Finite type invariants based on the band-pass and doubled delta moves

2.6 MB·English

by  James Conant

FINITE-TYPE KNOT INVARIANTS BASED ON THE BAND-PASS AND
DOUBLED-DELTA MOVES
JAMESCONANT,JACOBMOSTOVOY,ANDTEDSTANFORD
9
0 Abstract. Westudygeneralizationsoffinite-typeknotinvariantsobtainedbyreplacingthecrossingchange
0 intheVassilievskeinrelationbysomeotherlocalmove.
2 First, we represent the local moves by normal subgroups of the pure braid group P∞. Subgroups that
are stable under the “strand-tripling” endomorphisms are shown to produce finite-type invariants with
n
familiarproperties;inparticular,generalizedGoussarov’sn-equivalenceclassesofknotsformgroupsunder
a
theconnectedsum. (Similarresults,butwithadifferentapproach,havebeenobtainedbeforebyTaniyama
J
andYasuhara.)

4 Treating local moves as surgeries on claspers, we study two particular cases in detail: the band-pass
1 and the doubled delta move. While the band-pass move gives only one “new” invariant (namely, the Arf
invariant),theinvariantscorrespondingtothedoubled-deltamovecontaininformationwhichisnotavailable

] toanyfinitecollectionofVassilievinvariants.
T
The complete degree 0 doubled-delta invariant is the S-equivalence class of the knot. In this context,
G we generalize a result of Murakami and Ohtsuki to show that the only primitive Vassiliev invariants of S-
. equivalencetakingvaluesinanabeliangroupwithno2-torsionarisefromtheAlexander-Conwaypolynomial.
h
Tothisend,weintroduceadiscretelogarithmwhichtransformsthecoefficientsoftheConwaypolynomial
t
a intoprimitiveinteger-valuedinvariants. Asforthehigherdegreedoubled-deltainvariants,westartanalyzing
m thembyconsideringwhichVassilievinvariantsaredegree1asdoubled-deltainvariants. Wefindthatthere
isexactlyoneVassilievinvariantineachodddegreewhichisofdoubled-deltadegreeone, whereasineach
[ even degree there is at most a Z2-valued invariant, which we show exists in degree 4. For higher doubled
deltadegrees,weobservethattheEulerdegreen+1partofGaroufalidisandKricker’srationalliftofthe
2
Kontsevichintegralisadoubled-deltadegree2ninvariant. Finally,thedoubled-deltamoveisaspecialcase
v
ofGaroufalidisandRozansky’snull-moveofpairs(M;k)whereM isahomology3-sphereandk aknotin
9
M. A consequence of our work is that n-equivalence classes of pairs (S3;k) with respect to the null-move,
8
do,infact,formagroup.
1
1
1
5
0 1. Introduction
/
h
It has been observed that the theory of finite-type knot invariants can be generalized by replacing the
t
a crossing change appearing in the Vassiliev skein relation by some other local move.
m Vassiliev’sknot-spacedefinitionoffinite-typeinvariantsprovideslittlemotivationforsuchgeneralization.
: Indeed, a crossing change on a knot diagram corresponds to a codimension one singularity of a map from a
v
circle to a 3-sphere, while other local moves correspond to singularities of higher codimension.
i
X Nevertheless, the theories of finite-type invariants based on local moves other than crossing changes fit
r perfectly with the approach developed in the works of Goussarov [9, 10, 11] and later, Habiro [12] and
a
Stanford [18]. In particular, Goussarov’s notion of n-equivalence can be extended to a wide class of local
moves; this generalized notion of n-equivalence is consistent with suitably defined finite-type invariants, and
the n-equivalence classes of knots form groups under the connected sum operation. See Stanford [19] for
a particular example of such situation and Taniyama and Yasuhara [21] for a more general treatment of
theories based on local moves.
The local moves considered in [19] and [21] essentially consist of replacing one fixed subtangle of a link
by another fixed subtangle. A different approach to local moves can be taken by defining local moves as
surgeries on claspers. These local moves can change the ambient manifold. Finite-type invariants based on
2000 Mathematics Subject Classification. 57M25.
Key words and phrases. finite-typeinvariants,localmoves,band-passmove,doubled-deltamove.
ThefirstauthorwaspartiallysupportedbyNSFgrantDMS0305012.
1
2 J.CONANT,J.MOSTOVOY,ANDT.STANFORD
a move of this type (the “null-move”) were considered by Garoufalidis and Rozansky in [8]. It was shown in
[8, 6] that the rational lift of the Kontsevich integral is a universal rational finite-type (with respect to the
null-move) invariant of knots in integral homology spheres with trivial Alexander polynomial.
