29||2, 3, 8, 12||41||3, 6, 11, 12, 13|
|16||3||32||3, 5||45||2, 7, 8|
|17||3, 5||34||3, 5||48||7, 17|
|20||3||37||2, 5, 6, 8, 13||52||5, 7, 11|
The natural numbers for which supports a non-semisimple form, the first few of which appear in the table, will be called linked numbers. All other natural numbers will be called unlinked. Alternatively, these notions can be phrased in terms of the following:
For any finitely generated abelian group , define the Milnor set of to be the set of all natural numbers that can be realized as Milnor degrees of -manifolds with first homology ,
Infinity is excluded because it can always be realized, as noted in the introduction.
Now Corollary 1.4 shows that is linked if and only if (meaning there exist -manifolds of degree one with ).
For example, from the calculations in Example 1.8 it follows that the linked prime powers are exactly the -powers for and the -powers , and that any product of primes all congruent to is unlinked. Furthermore, it is clear from the definitions that if is a prime power, then every semisimple form on is simple, and so as a consequence:
If is a prime power , then
The preceding discussion has brought attention to the natural numbers for which every semisimple form on is simple. Such numbers will be called quasiprime, since as noted above they include all prime powers. The first few non-quasiprimes are , , , and . By Corollary 1.4, we have or for any quasiprime , according to whether is linked or not.
Pinning down the Milnor degree of non-simple manifolds with cyclic first homology of non-quasiprime order is much more difficult. In fact we do not at present have a finite upper bound for the Milnor degree of any such manifold; conceivably they all have infinite degree.
Do there exist -manifolds with finite cyclic first homology of finite Milnor degree greater than one?
Of course lower bounds for the Milnor degree can be established by displaying suitable surgery links. For example , the smallest non-simple quasiprime lens space, has degree at least . Indeed, it can be obtained by surgery on a two-component link of degree . This is shown in Figure 1 using the Kirby calculus [K], starting with Dehn surgery on the unknot. Unfortunately, that is all that we currently know about its Milnor degree. Note however that manifolds of infinite degree with the same linking form are easily constructed, e.g. surgery on the unlink with framings and .
2. Manifolds with torsion free homology
In this section, the Milnor degree of any 3-manifold with torsion-free homology is related to the lower central series of its fundamental group, and thence to the Massey products of its one-dimensional cohomology classes. As a result the degree of such a manifold can in principal be computed. Some aspects of this theory that hold in the presence of torsion will be discussed at the end of the section.
We begin by reviewing the definition and some basic properties of Milnor’s -invariants for links in the -sphere [M2], or more generally in any integral homology sphere (see e.g. [CO3], [FT2, §2], [HO, Appendix A]). By considering only the first nonvanishing invariants, we avoid any consideration of indeterminacy. Throughout, the lower central series of a group will be written where for .
Milnor’s link invariants
Let be an -component ordered oriented link in an integral homology sphere , and set . A presentation of the nilpotent quotient for each can be obtained as follows, generalizing a result of Milnor (Theorem 4 in [M2]).
Enlarge to a connected -complex by adjoining disjoint paths from its components to a common basepoint, and choose based meridians and longitudes in for the components of . Let be the free group of rank generated by the , and set . Then there is a commutative diagram
for each , where is the “meridional” map sending to itself, is induced by the inclusion , and the vertical maps are the natural projections.
Observe that is -connected on integral homology (it is clearly an isomorphism on , and ) and so is an isomorphism by Stalling’s theorem [St, Theorem 3.4]. In particular is surjective, so for each longitude we can choose an element that represents in the sense that
The particular choice of will not affect our subsequent discussion. Such elements will be called Milnor words of degree (or length ) for . They form a coset of in , since is injective.
Also observe that is surjective with kernel normally generated by the commutators (for ) since is obtained from by adding -cells along these commutators (and one -cell). It follows that the composition is an epimorphism with kernel normally generated by the cosets of the commutators , and so
Now by definition, Milnor’s invariants of degree are zero. Assuming inductively that the invariants of degree less than vanish, those of degree (or equivalently length ) are defined as follows:
For any sequence of integers between and , the integer invariant is the coefficient of in , where is the Magnus embedding of into the group of units in the ring of power series in noncommuting variables . The Magnus embedding has the property that if and only if is of the form where has only terms of degree . Thus fixing but letting vary, the collection of integers are all zero precisely when lies in . Allowing to vary, this implies
Let be an -component link with Milnor words in the free group of rank , as above. Then .
With a little work using the presentation of and properties of the Magnus embedding, it can be shown that
although this will not be needed below.
