A ONE-DIMENSIONAL EMBEDDING COMPLEX
KEVIN P. SCANNELL AND DEV P. SINHA
Abstract.We give the first explicit computations of rational homotopy gro*
*ups of spaces of öl ng knots"
in Euclidean spaces. We define a spectral sequence which converges to the*
*se rational homotopy groups
whose E1term is defined in terms of familiar Lie algebras. For odd k we e*
*stablish a vanishing line for this
spectral sequence, show the Euler characteristic of the rows of this E1te*
*rm is zero, and make calculations
of E2in a finite range.
1.Introduction
In this paper we introduce a spectral sequence which converges to the rationa*
*l homotopy groups of
Emb(I, Rkx I), for k 3, which is the space of embeddings of an interval in Rk*
*x I with fixed endpoints
and tangent vectors at those endpoints (essentially, the space of long knots in*
* Rk+1). Our starting point
is the work of [11], which defines such spectral sequences in terms of the topo*
*logy of configuration spaces.
The paper [11] in turn builds upon work of Goodwillie and his collaborators [4,*
* 5, 14], who have built a
powerful theory for studying spaces of embeddings in general.
The rational homotopy groups of configuration spaces, which comprise the E1 t*
*erm, are Lie algebras
which are well-known. Just as the study of cohomology of embedding spaces gives*
* rise to the study of graph
cohomology, which has been studied extensively [2, 7, 12, 13], our complexes of*
* Lie algebras are interesting
new objects in quantum algebra. Similar complexes were described by Kontsevich *
*in his plenary talk [8].
We start by reviewing the computation of the rational homotopy groups of orde*
*red configurations of
points in Euclidean space as a graded Lie algebra under Whitehead product, as w*
*ell as some basics of
free Lie algebras, which appear as subalgebras of these homotopy groups. At tha*
*t point, we will have
the necessary algebraic background to define the chain complexes which are the *
*rows of the E1 term of
our spectral sequence. It turns out that through E2, our spectral sequence for *
*the homotopy groups of
Emb(I, Rkx I) depends, up to regrading, only on the parity of k. We focus on od*
*d k. We prove some
fundamental facts about these complexes, such as the vanishing of their Euler c*
*haracteristic. We proceed
to describe algorithms for computing the homology of these chain complexes, and*
* in the final section
present the results of these computations in low dimensions. In some cases, the*
* classes which arise in
E2 must survive, implying the existence of non-trivial spherical families of em*
*beddings. Non-zero higher
differentials are also possible. We end with a brief discussion of the case of *
*k even, which pertains to the
theory of finite-type knot invariants.
The second author would like to thank Tom Goodwillie for many helpful discuss*
*ions and Ira Gessel for
help in simplifying the proof of Theorem 4.7.
2. The Rational Homotopy Groups of Configuration Spaces
We remind the reader of the computations of the rational homotopy groups of c*
*onfiguration spaces
[3] and their Lie algebra structure under Whitehead product. Throughout this pa*
*per, ß*(X) will denote
the homotopy groups of X tensored with the rational numbers. Let F(M, n) denote*
* the space of ordered
__________
Date: October 27, 2000.
1991 Mathematics Subject Classification. Primary: 57R40; Secondary: 55T99, 17*
*B70, 57M25, 57M27, 55R80.
The first named author was partially supported by NSF grant DMS-0072515.
1
2 KEVIN P. SCANNELL AND DEV P. SINHA
configurations of n distinct points in a manifold M. We consider the projection*
* æ : F(M, n) ! F(M, n-1)
defined by forgetting the last point in the configuration, which is in fact a f*
*iber bundle whose fiber is
M \ {(n - 1) points}.WLet ' denote the inclusion of the fiber. When M = Rk+1, t*
*he fibers are homotopy
equivalent to n-1Sk, and the projection map admits a section, by adding a poin*
*t (say in a fixed direction
at a large distance) to a configuration of n - 1 points. This section leads to *
*a splitting of the long exact
sequence of a fibration into split short exact sequences
`
0 ! ßi( Sk) ! ßi(F(Rkx I, n)) ! ßi(F(Rk+1, n - 1)) ! 0.
n-1
By induction, we find that additively
n-1M `
ßi(F(Rkx I, n)) ~= ßi( Sk).
j=1 j
We now compute the structure of the rational homotopy groups ß*(F(RkxI, n)) a*
*s a Lie algebra under
the Whitehead product.
Definition 2.1.Let Bon(respectively Ben) be the Lie algebra (super Lie algebra *
*for Ben) generated over Q
by classes xijfor 1 i, j n with relations
1.xij= xji(respectively, -xjifor Ben).
2.xii= 0.
3.[xij, x`m] = 0 if{i, j} \ {`, m} = ;.
4.[xij, xj`] = [xj`, x`i] = [x`i, xij].
We call Bonand Benconfiguration space Lie algebras.
Theorem 2.1.There is a Lie algebra isomorphism between ß*(F(Rkx I, n)) and Boni*
*f k is odd or Benif
k is even.
Proof.We first define classes which generate ß*(F(Rk x I, n)) as a Lie algebra.*
* Pick a basepoint in
F(Rkx I, n), say with zi= (2i, 0, . .,.0) for definiteness. There are n2genera*
*tors of ßk(F(Rkx I, n)),
corresponding to distinct pairs {i, j} {1, . .,.n}, which we now realize geom*
*etrically. We define bij2
ßk(F(Rkx I, n)) as the class represented by the composite of two maps. First, w*
*e collapse Sk onto Sk_ I
by sending the ös uthern hemisphereö f Sk to I through the height function. Ne*
*xt, choose a path flijfrom
zito the point (2j - 1, 0, . .,.0) in the complement of the other configuration*
* points, and let 'jdenote the
map which sends Sk to the unit sphere about the point zj. To define bijwe compo*
*se the collapse map above
with the map Sk_I to F(RkxI, n) which sends t 2 I to F(RkxI, n) as (z1, . .,.zi*
*-1, flij(t), zi+1, . .,.zn)
and Sk to F(Rkx I, n) as v 7! (z1, . .,.zi-1, 'j(v), zi+1, . .,.zn).
To see inductively that these classes are generatorsWof ßk(F(RkxI, n)), we si*
*mply note that binis equal
to the image under '* of the generator of ßk( n-1Sk) defined by the inclusion *
*of the ith wedge factor.
It is simple to check that these bijsatisfy the relations for xijin the defin*
*ition of Bon. Note that from
the usual graded commutativity of the Whitehead product, brackets in bijanti-co*
*mmute when k is odd
and commute when k is even. Note that bij= (-1)k+1bjiand bii= 0 so that relatio*
*ns (1) and (2) are
satisfied.
We next verify that the bijsatisfy relation (3). Recall that if {f} and {g} a*
*re elements of ßk(X) then
[{f}, {g}] = 0 if and only if f _g:Sk_Sk ! X extends to SkxSk. If {i, j}\{`, m}*
* = ;, the map bij_b`m
may be so extended by sending
v x w 7! (z1, . .,.zi-1, 'ij(v), zi+1, . .,.z`-1, '`m(w), z`+1,*
* . .).,
where 'ijis the composite of the collapse map of Sk onto Sk _ I with 'j_ flij. *
*Informally we say that zi
can travel around zjand z`can travel around zm without having their paths (the *
*images of Sk under the
projection onto the ith and `th coordinates) intersect.
A ONE-DIMENSIONAL EMBEDDING COMPLEX 3
Next, we verify that the bijsatisfy relation (4). Equivalently, we claim that*
* [bj`, bij+bi`] = 0. Informally,
we say that bij+ bi`is represented by a map in which zitravels around zj and z`*
* but no other points
in the configuration, and this may happen simultaneously as zj travels around z*
*`, giving an extension of
(bij+ bi`) _ bj`to Skx Sk similar to the one given for bij_ b`m.
We claim that relations (1) through (4) are a complete set of relations for ß*
**(F(RkxI, n)). This follows
from the fact that these relations may be used to reduce to an additive basis o*
*f Lie algebra monomials
of the form [. .[.bim, bjm] . .b.`m.].,.where i, j, ` < m n. We exhibit this *
*claim algorithmically when
discussing the computations in x5; see in particular Algorithm 5.2. _
|_|
The fiber sequence above leads us to identify some subalgebras of ß*(F(Rkx I,*
* n)) whichWare free Lie
algebras. Tensored with the rationals, the homotopy groups of wedges of spheres*
*, ß*+1( jSk) for k > 1,
are well known [6, 15] to form free LieWalgebra under Whitehead product, with j*
* generators in degree
*+1 = k. Since the inclusion map ' : n-1Sk ! F(RkxI, n) is injective on homoto*
*py, and by naturality
of Whitehead products, the image of these homotopy groups under '* in ß*(F(Rkx *
*I, n)) is a free Lie
algebra which is generated by the classes bin.
In the development of the spectral sequence we will in fact need the rational*
* homotopy groups of
FT(k, n) = F(Rkx I, n) x (Sk)n. We call FT(k, n) the space of tangential config*
*urations, thinking of the
points in Sk as unit tangent vectors at points of a configuration. Recall that *
*the homotopy groups of a
product of spaces is a direct sum of their homotopy groups, and all Whitehead p*
*roducts between these
summands are zero. Let o (respectively e) be the free Lie algebra (respective*
*ly super Lie algebra) on
one generator. Let BTondenote Bon n o and similarly for BTen.