Apart form the null-move, the only local moves for which the corresponding finite-type invariants have
been identified to some extent are the C -moves. Taniyama and Yasuhara proved in [20] that the space
k
of primitive C -finite-type invariants of order n and smaller coincides with the space of usual primitive
k
finite-type invariants of order kn and smaller.
The present paper consists of two parts. In the first part we show how local moves on knots can be
interpretedviapurebraidclosures. Thenotionofalocalmoveisreplacedherebythenotionofamodification
byanelementofanormalsubgroupofthepurebraidgrouponaninfinitenumberofstrandsP (Thisisthe
∞
direct limit of finite pure braid groups, each included in the next by adding a trivial strand.) Each normal
subgroup G ⊂ P gives rise to G-finite-type knot invariants and to the relation of γ G-equivalence on the
∞ n
isotopy classes of knots. (Here γ refers to the nth term of the lower central series.) Using a Markov-type
n
theorem for the short-circuit closure of [15], we show the following
Theorem 1. If the subgroup G is stable under “strand-tripling”, then
(i) two G-trivial knots are γ G-equivalent if and only if they cannot be distinguished by G-finite-type
n
invariants of orders n and smaller;
(ii) the set of G-trivial knots modulo γ G-equivalence is an abelian group.
n
Here a knot is said to be G-trivial if it is G-equivalent to the unknot. The precise definition of the
strand-tripling endomorphisms of P is given in Section 2.
∞
The above result generalizes the well-known theorems of Goussarov [9]. It is very similar, though not
immediately equivalent, to a theorem by Taniyama and Yasuhara [21]. It is probable that our result can be
obtained using methods of [21]; nevertheless, we think that our approach is of independent interest.
We also re-state Theorem 1 in terms of claspers; this enables us to apply it to finite-type invariants based
on the null move.
In the second part we study the finite-type invariants based on two particular local moves: the band-pass
move and the doubled-delta move.
We prove that the finite-type invariants with respect to the band-pass move essentially coincide with the
Vassiliev invariants. More precisely, primitive finite-type invariants with respect to the band-pass move of
order n coincide with primitive Vassiliev invariants of order n for n ≥ 1. The Arf invariant is the unique
band-pass-finite-type invariant of order 0.
The case of the doubled-delta move turns out to be more interesting.
Inwhatfollows,weshallabbreviate“doubled-delta-finite-typeinvariantsofordern”to“∆∆ -invariants”.
n
These are quite interesting already for n = 0: it has been proved by Naik and Stanford in [16] that two
knots are S-equivalent if and only if one can be transformed to the other by a sequence of doubled delta
moves. Therefore, ∆∆ -invariants coincide with the invariants of S-equivalence. Note that the Seifert form
0
of a knot is a complete invariant of S-equivalence; in particular, S-trivial knots are exactly the knots with
trivial Alexander polynomial.
OnemayaskwhichVassilievinvariantsarealso∆∆ -invariants. ItwasshownbyMurakamiandOhtsuki
0
in [14] that all Q-valued Vassiliev invariants that are invariants of S-equivalence are polynomials in the
coefficients of the Conway polynomial. Here we extend their result to primitive Vassiliev invariants with
values in any abelian group A with no 2-torsion.
To this end, we define a logarithm-like function
log : 1+z·Z[[z]]→z·Z[[z]]
Z
whichtakesmultiplicationtoaddition. Thisis,asfarasweknow,anovelconstructionwhichshouldbequite
useful in studying primitive Vassiliev invariants over the integers. Applying log to the Conway polynomial
Z
C(z), the coefficients are non-zero only for even degrees of z. Denote by pc the coefficient at z2n.
2n
FINITE-TYPE INVARIANTS BASED ON LOCAL MOVES 3
Theorem 2. For each n > 0, pc is both a ∆∆ -invariant and a primitive Vassiliev invariant. Every
2n 0
primitive Vassiliev knot invariant taking values in an Abelian group with no 2-torsion, which is also a ∆∆ -
0
invariant, is a linear combination of the pc . As a Vassiliev invariant, pc has order 2n, and pc mod 2
2n 2n 2n
has order 2n−1.
Here we should clarify what it means for an A-valued invariant f to be a linear combination of the pc :
2n
this is said to be the case if there exists a homomorphism φ : Zk → A for some k, such that f is the
composition of the direct sum of the pc , for 0<n≤k, with φ.
2n
Proceeding to ∆∆ invariants, we show
1
Theorem 3.
• The associated graded Q-module of primitive rational-valued Vassiliev invariants which are also
∆∆ -invariants, has exactly one generator in degree 2n+1 for each n ≥ 1, and no generators in
1
even degrees.
• Any integer-valued Vassiliev invariant of order 4 is, modulo 2, a ∆∆ -invariant.
1
It is possible that there are more 2-torsion invariants in larger even degrees; we shall see, however that at
most one copy of Z/2Z can exist in each even degree.
OnecouldassembletheZ-valuedinvariantsintoapowerserieswhichisa∆∆ -invariantbutnotaVassiliev
1
invariant, but we leave open the question of whether there exist ∆∆ -invariants which do not come directly
1
from Vassiliev invariants in this way.
We shall not discuss ∆∆ -invariants with n > 1 in detail, though not for the lack of examples. Indeed,
n
the ∆∆-move is a special case of a null-move that preserves the ambient 3-manifold (see [6, 8]). Above we
have mentioned that the rational lift of the Kontsevich integral, Zrat, is a universal rational null-finite-type
invariant of S-trivial knots. This implies the following theorem.
Theorem 4. Zrat is a ∆∆ -invariant. Therefore, the vector space of rational ∆∆ -invariants modulo
i+1 2i 2n
∆∆ -invariants is infinite-dimensional.
2n+1
Some notation. The band-pass and the doubled-delta moves will be denoted by BP and ∆∆ respectively;
V willsometimesbeusedforthecrossingchange. (Here“V”standsfor“Vassilievinvariants”.) Thenotation
Z[X]willbeusedforthefreeZ-algebraonamonoidX. ThelowercentralseriesofagroupH willbedenoted
by γ H, indexed so that γ H =H. P will denote the pure braid group on N strands; the trivial braid on
k 1 N
N strands will be written as 1 .
N
2. Local Moves and Finite-Type invariants
There are at least three different ways to define local moves on knots. The definitions we give below are
not equivalent; however, all “interesting” local moves, such as the band-pass, doubled delta, C -moves and
k
many other moves can be obtained from all three constructions.
2.1. Local moves and tangles. The most intuitive approach to local moves is via tangles. The following
definition can be found, for example, in [21].
Let T and S be two tangles in an oriented closed ball U. Assume that neither T nor S contain closed
components and denote by t ,t ,...,t and s ,s ,...,s the arcs of T and S respectively. Suppose that
1 2 k 1 2 k
for each t there exists some s such that ∂t = ∂s . Then the ordered pair (T,S) is called a local move.
i j i j
Two local moves (T,S) and (T(cid:48),S(cid:48)), defined in 3-balls U and U(cid:48) respectively, are equivalent if there exists
an orientation preserving homeomorphism h:U →U(cid:48) such that h(T)=T(cid:48) and h(S) is ambient isotopic to
S(cid:48) relative to ∂U(cid:48).
Let K and K be knots in an oriented 3-manifold M. We say that K is obtained from K by applying
1 2 2 1
a local move (T,S) if there exists a ball U in M such that
(i) ∂U intersects both knots transversely,
(ii) K and K coincide outside U
1 2
(iii) the pair (U ∩K ,U ∩K ) is a local move equivalent to (T,S).
1 2
4 J.CONANT,J.MOSTOVOY,ANDT.STANFORD
↔
↔
Delta move ∆
Crossing change V
↔ ↔
Band-pass move BP Doubled-delta move ∆∆
Figure 1. Basic local moves on knots.
(cid:15)(cid:15)
(cid:31) (cid:47)(cid:47)
(cid:15)(cid:15)
Figure 2. The short-circuit closure of a braid.
A theory of local moves based on this definition is developed in [20, 21]. We shall not give further details
here.
2.2. Local moves and braids. There are several kinds of braid closures. We shall use the “short-circuit”
closure of [15]; this is a version of the plat closure which sends pure braids on an odd number of strands
to (long) knots. See Figure 2. The short-circuit closure commutes with adding two unbraided strands to a
braid, hence, there is a map
S :P →Knots.
∞
FINITE-TYPE INVARIANTS BASED ON LOCAL MOVES 5
A Markov-type theorem for the short-circuit closure is proved in [15]: it says that there are two subgroups
HT and HB of P such that the map S identifies the set of isotopy classes of knots with the two-sided
∞
quotient HT\P /HB.
∞
Let G ⊂ P be a normal subgroup. Two knots a and b are G-equivalent if there exist x,y ∈ P and
∞ ∞
h∈G such that a=S(x), b=S(y) and x=hy. The Markov Theorem for the short-circuit closure implies
that G-equivalence is indeed an equivalence relation on the set of isotopy classes of knots. All knots are
P -equivalent.
∞
An informal explanation of this definition in terms of local moves is that G should be thought of as the
subgroupofP ,consistingofthebraidsthatcanbeundonebysomefixedlocalmove. Forexample,crossing
∞
changes can undo all braids in P ; delta moves undo the commutator subgroup [P ,P ] (see [19]). Then
∞ ∞ ∞
one can think of G-equivalent knots as those that can be transformed into each other by a sequence of local
moves that undo G.
Related to G-equivalence is the notion of a G-finite-type invariant. Let I be the augmentation ideal of
G
G inside Z[G], and denote by I(cid:98)G the ideal in Z[P∞] generated by IG. The ideal I(cid:98)G is the kernel of the ring
homomorphism Z[P ]→Z[P /G] sending elements of P to P /G.
∞ ∞ ∞ ∞
A linear function Z[Knots]→A, where A is an abelian group, is a G-finite-type invariant of order n if it
vanishes on all elements of Z[Knots] which are of the form S(a) with a∈I(cid:98)Gn+1. For G =P∞ this defines the
usual Vassiliev invariants. Notice that all the knots within the linear combination S(a) are G-equivalent, so
it makes sense to define a G-finite-type invariant on a single G-equivalence class. In particular, G-finite-type
invariants of order n for G-trivial knots can be defined as those that vanish on S(In+1).
G
Denotebyτ :P →P thehomomorphismofshiftingthebraidbytwostrands“totheright”. Inother
0 ∞ ∞
words, τ sends the braid x to 1 ⊗x. For k >0 let τ be the homomorphism of P into itself that triples
0 2 k ∞
the k-th strand.
Theorem 5. Let G be a normal subgroup of P such that τ (G)⊂G for all k (cid:62)0. Then
∞ k
(i) G-trivial knots considered modulo γ G-equivalence form a group under connected sum;
n
(ii) two G-trivial knots are γ G-equivalent if and only if they cannot be distinguished by G -invariants.
n <n
The proof of this theorem will be given in Section 3.
2.3. Local moves and claspers. Weshallassumethatthereaderisfamiliarwiththelanguageofclaspers.
For definitions and properties of claspers we refer to [12, 4]. We shall use the terminology of [4].
Let T be an abstract (i.e. not embedded) clasper of a fixed tree type. Then a basic T-move on a pair
(M,k) where k is a knot embedded in a three-manifold M, is a surgery on an embedded T-clasper in M\k,
which produces a new pair (M(cid:48),k(cid:48)). A cT-move is a T-clasper surgery where the leaves bound disjoint disks
(called caps) which may hit the knot. Note that such moves preserve the ambient 3-manifold. An nT-move
is a T-move where the clasper’s leaves link the knot homologically trivially. (Here “n” stands for “null.”)
Note that this makes sense when M is a homology sphere. Finally an ncT-move is a T-move where the
leaves bound disjoint caps that intersect the knot algebraically trivially.
LettheclasperwithtreetypeanintervalbedenotedI,andtheclasperwithtreetypea“wye”bedenoted
Y. Then what follows is a table of other names for the above moves. See Figure 1.
c n nc
I Crossing change (V) Band-pass move (BP)
Y Delta move (∆) Null-move (N) Doubled-delta move (∆∆)
ThistableshouldbeinterpretedinthesensethateverycI-movecanberealizedbyasequence ofcrossing
changes and similarly for the other entries.
Let µ denote a move type of the form cT or ncT. Then there is a descending filtration
Z[Knots]=Fµ ⊃···⊃Fµ ⊃Fµ ⊃··· ,
0 n n+1
6 J.CONANT,J.MOSTOVOY,ANDT.STANFORD
where Fµ is defined as follows. Given a knot k, and n disjointly embedded µ-claspers C ,...,C , with
n 1 n
disjointly embedded caps. Let
(cid:88)
[k;C ,...,C ]= (−1)|σ|k ,
1 n σ
σ⊂[n]
where k means “modify k by the claspers C ” where i ∈ σ. Then Fµ is defined to be the submodule
σ i n
generated by such elements [k;C ,...,C ] for all choices of k and {C }. By Habiro’s zip construction and
1 n i
standard arguments, in the cT case the submodule Fµ is actually generated by such alternating sums where
n
each clasper is simple. (That is, each of the caps intersect the knot in a single point.) In the ncT case one
may assume each cap intersects the knot in a pair of algebraically canceling points.
Let Z[Knots] be the quotient of Z[Knots] by the subspace generated by the vectors
k +k −k #k
1 2 1 2
for all pairs of knots. Let Fµ ⊂ Z[Knots] be the image of Fµ. Linear functions Z[Knots] → G are exactly
n n
the primitive (that is, additive) knot invariants.
Definition 6.
(i) Aninvariantf: Knots→A,whereAisanabeliangroup,issaidtobeµ-finite-typeofdegree(order)
n if f(Fµ )=0. We also say that f is a µ -invariant.
n+1 n
(ii) Two knots k and k are µ -equivalent if k −k ∈Fµ . In other words, µ -equivalent knots are
1 2 n 1 2 n+1 n
those that cannot be distinguished by µ invariants. By Gµ we denote the set of knots which are
≤n n
µ -equivalent to the unknot, considered modulo µ -equivalence.
0 n
Theorem 5 implies the following statement.
Theorem 7. Given a tree T which is a Y or an I, let µ be a move of the type cT or ncT. Then Gµ is an
n
abelian group for all n.
The language of claspers facilitates the treatment of the null-move N. One can define a null-finite-type
invariant of degree n to be an invariant (with values in some abelian group) of pairs (M,k) which vanishes
on alternating sums of n+1 null moves, and analogously define N -equivalence. Let GN be the set of pairs
n n
(M,k) which are N -equivalent to the pair (S3,unknot), considered modulo N -equivalence. Note that GN
0 n n
is a monoid under the operation of connected sum, where the 3-ball that is removed hits the knot in a
standard arc. Let GN(S3) be the submonoid of pairs (S3,k).
n
Theorem 8. GN(S3) is a group for all n.
n
Proof. LetX denotethesetofpairs(S3,k)whichareN -equivalenttothepair(S3,unknot). Weclaimthat
0
X is the set of all pairs (S3,k) where k has trivial Alexander polynomial. Every such pair is in X by [16].
Conversely, any element in X will have trivial Alexander polynomial.
ButthenX ispreciselythesetofknotsinS3 whichare∆∆ equivalenttotheunknot. G∆∆ isanabelian
0 n
group, by Theorem 7, which projects onto the monoid GN(S3). The latter is then forced to be a group. (cid:3)
n
Problem 1. Is GN a group?
n
On the level of manifolds, this is true. The monoid of homology 3-spheres is a group after modding out
by alternating sums of Y moves as above [10, 12].
2.4. Translating between claspers and braids. Given a tree T, a simple cT clasper (with respect to a
given knot) is defined to be a cT clasper where each leaf bounds a disk which hits the knot in one point.
Definition 9. A clasper C embedded in the complement of the trivial braid 1 is said to be braid-like, if
n
(i) Surgery along the clasper (1 ) is a braid.
n C
(ii) Regarding 1 as a subset of a cube in the usual way, the complement of 1 in the cube is a regular
n n
neighborhood of the clasper.
Lemma 10. If T is an interval or a Y, then there is a braid-like T clasper.
FINITE-TYPE INVARIANTS BASED ON LOCAL MOVES 7
Proof. The case of an interval is obvious. A braid-like Y-clasper is illustrated below.
↔
(cid:3)
Definition 11.
(i) Let GcT be the subgroup of P normally generated by elements (1 ) , where C ∪···∪C
∞ N C1∪···∪Ci 1 i
is a union of cT claspers which have the property that (1 ) is a braid.
N C1∪···∪Ci
(ii) Denoting by η: P → P the homomorphism which doubles every strand, define GncT as the
∞ ∞
group normally generated by η(GcT) inside B , the direct limit of braid groups on a finite number
∞
of strands. Note that GncT is indeed a subgroup of P .
∞
Proposition 12. The subgroups GncT and GcT are invariant under the strand tripling homomorphism τ .
k
Proof. Consider a generating element (1 ) of GcT. If we triple the kth strand, this will either
N C1∪···∪Ci
miss the caps of the claspers or it will convert a single intersection point to three intersection points. The
claspers will remain capped. Also, the homomorphism τ commutes with the clasper surgery, indicating
k
that τ ((1 ) ) is again a braid. We note that the inverse of a generator is also a generator, since
k N C1∪···∪Ci
it is formed by mirror reflection, and the mirror reflection of a clasper surgery is also a clasper surgery of
the same tree type. (Note though that the clasper itself is not the mirror reflection, since orientation data
is important.)
Now consider a generator η((1 ) ) of GncT. Applying τ , we are tripling one strand of the double
N C1∪···∪Ci k
of a strand in η((1 ) ). That is we are doubling all the strands of the braid except one which we
N C1∪···∪Ci
are quadrupling. So τ η((1 ) ) = η((1 ) ) where the claspers C(cid:48) are formed from C by
doubling the appropriakte stNranCd1∪o·f··∪1Ci. N+1 C1(cid:48)∪···∪Ci(cid:48) i i (cid:3)
N
Theorem 13. Suppose T is a tree for which a braid-like clasper exists. For µ = cT or µ = ncT, µ -
n
invariants coincide with Gµ-invariants.
n
Proof. Let us first consider the case µ = cT. Suppose f: Knots → G is a cT -invariant. Note that Iˆn+1 is
n GcT
generated by elements of the form
g (h −1)g (h −1)···g (h −1)g ,
1 1 2 2 n+1 n+1 n+2
where h ∈GcT and g ∈P . One may assume that each h is a normal generator of GcT: h =1 , where
i i ∞ i i Ci
C is a union of cT claspers.
i
Letting
(cid:16) (cid:17)
x=S g (1 −1)g (1 −1)···g (1 −1)g
1 C1 2 C2 n+1 Cn+1 n+2
we must show that f(x)=0. But this follows since
x=(−1)n+1[S(g ...g );C ,...,C ],
1 n+2 1 n+1
and in the usual way one can write an alternating sum of surgeries on unions of claspers as a linear combi-
nation of alternating sums of surgeries on individual claspers.
Conversely, let C ,...,C be a union of simple cT claspers on a knot k. We want to deform the Morse
1 n+1
function on R3 to get a knot S(b) where the claspers C satisfy the following properties:
i
• Each C sits in an interval [s ,t ] with respect to the Morse function, and all these intervals are
i i i
pairwise disjoint.
8 J.CONANT,J.MOSTOVOY,ANDT.STANFORD
• The restriction of the knot to each interval is the trivial braid.
• Each C is braid-like.
i
To do this, using condition (2) in the definition of “braid-like,” position the claspers C so that they match
i
their braid-like embedding with respect to the morse function for 1 . Now poke the maxima and minima of
n
the knot up and down, to get a short-circuit closure. Since a surgery on a braid-like clasper is equivalent to
a modification by an element of GcT, the case µ=cT is settled.
Consider now the case µ=ncT. Showing that an ncT invariant vanishes on S(Iˆn+1 ) is proved exactly
n GncT
analogously. We have that Iˆn+1 is generated by elements
GncT
g (h −1)g (h −1)···g (h −1)g
1 1 2 2 n+1 n+1 n+2
where h ∈ η(GcT), and g ∈ B . So each h is equal to 1 , where C is a union of ncT-claspers, and the
i i ∞ i Ci i
argument indeed proceeds as above.
Now for the converse. Let C ,...,C be a union of ncT claspers on k where each leaf bounds a disk
1 n+1
hitting k in precisely two algebraically canceling points. As above, we can deform the Morse function so
that each C sits in between two heights s ,t , all the intervals [s ,t ] are disjoint, in each interval C sits in
i i i i i i
the complement of a trivial braid such that C is obtained from a braid-like embedding by doubling some
i
strands. So each C converts the identity element 1 to a pure braid b. Assume that the strands of 1 are
i N N
numbered from left to right. The leaves of C each grab two antiparallel strands. If this pair of strands is
i
of the form (2j−1,2j) we have 1 = η(1 ) with C(cid:48) a braid-like clasper. If this is not the case, let p be
Ci Ci(cid:48) i
a braid such that in p1 p−1, the leaves of C do grab pairs of consecutive strands of the form (2j−1,2j).
Ci i
So then C turns 1 =p−1p1 p−1p into b(cid:48) =p−1(pbp−1)p. Now pbp−1 =η(1 ) for a braid-like cT-clasper
i N N i C(cid:48)
i
C(cid:48), and so b(cid:48) ∈GncT. As before, the argument is now straightforward. Assume, for notational ease, that the
i i
claspers C occur in order on the braid which closes up to give k. Write
i
k =S(g ·1·g ···1·g )
1 2 n+2
where C turns the 1 following g into the braid b(cid:48) ∈GncT. Then
i i i
[k;C ,...,C ]=S(g (1−b(cid:48))···g (1−b(cid:48) )g )∈Iˆn+1 .
1 n+1 1 1 n+1 n+1 n+2 GncT
(cid:3)
3. Groups of knots via the short-circuit closure
3.1. Proof of Theorem 5. We shall think of P as consisting of braids on finite odd number of strands.
∞
Given x ∈ P , the long knot S(x) is obtained by joining together lower and upper ends of the strands in
∞
turns, andextendingtoinfinitytheupperendofthefirststrandandthelowerendofthelaststrand. There
are two subgroups HT,HB ⊂P such that x,y ∈P satisfy S(x)=S(y) if and only if there exist t∈HT
∞ ∞
and b∈HB with x=tyb, see [15].
Recall the definition of the endomorphisms τ of P : τ shifts the braids by two strands “to the right”
i ∞ 0
and, for k >0, τ triples the k-th strand. It is clear that S(τ (x))=S(x) for all x∈P , thus τ (x)=txb,
k 0 ∞ 0
where t∈HT and b∈HB.
Lemma 14. For x∈G there exist t∈HT ∩G and b∈HB ∩G such that τ (x)=txb.
0
Proof. Let t = τ (x)(τ (x))−1, and let b = (τ (x))−1τ (x). Notice that t ∈ HT ∩G and
2k−1 2k−1 2k 2k 2k+1 2k 2k−1
b ∈HB ∩G. We have
2k
τ (x)=t τ (x),
2k−1 2k−1 2k
τ (x)=τ (x)b .
2k 2k+1 2k
There exists N such that τ (x)=x. Thus the following equality holds:
2N+1
τ (x)=t ···t xb ···b ,
0 1 2N−1 2N 0
and this completes the proof. (cid:3)
Denote by G the set of γ G-equivalence classes of G-trivial knots. The operation of connected sum
n n
descends to an addition on G as τ (γ G)⊂γ G.
n 0 n n
FINITE-TYPE INVARIANTS BASED ON LOCAL MOVES 9
Lemma 15. G is an abelian group.
n
Proof. The connected sum of knots is commutative and associative and the neutral element is provided by
the equivalence class of the unknot. Thus it is only necessary to check that every element of G has an
n
inverse.
Let k = S(y) be the short-circuit closure of y ∈ γ (G ∩P ). If m (cid:62) n, the knot k is γ -equivalent
m 2N−1 n
to the trivial knot and defines the identity in G . Assume that m < n. Consider the knot k#S(y−1) =
n
S(yτN(y−1)). By Lemma 14 the braid τN(y−1)) can be written as ty−1b with t∈HT ∩G and b∈Hb∩G,
0 0
so
yτN(y−1)=yty−1b=t(t−1yty−1)b
0
and
S(yτNy−1)=S(t(t−1yty−1)b)=S(t−1yty−1)=S(y )
0 1
where y ∈γ G∩P .
1 m+1 4N−1
Repeating the construction one sees that k#S(y−1)#S(y−1)#...#S(y−1 ) is γ G-equivalent to the
1 q−1 (m+q)
unknot. In particular, the knot S(y−1)#...#S(y−1 ) is an inverse for k in G . (cid:3)
n−m−1 n
In fact, the same proof works to show that G-trivial knots modulo the derived series of G form a group.
Problem 2. Arethesegroupsmodulothederivedseriesrelatedtoknownknotinvariants? Dotheycontain
any concordance information?
The value of any G -invariant on a knot depends only on the γ G-equivalence class of the knot. Indeed,
<n n
γ G−1⊂In.
n G
InordertoproveTheorem5itremainstoshowthatdifferentγ G-equivalenceclassescanbedistinguished
n
by finite-type invariants. This is established by the following lemma.
Lemma 16. The quotient map S(G)→G is a G -invariant.
k k
Proof. We have to show that S(In) is mapped to zero in G .
G n−1
Define a relator of degree n and length m as an element of Z[S(G)] of the form
(∗) S((x −1)(x −1)...(x −1)y)
1 2 m
(cid:80)
with y ∈ G, x ∈ γ G and n = n. The greatest n such that a relator is of degree n will be called the
i ni i
exact degree of arelator. A composite relator is anelementof Z[S(G)] ofthe form k #k −k −k +1where
1 2 1 2
k ,k are knots. Notice that a connected sum of two relators of non-zero degree is a linear combination of
1 2
composite relators.
The kernel of the map Z[S(G)] → G contains all the relators of length 1 and degree n and all the
n
composite relators. On the other hand, an element of S(In) is a linear combination of relators of length n
G
and, hence, of degree n. Thus if we show that any relator of degree n is a linear combination of relators of
degree n and length 1 and composite relators, the lemma is proved.
Suppose that there exist relators of degree n which cannot be represented as linear combinations of the
above form. Among such relators choose the relator R of minimal length and, given the length, of maximal
exact degree. ( If one of the n in a relator exceeds n, then clearly the relator is in the kernel of the quotient
i
map, so there is indeed an upper bound on the exact degree of such a relator.)
AssumethatR isoftheform(∗)asabove, withy,x ∈P ∩G. Chooset∈HT andb∈HB suchthat
i 2N−1
the braid tx b coincides with the braid obtained from x by shifting it by 2N strands to the right, that is,
1 1
with τN(x ). By Lemma 14 the braids t and b can be taken to belong to the same term of the lower central
0 1
series of G as the braid x . The relator
1
R(cid:48) =S((tx b−1)(x −1)...(x −1)y)
1 2 m
is a connected sum of two relators and, hence, is a combination of composite relators. On the other hand,
R(cid:48)−R = S((tx b−x )(x −1)...(x −1)y)
1 1 2 m
= S(x (b−1)(x −1)...(x −1)y)
1 2 m
10 J.CONANT,J.MOSTOVOY,ANDT.STANFORD
Notice now that (b−1) can be exchanged with (x −1) and y modulo relators of shorter length or higher
i
degree. Indeed,
(b−1)y =y(b−1)+([b,y]−1)yb
and
(b−1)(x −1)=(x −1)(b−1)+([b,x ]−1)(x b−1)+([b,x ]−1).
i i i i i
Thus, modulo relators of shorter length or higher degree
S(x (b−1)(x −1)...(x −1)y)=S(x (x −1)...(x −1)y(b−1))=0.
1 2 m 1 2 m
and this means that R is a linear combination of composite relators and relators of length 1 and degree
n. (cid:3)
4. Band-pass invariants
In this section we analyze band-pass invariants, completely characterizing the primitive ones.
Theorem 17. For n≥1, primitive BP -invariants coincide with primitive Vassiliev invariants of order n.
n
The unique primitive BP -invariant is the Arf invariant. In other words, GBP is the index 2 subgroup of GV
0 n n
consisting of those knots with trivial Arf invariant.
We first prove some lemmas.
Lemma 18.
(i) For all n, FBP ⊂FV
n n
(ii) For n>2, FV ⊂FBP .
n n−2
Proof. Part (i) follows by writing a band-pass move as a union of crossing changes.
For part(ii), we use [12, Prop. 6.10]. The quoted result implies that FV is generated by elements of the
n
formk−k whereC isasimple(whichimpliescapped)treeclasperofdegreenonaknotk,andbyelements
C
whichlieinthesecondpoweroftheaugmentationideal. However,theselatterelementsvanishuponpassing
to Z[Knots]. By [4, Thm. 28], we can assume the underlying tree of C is a tree where every trivalent vertex
shares an edge with at least one univalent vertex.
Consider such a tree clasper C on a knot k. Decompose C in the following way: insert two Hopf pairs of
leavesalongalledgeswhicharenotincidenttoaunivalentvertex. ThisgivesusaunionofI andY claspers.
Perform a surgery of k along the Y claspers to get a knot k . Note that k is isotopic to k. There are
Y Y
exactly n−2 of the I claspers, denote them by I ,...,I . Then
1 n−2
(−1)n−2[k ;I ,...,I ]=k −k.
Y 1 n−2 C
Since the I are of the form ncI, k −k ∈FBP , completing the proof. (cid:3)
j C n−2
Lemma 19.
(i) For n>2, FBP surjects onto FV/FV .
n n n+1
(ii) FBP surjects onto 2Z⊂Z∼=FV/FV.
2 2 3
Proof. Assume first n>2. Then FV/FV is generated by chord diagrams with n chords (see, for example,
n n+1
[2]). By[17],thisisgeneratedbywheelsattachedtotheWilsonloopbysomepermutation,andbyseparated
diagrams. Passing to FV/FV , the separated diagrams vanish. Thus it suffices to realize a wheel with n
n n+1
legs attached to the Wilson loop by some alternating sum of band-pass moves. An easy way to do this is to
realize the wheel by an embedded clasper, producing band-pass moves as in the proof of Lemma 18 (ii).
Ifn=2thesameargumentshowsthatthesubspacegeneratedbythewheelwithtwolegsisrealized. This
is indeed equal to twice the generator. Finally, the generator itself cannot be realized since it has nontrivial
Arf invariant, and Arf is a band-pass degree zero invariant [13], and so vanishes on FBP. (cid:3)
2
Proposition 20. For n>2 we have FBP =FV.
n n