Zero surgery and nilpotent quotients
Any 3-manifold whose first homology is can be obtained by zero surgery on an -component diagonal link – meaning pairwise linking numbers vanish – in some homology sphere, as noted in Corollary 1.5 (cf. [Les, Lemma 5.1.1]). The main result of this section is that the Milnor degree of can be calculated from any such framed link description. Thus for manifolds with torsion-free homology, the Milnor degree can theoretically be computed. This is in sharp contrast to the situation when torsion is present.
If is zero surgery on a diagonal link in an integral homology sphere, then .
Before giving the proof, we derive the following consequence, the first of several realization results that will be established. Recall that denotes the set of all natural numbers that arise as Milnor degrees of -manifolds with first homology .
Proof. Let denote the set of all natural numbers that arise as the Milnor degrees of -component diagonal links,***Note that since we are restricting to diagonal links. Also, we exclude from since it is realized for any by the -component unlink. so by the theorem. Clearly , and
for , where is the number of linearly independent Milnor invariants of degree distinguishing -component links in the -sphere whose lower degree invariants vanish.
The Milnor numbers were computed by Orr [O1] to be
where denotes the number of basic commutators of length in variables, given classically by Witt’s formula
Here is the Möbius function, defined to be if or is a product of an even number of distinct primes, to be if is a product of an odd number of distinct primes, and to be otherwise. Therefore the corollary reduces to the following number theoretic result, whose proof is deferred to Appendix B.
The Milnor number is positive, or equivalently , for all integers except when and , or .
(a) A little more can be said for -manifolds with , namely that the Lescop invariant of [Les] is nonzero. Indeed is equal to the negative of the Sato-Levine invariant of any -component link whose zero surgery produces [Les, Prop. T5.2], which in turn equals of the link [C3, Th. 9.1], the unique (up to sign) -invariant of degree [C4, App. B][O1].
(b) Our proof of Corollary 2.4 (via Lemma 2.5, proved in the appendix) is non-constructive. Using the techniques of [C4], however, one can give explicit examples for each of -component links of any given Milnor degree , and thus by doing zero surgery on these links, of -manifolds with first homology of Milnor degree .
For example, one such link is the split union of the -component link in Figure 2 with the -component unlink. Note that is obtained from the iterated Bing-double of the Hopf link (denoted in §3 below) by banding together some of its components, following the procedure of [C4, §7.4]. By [C4, §6], the Milnor invariant of degree is equal to the single self-linking number . For , this linking number is equal to while the invariants of degree less than vanish (see 7.2 and 7.4 in [C4]). Therefore .
For , Milnor [M2, Fig. 1] has given examples (without proof) in each odd degree , shown below in Figure 3. It was confirmed in [C4, Example 2.7] that these do indeed have degree . It should be feasible using the same methods to produce such examples for even as well, although we have not done so here.
We now turn to the proof of Theorem 2.3. It is based on the following characterization of the Milnor degree of a diagonal link:
Let be an -component diagonal link in an integral homology sphere , be the -manifold obtained by zero surgery on , be the fundamental group of , and be a free group of rank . Then
Set . Then as shown above
for all , where are the associated Milnor words. Zero surgery adds the relations , and so for each
evidently a quotient of by the . It is clear from this presentation that if for all , then . Conversely, if , then the quotient map above will be an isomorphism (since is nilpotent and hence Hopfian, as are all finitely generated nilpotent groups [MKS, Theorem 5.5]), and so for all . Therefore
and the formula for follows from Lemma 2.1.
of Theorem 2.3.
By definition , so we must prove . Let be any framed link with whose surgery produces . It suffices to show . Since is a diagonal link, must be as well. Let be the zero-framed sublink of . Then , and is surgery on in the homology sphere obtained by surgery on (all of whose framings are since is torsion free). But Lemma 2.7 shows that is characterized by a property of the fundamental group of , and so will be the same for any other link whose zero surgery produces , such as . Therefore as desired.
The following characterization of the Milnor degree of a -manifold with torsion free homology in terms of the lower central series of its fundamental group is an immediate consequence of Lemma 2.7 and Theorem 2.3.
Let be a -manifold with first homology and be a free group of rank . Then .
By Proposition 6.8 in [CGO], this lower central series condition is also a characterization of the Massey degree of , defined to be the length of the first non-vanishing Massey product of -dimensional cohomology classes in , or if all such Massey products vanish:
If is a -manifold with torsion free homology, then its Milnor degree is equal to its Massey degree .
The Milnor degree is also related to the notion of -surgery equivalence [CGO], the equivalence relation on -manifolds generated by -framed surgery on links whose components lie in (for ). In particular, Theorem 6.10 in [CGO] states that if , then if and only if . It follows that
Modifications in the presence of torsion
Some of the ideas used in the proof of Theorem 2.3 carry over in the presence of torsion, and can be used to extend the realization result, Corollary2.4 (this result can also be deduced easily from [St, Theorem 7.3]).