Corollary 2.2.There is a Lie algebra isomorphism between ß*+1(FT(k, n)) and BTo*
*nif k is odd or BTen
if k is even.
These isomorphisms respect the gradings involved. We may grade BTonaccording *
*to the number of
generators appearing in a bracket. The dth graded summand of BToncoincides with*
* ßd(k-1)+1(FT(k, n)).
3.Free Lie algebras
Let L(A) denote the free Lie algebra over Q on a set A of symbols. For our ex*
*plicit computations, we
must choose an additive basis for L(A). Natural labels for elements of free Lie*
* algebras can be obtained
from rooted, planar binary trees (hereafter, referred to as simply a trees) wit*
*h leaves labeled by elements
of A. Such a tree prescribes a bracketing of the elements which label the leave*
*s. The number of leaves is
the degree of the tree. Trees with a root but no branches (degree one) are iden*
*tified with the set of symbols
A. When the context is clear, we will identify trees with the free Lie algebra*
* elements they produce.
The obvious product of two trees x and y (a tree with a new root, left subtree *
*x, and right subtree y)
corresponds to the product in the Lie algebra and will therefore also be denote*
*d [x, y].
A set H of trees is called a Hall set for L(A) [9, x4.1] if the following con*
*ditions hold:
1.H has a total order .
2.A H.
3.If h = [h1, h2] 2 H then h22 H and h < h2.
4.For any tree h = [h1, h2] of degree at least two, we have h 2 H if and only*
* if h1, h2 2 H, h1 < h2,
and either h12 A or h1= [x, y] with h2 y.
It is straightforward to show that a Hall set forms an additive basis of L(A)*
* [9, Thm. 4.9] (cf. Algorithm
5.1 below). The basis elements comprising a fixed Hall set will be called Hall *
*trees. It is easy to see that
(many) Hall sets exist [9, Prop. 4.1]; for completeness, we give a quick descri*
*ption of an algorithm for
creating one.
4 KEVIN P. SCANNELL AND DEV P. SINHA
Algorithm 3.1 (Generating a Hall set). Given an ordered list of symbols A and *
*a positive integer d, this
algorithm outputs a list Hd consisting of the elements of degree less than or e*
*qual to d in a Hall set for L(A).
The list Hd will be sorted according to the total order on the Hall set.
1. Set a counter n = 1. Copy the list A into Hd.
2. If n = d, terminate the algorithm and output Hd. Otherwise, proceed.
3. Form all products [h1, h2] such that h1, h22 Hd, h1< h2, the degree of [h1*
*, h2] is n + 1, and condition
(4) is satisfied in the definition of Hall set. The only choice to be made *
*is where to insert this new element
in the ordering on Hd; for definiteness in performing the calculations in s*
*ection 6, we insert [h1, h2] into
Hd as the immediate successor to h1, thus h1< [h1, h2] < h2 as required.
4. Increment n and go to (2).
Example. The output of Algorithm 3.1 with A = {a, b} and d = 5 is the following*
* list:
a [a, [[[a, b], b], b]] [a, [a, [a, [a, b]]]] [a, [a, [[a, b], *
*b]]] [a, [[a, b], b]]
[a, [a, [a, b]]] [a, [a, b]] [[a, [a, b]], [a, b]] [a, b]
[[a, b], [[a, b], b]] [[a, b], b] [[[a, b], b], b] [[[[a, b], b*
*], b], b] b
The following result will be used in computing the Euler characteristic of th*
*e chain complexes which
appear as the rows of the E1 term of our spectral sequence.
Lemma 3.1.[9, Cor. 4.14] The number of Hall trees for L(A) of degree d equals
1_X ~(j)|A|d=j
d j|d
where ~ is the Möbius function.
4.The spectral sequence
In this section we present an explicit realization of the spectral sequence i*
*ntroduced in [11, x4] which
converges to the rational homotopy groups of Emb(I, Rkx I). The spectral sequen*
*ce arises from models
of Emb(I, Rk x I) which are reminiscent of cosimplicial spaces, but whose combi*
*natorics are based on
Stasheff polytopes instead of simplices. The entries are (Fulton-MacPherson com*
*pactified versions of) the
ordered tangential configuration spaces FT(k, n) = F(Rkx I, n) x (Sk)n
We can now describe the E1 term of this spectral sequence. We first describe *
*an unreduced version
(which will be denoted throughout by the addition of a tilde ~E1), followed by *
*the reduced version (denoted
simply E1). Recall that one way to obtain the E1 term of the homotopy spectral *
*sequence of a cosimplicial
space is by first passing to homotopy groups of the entries, which if all entri*
*es are simply connected defines
a cosimplicial abelian group. The E1 term is then the chain complex associated *
*to this cosimplicial abelian
group, which is bi-graded because the homotopy groups themselves are graded. We*
* show in [11] that even
though our models are based on Stasheff polyhedra, applying homotopy groups to *
*these models gives rise
to cosimplicial abelian groups. Hence the ~E1term of our spectral sequence is t*
*he chain complex of the
cosimplicial abelian group:
ß*(FT(k, 0)) = pt. ) ß*(FT(k, 1)) V(ß*(FT(k, 2)) . . .
Here the coface maps di*are induced by maps di on configuration spaces (or ra*
*ther their Fulton-
MacPherson compactifications) which are öd ubling" the ith point in a tangentia*
*l configuration in the
direction of the unit tangent vector determined by the ith factor of Sk, or if *
*i = 0 or n by adding a point
to the configuration at (~0, 0) or (~0, 1) 2 Rkx I. The codegeneracy maps siare*
* defined by forgetting a
point in the configuration.
A ONE-DIMENSIONAL EMBEDDING COMPLEX 5
Theorem 4.1 (see [11]).There is a second-quadrant spectral sequence whose E1 te*
*rm is given by ~E1-p,q=
ßq(FT(k, p)) and d1 given by i(-1)idi*which converges to ß*(Emb(I, Rkx I)).
We now make the coface and codegeneracy maps algebraically explicit. Recall f*
*rom Corollary 2.2 that
the rational homotopy groups of FT(k, n) are isomorphic to the Lie algebra BTon*
*(or BTen) generated by
classes xijand yifor 1 i, j n satisfying xii= 0, xij= (-1)k+1xji, the other*
* relations of Definition 2.1,
and so that [yi, xj`] = 0 for all i, j, ` and [yi, yj] = 0 for i 6= j.
Definition 4.1.Define oe`(i) to be i if i < ` and i+1 if i > `. For 0 ` n+1*
* define @`:BT on! BTon+1
(respectively from BTento BTen+1) to be the Lie algebra homomorphism defined on*
* generators as follows.
(
@`(xij) = xoe`(i)oe`(j) if i, j 6= `
xioe`(j)+ xi+1,oe`(j)if i = `
(
@`(yi) = yoe`(i) if i 6= `
xi,i+1+ yi+ yi+1if i = `
Definition 4.2.For 1 ` n define OE`:BTon! BTon-1(respectively from BTento B*
*Ten-1) to be the
Lie algebra homomorphism defined on generators as follows.
(
OE`(xij) = xoe`(i)oe`(j)if i, j 6= `
0 if i or j = `
(
OE`(yi) = yoe`(i)if i 6= `
0 if i = `
The following proposition is immediate from the definitions of the classes bi*
*jand the maps d`and s`.
Proposition 4.2.Under the isomorphisms of Corollary 2.2 the homomorphisms d`*an*
*d s`*coincide with
@` and OE`respectively.
Making Theorem 4.1 algebraically explicit using Corollary 2.2 and the previou*
*s proposition leads us to
the following spectral sequence whose E1 term is defined in terms of configurat*
*ion space Lie algebras.
Corollary 4.3.There is a second-quadrant spectral sequence which converges to ß*
**(Emb(I, Rkx I)) such
that ~E1-n,d(k-1)+1is isomorphic to the dth graded summand of BTon(respectively*
* BTen), E1-n,q= 0 when
q - 1 is not a multiple of k - 1, and d1 is given by i(-1)i@i.
For the rest of the paper, we focus on the case in which k is odd.
Now we will describe a reduced version of the preceding spectral sequence. Th*
*e standard reduction
of a cosimplicial abelian group proceeds by replacing the nth group by the inte*
*rsection of the kernels of
the codegeneracy maps. Such a reduction does not change the homology of the ass*
*ociated chain complex.
First note that the codegeneracy maps OE`:BTon! BTon-1respect the direct sum de*
*composition BTon=
Bon n o. Restricted to the o factors, the intersection of the kernel of the O*
*E`is zero unless n is equal
to one, in which case it is all of o. Restricted to the Bonfactor, the kernel *
*of the codegeneracy map
OEn:Bon! Bon-1is the subalgebra generated by the classes xin, which is in fact *
*a free Lie algebra (see the
remarks following the proof of Theorem 2.1). We identify the kernel of all of t*
*he OE`as a submodule of this
free Lie algebra.
Definition 4.3.For n > 1, let Md,nbe the submodule of of the degree d summand o*
*f BTongenerated by
brackets of the classes xinsuch that each i from 1 to n - 1 appears as an index*
*. Let M1,1= BTo1.
6 KEVIN P. SCANNELL AND DEV P. SINHA
Theorem 4.4.There is a spectral sequence which converges to ß*(Emb(I, RkxI)) wh*
*ose E1 term is given
by E1-n,d(k-1)+1= Md,nand whose d1 is the restriction to this submodule of the *
*d1 of Corollary 4.3.
Note that Md,n= 0 for d < n - 1, which leads to the following vanishing theor*
*em.
Corollary 4.5.In the spectral sequence of Theorem 4.4, E1-p,q= 0 if q < p(k - 1*
*) + 2 - k.
It is interesting to note that while the modules Md,nmay be defined purely in*
* terms of the free Lie
algebra (on n - 1 generators), the boundary maps between them require extending*
* the free Lie algebra to
a configuration space Lie algebra. From the algebraic definition of d1 it is no*
*t obvious that its restriction
to Md,nmaps to Md,n+1.
Since computing the E2 term amounts to computing the cohomology of the comple*
*xes Md,*, as a
warmup we will compute the rank of Md,n, which we denote R(d, n), and will show*
* that Ø(Md,*)P= 0.
Recall that the number of Hall trees of degree d with n symbols is equal by Lem*
*ma 3.1 to 1_dj|d~(j)nd=j.
We may produce a basis of Md,nby first considering all brackets of degree d and*
* throwing away ones in
which fewer than n elements appear. We find that
Xn `n'X
R(d, n) = 1_d (-1)n-i ~(j)id=j.
i=1 i j|d
P n n
We pause to define S(d, n) = i=0(-1)n-ii id, which are essentially Stirling*
* numbers. There is
a combinatorial interpretation of S(d, n) as the number of surjections from a d*
* element set onto an n
element set (to verify this, count all set maps and subtract the non-surjection*
*s). Note as well that the
S(d, n) have a generating function, as
1X xm
S(m, n)___= (ex- 1)n.
m=0 m!
Reordering the summations of R(d, n) we find the following:
P
Proposition 4.6.R(d, n) = 1_dj|d~(j)S(d=j, n).
We may give R(d, n) a combinatorial interpretation in line with this equality*
* as the number of surjections
of a d element set to an n element set which are not invariant under any cyclic*
* permutation of the d element
set, modulo cyclic permutations of the d element set. It would be interesting t*
*o find a bijection between
such equivalence classes of surjections and a basis of Md,n. Such a combinatori*
*al interpretation would be
particularly interesting for Mn,nwhich, along with its n action by permuting t*
*he letters, is known as
Lie(n) and arises in the calculus of functors approach to homotopy theory [1].
Theorem 4.7.The Euler characteristic of Md,*is zero for d > 2.
Pd
Proof.The Euler characteristic of the complex Md,*is by definition `=1(-1)`R(d*
*, `), which after applying
Proposition 4.6, reversing the order of summation, and ignoring zero terms, is *
*equal to
1_X~(j) d=jX(-1)`S(d=j, `).
dj|d `=1
Pm
WePclaim that `=1(-1)`S(m, `) = (-1)m, which can be verified by computing the *
*coefficient of xm=m! of
m p x p 1_P d=j
P p=0(-1) (e -1) . Hence the Euler characteristic isPequal to d j|d~(j)(-1) .*
* Let t(d) = dØ(Md,*) =
j|d~(j)(-1)d=j. Applying Möbius inversion we find j|dt(j) = (-1)d. Computin*
*g that t(1) = -1 and_
t(2) = 2, we see by induction that t(d) = 0 if d > 2, proving the theorem. *
* |_|
A ONE-DIMENSIONAL EMBEDDING COMPLEX 7
5.Algorithms
In this section we provide a detailed description of the methods used to comp*
*ute the boundary operators
in the complexes described above. These algorithms can be performed by hand for*
* the complexes of small
degree d, but are best implemented on the modern electronic computer otherwise.
Because the product of two Hall trees is not necessarily a Hall tree, one mus*
*t have an algorithm which
takes an arbitrary tree representing a free Lie algebra element, and expresses *
*it as a linear combination of
Hall trees. The proof that this algorithm terminates and produces the desired r*
*esult is contained in the
proof of Theorem 4.9 in [9].
Algorithm 5.1 (Hallification). Given an integral linear combination of trees r*
*epresenting an element of
L(A), this algorithm outputs a linear combination of Hall trees representing th*
*e same element of L(A).
1. If each tree appearing with a non-zero coefficient in the linear combinati*
*on is Hall, terminate the algorithm
and output the linear combination. Otherwise, choose t to be the first non-*
*Hall tree appearing in the
linear combination and proceed.
2. Find a subtree s = [s1, s2] of t which is not Hall but whose children s1 a*
*nd s2 are Hall. This can be
achieved by a simple recursion, noting that the degree one trees (single le*
*tters) are Hall.
3. If s1= s2, then remove t from the linear combination and go to step (1).
4. If s1> s2, then switch s1 and s2 in t, multiply the coefficient of t by -1*
*, and go to step (1).
5. We have s1< s2. In this case, s1 cannot be a single letter, or else s woul*
*d be Hall. So s1= [x, y]. We
must have y < s2again using the fact that s is not Hall. Replace t in the l*
*inear combination by the sum
of two trees obtained by replacing s = [[x, y], s2] by [[x, s2], y] and [x,*
* [y, s2]] respectively and go to step
(1).
The following algorithm uses the relations for Bonfrom Definition 2.1 and the*
* Jacobi identity to express
elements of Bonin a standardized form. It will be used in the computation of th*
*e boundary operator @n in
Algorithm 5.3 below.
Definition 5.1.We say that a bracket in the classes xijfor 1 i, j n is pure*
* if either all xijwhich
appear are of the form xinor none are of this form.
Algorithm 5.2 (Standard basis for Bon). Given an element x of Bonexpressed as *
*a linear combination of
brackets in the xij, this algorithm computes a linear combination of pure brack*
*ets also representing x.
1. If each bracket appearing with a non-zero coefficient in the linear combin*
*ation is pure, terminate the
algorithm and output the linear combination. Otherwise, choose t to be the *
*first bracket in the linear
combination which is not pure and proceed.
2. Find a smallest degree sub-bracket s = [s1, s2] of t which is not pure. A *
*simple recursion finds this
sub-bracket.
3. If the degree of s is two, go to step (4), otherwise go to step (7).
4. Since the degree of s is two, we have s1= xijand s2= x`m with either j = n*
* or m = n. If j = n, go
to step (5) and if m = n, go to step (6).
5. If i = `, then replace t in the linear combination by a new bracket obtain*
*ed from t by replacing
s = [xin, xim] with [xmn, xin], using relation (4) in the definition of Bon*
*. If i = m, then do the same
thing, replacing s = [xmn, x`m] with [x`n, xmn] by the same relation. In al*
*l other cases, remove t from
the linear combination (applying relation (3) in the definition of Bon). St*
*art over at step (1).
6. If i = `, then replace t in the linear combination by a new bracket obtain*
*ed from t by replacing
s = [xij, xin] with [xin, xjn]. If ` = j, then do the same thing, replacing*
* s = [xij, xjn] with [xjn, xin].
In all other cases, remove t from the linear combination. Start over at ste*
*p (1).
7. If the degree of s1is greater than one, say s1= [x, y], we use the Jacobi *
*identity to replace t in the linear
combination by the sum of two brackets obtained from t by replacing the sub*
*-bracket s = [[x, y], s2] by
8 KEVIN P. SCANNELL AND DEV P. SINHA
[[s2, y], x] and [[x, s2], y] respectively. If s1 has degree one, then s2 m*
*ust have degree at least two, say
s2= [x, y], and we do the same thing, replacing s = [s1, [x, y]] by [x, [s1*
*, y]] and [y, [x, s1]], respectively,
and adding the results. In either case, start over at step (1).
A simple induction argument shows that this argument terminates and produces *
*the desired result.
Namely, we associate to a bracket t the pair (a, b) where a is the number of ge*
*nerators xijwith j < n
appearing in t, and b is the degree of the smallest impure sub-bracket found in*
* step (2). We order such
pairs lexicographically, with the minimum (0, 0) being achieved by pure bracket*
*s. At every step, this
algorithm produces brackets whose associated pairs are less than that of the or*
*iginal. Steps (5) and (6),
corresponding to b = 2, clearly reduce a. Step (7) leaves a unchanged but reduc*
*es b, since the sub-bracket
[x, y] of s which is initially pure becomes impure in all terms which occur aft*
*er applying the Jacobi identity.
Finally note that since b in such an associated pair is bounded by the degree o*
*f the bracket, there are
only finitely many pairs less than a given one, so the algorithm must terminate*
* after a finite number of
recursive steps.
Note that the terms in the linear combination output by Algorithm 5.2 which d*
*o not involve any xin
can be run recursively through the algorithm as elements of Bon-1, yielding the*
* standard form claimed in
the proof of Theorem 2.1.
The final algorithm is the heart of the calculation; it computes @` for ` = 0*
*, . .,.n, exploiting the fact
that these maps are Lie algebra homomorphisms. Observe that @n+1is simply the n*
*atural inclusion of
BT oninto BTon+1and therefore requires no detailed description.
Algorithm 5.3 (Boundary operator). Given a basis element t of Md,n(expressed a*
*s a Hall tree) and an
integer ` between 0 and n, this algorithm computes @`(t) as a linear combinatio*
*n of degree d elements of Bon+1
in the standard form given by Algorithm 5.2.
1. If the degree of t is greater than one, say t = [t1, t2], then recursively*
* call Algorithm 5.3 to compute
@`(t1) and @`(t2). Set @`(t) = [@`(t1), @`(t2)] and proceed to step (2). If*
* the degree of t is one, go to
step (4).
2. If ` = n, then use Algorithm 5.2 to express the answer @`(t) in standard f*
*orm. Proceed to step (3).
3. Use Algorithm 5.1 to express @`(t) in terms of Hall trees. Terminate the a*
*lgorithm and return @`(t).
4. If ` < n, proceed to step (5), otherwise go to step (6).
5. Assume t = xin. If i < `, set @`(t) = xi,n+1. If i > `, set @`(t) = xi+1*
*,n+1. If i = `, set
@`(t) = xi,n+1+ xi+1,n+1. Go to step (3).
6. Assume t = xin. Set @`(t) = xin+ xi,n+1and go to step (2).
An example of this algorithm is worked out by hand in the next section.
6.Results
In this section we present some results of the computations described in the *
*previous section. We will
choose the gradings to correspond to the case k = 3, i.e. embeddings in R4.
First we note that in the degree one case, we have E1-1,3= Q, generated by y1*
*, E1-2,3= Q generated
by x12, and d1 is an isomorphism. In degree two, the only non-zero entry is E1-*
*3,5= Q, generated by
[x13, x23], implying E2-3,5= Q.
We proceed by working out the first non-trivial boundary operator d1: E1-3,7!*
* E1-4,7by hand. These
spaces are by definition M3,3and M3,4. Bases are obtained by creating, with Alg*
*orithm 3.1, Hall bases for
the free Lie algebra generated by {x13, x23} (resp. {x14, x24, x34}) and select*
*ing the elements which have
degree 3 and such that all possible values of i appear. It turns out that each *
*space is two-dimensional; the
first is generated by [x13, [x13, x23]] and [[x13, x23], x23] and the second by*
* [x14, [x24, x34]] and [[x14, x34], x24].
A ONE-DIMENSIONAL EMBEDDING COMPLEX 9
Algorithm 5.3 is straightforward for ` 6= 3; in these cases we have:
@0[x13, [x13, x23]] = [x24, [x24, x34]]
@0[[x13, x23], x23] = [[x24, x34], x34],
while
@1[x13, [x13,=x23]][x14+ x24, [x14+ x24, x34]]
= [x14+ x24, [x14, x34] + [x24, x34]]
= [x14, [x14, x34]] + [x14, [x24, x34]] + [x24, [x14, x34]]*
* + [x24, [x24, x34]]
= [x14, [x14, x34]] + [x14, [x24, x34]] - [[x14, x34], x24]*
* + [x24, [x24, x34]]
and
@1[[x13, x23],=x23][[x14+ x24, x34], x34]
= [[x14, x34] + [x24, x34], x34]
= [[x14, x34], x34] + [[x24, x34], x34],
where the last line in the first case comes from an application of Algorithm 5.*
*1 for the free Lie algebra
over {x14, x24, x34}. Similarly we have for ` = 2
@2[x13, [x13,=x23]][x14, [x14, x24]] + [x14, [x14, x34]]
@2[[x13, x23],=x23][[x14, x24], x24] + [[x14, x24], x34] + [[x14, x34], x*
*24] + [[x14, x34], x34]
= [[x14, x24], x24] + [[x14, x34], x34] + 2 * [[x14, x34], x2*
*4] + [x14, [x24, x34]],
where again the last line comes from Algorithm 5.1. Finally, as noted above, @4*
* is the natural inclusion:
@4[x13, [x13, x23]] = [x13, [x13, x23]]
@4[[x13, x23], x23] = [[x13, x23], x23].
The case ` = 3 is much more computationally taxing, as it requires the use of A*
*lgorithm 5.2:
@3[x13, [x13,=x23]][x13+ x14, [x13+ x14, x23+ x24]]
= [x13, [x13, x23]] + [x14, [x14, x24]] + [x24, [x34, x14]] + [x1*
*4, [x34, x24]] (Alg. 5.2)
= [x13, [x13, x23]] + [x14, [x14, x24]] + [[x14, x34], x24] - [x1*
*4, [x24, x34]] (Alg. 5.1)
@3[[x13, x23],=x23][[x13+ x14, x23+ x24], x23+ x24]
= [[x13, x23], x23] + [[x14, x24], x24] + [[x14, x34], x24] + [[x*
*24, x34], x14] (Alg. 5.2)
= [[x13, x23], x23] + [[x14, x24], x24] + [[x14, x34], x24] - [x1*
*4, [x24, x34]] (Alg. 5.1)
Since d1= i(-1)i@i, we have from the above calculations that
d1[x13, [x13,=x23]]0
d1[[x13, x23],=x23]2 * [x14, [x24, x34]] + [[x14, x34], x24].
and so the matrix for the boundary operator with respect to our chosen bases is*
* given by (0201). We
conclude that the boundary operator has rank one, and so E2-3,7~=E2-4,7~=Q. Fu*
*rther (computer)
calculations yield the E1 and E2 terms for k odd given in Tables 1 and 2.
10 KEVIN P. SCANNELL AND DEV P. SINHA
Q120|Q300|Q260|Q89|_Q9_|___|___|13_|
_____|____|____|___|____|___|___|12_|2448306
_____|Q___|Q___|Q__|Q__|____|___|11_|
_____|____|____|___|____|___|___|10_|693
_____|____|Q__|_Q__|Q__|____|___|9_|
_____|____|____|___|____|___|___|8_|22
_____|____|____|Q__|Q__|____|___|7_|
_____|____|____|___|____|___|___|6_|
_____|____|____|___|_Q__|___|___|5_|
_____|____|____|___|____|___|___|4_|
_____|____|____|___|____|Q__|Q__|3_|
__-7__|-6__|-5__|-4_-|3_-2|_|-1__|_|
Table 1. E1 term for k = 3
_Q_|_Q_|__|___|___|__|13_|
____|__|___|___|__|___|12_|2
____|Q__|Q_|___|Q_|___|11_|
____|__|___|___|__|___|10_|
____|__|___|___|__|___|9_|
____|__|___|___|__|___|8_|
____|__|___|Q_|_Q_|___|7_|
____|__|___|___|__|___|6_|
____|__|___|___|Q_|___|5_|
____|__|___|___|__|___|4_|
____|__|___|___|__|___|3_|
_-7__|-6-|5_-|4-3|_|-2__||_
Table 2. E2 term for k = 3
These low-dimensional computations do not reveal any regular behavior. Note t*
*hat, as allowed because
the Euler characteristic of the rows is zero, some rows vanish while most do no*
*t. Note as well that there
is no additional vanishing along the edge of the vanishing line of Corollary 4.*
*5.
All of the classes in Table 2 survive to E1 except perhaps those in bidegrees*
* (-6, 13) and (-3, 11)
which could support a d3 differential.
Theorem 6.1.There are non-trivial classes in ßn(Emb(I, R3x I)) for n = 2, 3, 4,*
* 5, 6.
It would be interesting to find explicit spherical families of embeddings whi*
*ch represent these classes.
One expects the evaluation map
nx Emb(I, Rkx I) ! F(Rkx I, n) x (Sk)n
to play a central role in relating these homotopy groups to those of F(Rkx I, n*
*) x (Sk)n which appear in
our spectral sequence.
We conclude with a brief description of some of the methods used to verify th*
*e computer calculations
(beyond merely computing examples by hand and comparing with the computer outpu*
*t, which was done
extensively). Algorithm 3.1 was checked by an independent function which verifi*
*ed that the generated
trees were Hall, and checked the number of elements in the resulting Hall set a*
*gainst the dimension count
A ONE-DIMENSIONAL EMBEDDING COMPLEX 11
given by Lemma 3.1. It was verified in the course of computing the E2 term in T*
*able 2 that d2 = 0 for
each of the chain complexes comprising E1. A similar mathematical fact which wa*
*s not hard-coded into
the application is that the image of Md,nunder d1 lands in Md,n+1despite the fa*
*ct that this is not the
case for the individual homomorphisms @`. The ranks of the boundary operators w*
*ere verified using the
linear algebra capabilities of a symbolic mathematics package (Maple). Finally,*
* a nice check of the system
as a whole was provided by varying the algorithm for generating Hall sets (noti*
*ng the choices made in
Algorithm 3.1) and verifying that the ranks of all boundary operators remained *
*unchanged.
7.Further Work
In further work [10] we will investigate the case of k even, which includes t*
*he case of classical knots.
Though the spectral sequences of [11] do not necessarily converge, one can use *
*those methods to produce
knot invariants, which we show are of finite type. In particular, an optimistic*
* view of rational homotopy
theory predicts that the module of classes along the vanishing line of our spec*
*tral sequence (which for k = 2
is the anti-diagonal) is isomorphic to the module of primitives in the Hopf alg*
*ebra of finite type invariants
[2]. To prove such a conjecture would involve relating the combinatorics of con*
*figuration space Lie algebras
to those of Feynman diagrams, which could give a satisfactory explanation in te*
*rms of algebraic topology
of the appearance of Feynman diagrams in the study of knots.
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Department of Mathematics and Computer Science, Saint Louis University, St. L*
*ouis, MO 63103
E-mail address: scannell@slu.edu
Department of Mathematics, Brown University, Providence, RI 02906
E-mail address: dps@math.brown.edu