Let be a -manifold with first Betti number and be a free group of rank . Set . Then any map that induces an isomorphism on Torsion induces a monomorphism
for every not exceeding the Milnor invariant .
The conclusion is trivially true for , so assume that . This means that for some , can be obtained by surgery on an -component diagonal link in with exactly zero-framed components and with .
A presentation for can be calculated along the same lines as in the proof of Lemma 2.7. First set , and let be the free group of rank generated by the meridians of , the first of which correspond to the zero framed components of . As above
for any choice of Milnor words representing the longitudes . By Lemma 2.1, , and so the commutator relations can be ignored. The surgery adds relations for and for , where the are the non-zero framings of the last components of . Since all , we conclude that
Now killing the for defines a surjection , where is free on the for , that induces an isomorphism on Torsion. Therefore the composition
induces an isomorphism on . Since is nilpotent, is surjective***This is a well-known property of nilpotent groups. The idea is that if say generates the target then any commutator can be written as a product of conjugates of terms of the form where . Modulo the conjugations and the and can be ignored using basic properties of commutators. In this way it is inductively shown that is generated by a set of -order commutators in the set . Thus generates for any . Moreover is Hopfian, since it is finitely generated nilpotent, and so is in fact an isomorphism. This implies that is a monomorphism.
If where is torsion free and is a rational homology sphere, then .
Suppose that has Milnor degree and first Betti number . Let , , and be the free group of rank . Then so in particular there is a map .
Express as zero surgery on a link in a homology sphere . Choose a meridional map . Then the composition induces an isomorphism on Torsion. Thus for any , the composition is a monomorphism, by Theorem 2.11. It follows that is a monomorphism. But we saw in the proof of Lemma 2.7 that is always an epimorphism, and so it is in fact an isomorphism. Therefore , and so by Theorem 2.3, as claimed.
We conclude with a strengthening of the realization result, Corollary 2.4, by proving the existence of -manifolds of arbitrary Milnor degree with given first homology of rank , or with three exceptions. Note that we do not address the case because of the number theoretic issues raised in §1.
For any finitely generated abelian group of rank and any with the exception of or when there exists a -manfold with and Milnor degree .
3. Torsion and quantum invariants
It was noted in the introduction that every integral homology sphere has infinite Milnor degree. In contrast, there exist rational homology spheres of arbitrary Milnor degree, as will be seen below using quantum topology techniques. In fact, the same techniques will yield examples of -manifolds with any prescribed Milnor degree and first Betti number, complementing the realization results of the previous section.
The main result in [CM1] relates the Milnor degree of any 3-manifold with its quantum -order (we assume that the reader is familiar with [CM1], and adopt the notation used there) and its mod first Betti number
for any prime . For notational economy we use a rescaling of by dividing by . Then by [CM1, §4.3],
Solving for yields a useful upper bound for the Milnor degree:
For any prime , the Milnor degree of -manifolds satisfies the inequality where and are as defined above.
To apply this result, one must restrict to -manifolds whose -orders can be calculated, or at least estimated. Among these are the manifolds obtained by surgery on iterated Bing doubles of the Hopf link, whose -orders are computable using the methods of [CM1] as shown below.
The (untwisted) Bing double of a link along one of its components is obtained from as follows: First add a -framed pushoff of , and then replace with a pair of linked unknotted components as shown in Figure 4.
To understand how the quantum invariants of surgery on a (framed) link are affected by Bing doubling, we shall appeal to the following quantum calculation:
If is a framed link with a -framed component , and is the Bing double of along with both new components -framed, then .
By equation (21) in [CM1], it suffices to show where
is the -bracket of (see [CM1, §1] for the definitions of the framed quantum integers and the colored Jones polynomials ). Here and are multi-indices of integers, specifying respectively the framings and colorings on the components of .
Allowing colorings from the group ring , as explained in [CM1, §5], the -bracket can be written as a single colored Jones polynomial for a suitable multi-index of elements in , and similarly . In particular , and should all be colored with
since they are -framed, and so setting we have and . In fact there is an alternative color that can be used for any (or all) of the -framed components, namely
For example, . This is a consequence of the “symmetry principle” established in [KM, §4].
Now following [CM1], we say that two -colored framed links (for ) are equivalent, written , provided , and extend this to an equivalence relation on the set of -linear combinations of -colored framed links. We also consider the notion of weak equivalence , defined by the condition .
Noting that , where denotes the distant union with an unknot, it suffices to show This is seen by a sequence of (weak) equivalences. First observe that
where the first equivalence is a special case of equation (24) in [CM1, §5], the second follows from a well known cabling principle (see for example [KM, §3.10]), and the last equality holds since each odd occurs exactly times in the double sum. These equivalences are illustrated below: