On the Goussarov-Polyak-Viro Finite-Type Invariants and the Virtualization Move

0.21 MB·English

by  Micah W. Chrisman

ON THE GOUSSAROV-POLYAK-VIRO FINITE-TYPE INVARIANTS AND THE
VIRTUALIZATION MOVE
MICAHW.CHRISMAN
0
Abstract. In thispaper,it isshown that thereare nononconstantGoussarov-Polyak-Viro finite-type
1
invariants that are invariant underthe virtualizationmove. As an immediatecorollary, we obtain the
0 theorem of [1] which states none of the Birman coefficients of the Jones-Kauffman polynomial are of
2 GPVfinitetype.

n
a
J
1. Introduction
3
1 1In the realm of virtual knots, there are two notions of finite-type invariant. One method, proposed by

Kauffmanin[4]iscloselyrelatedtotheVassilievtheoryforclassicalknots. Theclassoffinite-typeinvariants,
]
T duetoGoussarov-Polyak-Viro[3](abbreviatedGPVinwhatfollows),arethemselvesallKauffmanfinite-type
invariants. However, they are defined by a different filtration in the set of virtual knots.
G
The two classes of invariants possess similar structures. In the classical case, the Vassiliev finite-type
.
h invariantshaveweightsystemsarisingfromsemisimpleLiealgebras.Inthevirtualcase,theGPVfinite-type
t invariants haveweight systems arising from the Lie bialgebras associated to semisimple Lie algebras [5].
a
Ontheotherhand,notallKauffmanfinite-typeinvariantsareofGPVfinite-type. Thiswasfirstobserved
m
for small orders by Kauffman in [4]. The result was sharpened in [1] to show that while all of the Birman
[ coefficients v are Kauffman finite-typeof order ≤n, none are of GPV finite-typeof order ≤m for any m.
n
Akeyplayerintheproofofthisresultisthevirtualizationmove. Inasmallneighborhoodofthecrossing
1
v thevirtualization move is given by:
7
4 ↔
2
2
. Figure 1. The Virtualization Move
1
0
Whilethismoveisnotavirtualisotopy moveinandofitself, therearenumerousvirtualknotinvariants
0
which are invariant under the virtualization move. The virtualization move was discovered by Kauffman
1
in [4] in his investigation of the Jones polynomial. The Jones-Kauffman polynomial is invariant under the
:
v virtualizationmove. Therefinedtheoremin[1]usesthisinvariancetogetherwithatwistsequenceargument
i to show that theBirman coefficients are not of GPV finitetype.
X
Thisleadstothequestionwhichisthesubjectofthispaper: ArethereanynonconstantGPVfinite-type
r
a invariantswhich are invariant underthevirtualization move? Weanswer this question in thenegative with
thefollowing theorem.
Theorem1. IfvisaGPVfinite-typeinvariantofvirtualknotsorlongvirtualknotswhichisinvariantunder
thevirtualizationmove,thenvisconstant. Morespecifically,ifvisanonconstantGPVfinite-typeinvariant,
then there are knots K and K′ such that K and K′ are obtained from one another by a virtualization move
and v(K)6=v(K′).
The proof of this theorem is surprisingly elementary. It uses only a few basic facts about the Polyak
algebra, the algebra of arrow diagrams, and module theory. We present here a proof with all details laid
bare.
There is a well-known conjecture about the virtualization move called the Virtualization Conjecture [2].
It states that if K and K are classical knots which are obtained from one another by a sequence of
1 2
virtualization movesand generalized Reidemeister moves,thenK andK areclassically isotopic. Thefact
1 2
1This is a preprint of a paper submitted for consideration for publication to the Journal of Knot Theory and its
Ramifications.
1
2 MICAHW.CHRISMAN
that thereare manyinvariantswhich are unchangedby thevirtualization moveispositive evidencefor this
conjecture. In light of theconjecture, Theorem 1 is quitecurious indeed.
Thispaperisorganizedasfollows. IntheremainderofSection1,wereviewtheconstructionoftheGPV
finitetypeinvariants. Thegoal of Section 2 is to proveTheorem 1.
TheauthorwouldliketoexpresshisdeepgratitudetoVassilyManturovwhoencouragedthisinvestigation
andpatientlyrespondedtotheauthor’snumerous(andcharacteristically)badideas. Theauthorwouldalso
liketothanktheMonmouthUniversityMathematicsDepartmentfortheirgenerousfinancialsupportofthis
research.
1.1. Review of GPV Finite-Type Invariants. In this section, we review the two notions of finite-type
invariants for virtual knots. First recall how one obtains a Gauss diagram D of an oriented virtual knot
diagram K. Traverse the circle in specified direction. Every time one arrives at a classical crossing, mark
a corresponding point on a copy of S1 (called the Wilson loop). If two marked points on S1 correspond
to the same classical crossing, connect them with an arrow. The arrowhead is incident to the arc on S1
which correspondstotheunderpassingarcon K. Moreover, weattach asign toeach arrowwhich givesthe
orientation of thecrossing:
Arrow carries sign: ⊕ Arrowcarries sign: ⊖
It is known that if two knots havethe same Gauss diagram, then they are virtually isotopic via a sequence
of virtual moves (see [3]). The same construction works just as well for long virtual knots. Instead of the
Wilson loop, we use theWilson line. It is just a copy of R.
Thevirtualization moveis given in Figure 1. Noticethat for any of theways in which thearcs might be
directed, thelocal crossing numberremains thesame. However,thecrossing changes from overtounderor
vice versa. The affect on theGauss diagram is easy to describe:
ε ↔ε
We now proceed to the defintion of Kauffman finite-type invariants. First, the class of virtual knots is
extended to the class of four valent graphs modulo rigid vertex isotopy (see [4] for precise definition).
Vertices take the place of the singular crossings that appear in the Vassiliev theory of classical knots. Any
virtual knot invariant can beextended to these graphs byapplying thefollowing relation:
v =v −v
! ! !
A knot invariant is said to beof Kauffman finitetype≤nif v(K )=0 for all graphs K with greater than
† †
n vertices. The coefficient of xn in the power series expansion of the Birman substitution (i.e. A→ex) of
theJones-Kauffman polynomial is known to be of Kauffman finite-type≤n(see [4]).
Thesecondkindoffinite-typeinvariantofvirtualknotsisduetoGoussarov,Polyak,andViro[3]. Virtual
knots are extended to knots having semivirtual crossings. Any virtual knot invariant can be extended to
this larger class byapplying therelation:
v =v −v
! ! !
Semivirtual crossings are represented in Gauss diagrams by dashed arrows. Schematically, we have:
ε ε
= −
A virtual knot invariant is said to be of GPV finite-type ≤ n if v(K ) = 0 for all diagrams K with
◦ ◦
greater than n semivirtual crossings. There is a simple and elegant algorithm for constructing all GPV
finite-type invariants of order ≤ n that is bounded only by how much computing power one has readily
available. Moreover,everyGPVfinite-typeinvariantoforder≤nisofKauffmanfinite-type≤n. However,
ON THE GOUSSAROV-POLYAK-VIROFINITE-TYPE INVARIANTS AND THE VIRTUALIZATION MOVE 3
the aforementioned coefficients obtained from the Jones-Kauffman polynomial are not of GPV finite-type
≤n for any n (see [1]).
Wenowrecall theconstruction of therational GPV finite-typeinvariantsfound in [3]. Let Ddenotethe
set of all Gauss diagrams of virtual knots. A subdiagram of D ∈D is a Gauss diagram consisting of some
subset of the edges of D. Denote by A the free abelian group generated by dashed Gauss diagrams. The
elements of A are just Gauss diagrams with all arrows drawn dashed. Definea map i:Z[D]→A to bethe
map which makes all thearrows of a Gauss diagram dashed. DefineI :Z[D]→A by:
GPV
I (D)= i(D′)
GPV
DX′⊂D
where the sum is over all subdiagrams of D. The Polyak algebra is the quotient of A by the submodule
∆P=h∆PI,∆PII,∆PIIIi generated by therelations in Figure 2. The Polyak algebra is denoted P.
ε
ε −ε
∆PI: ε =0,∆PII: + + =0,
−ε
ε ε ε ε
ε ε
∆PIII: + + ε + ε =
ε
ε ε ε
ε
ε + ε + ε + ε
ε
Figure 2. Polyak Relations
Denote by K the set of virtual knots (where the elements are virtual isotopy classes of knots). The
following theorem shows that thetheory of virtual knotsis entirely encoded in thePolyak algebra.
Theorem 2 (Goussarov, Polyak, Viro, [3]). The map IGPV : Z[D] → A is an isomorphism. The inverse
can be defined explicitly:
I−1 (A)= (−1)|A−A′|i−1(A)
GPV
AX′⊂A
Here, |A−A′| means the number of arrows in A that are not in A′. Furthermore, if D ∈Z[D] has dashed
arrows, then every element in the sum defining I(D) also has every dashed arrow of D. Finally, the map
extends to an isomorphism of the quotient algebras IGPV:Z[K]→P.
Let A denote submodule of A generated by those diagrams having more than n arrows. Define P =
n n
A/(A +∆P). Let ϕ :P→P denote the natural projection onto the quotient. The following important
n n n
theorem characterizes all rational valued GPV finite-typeinvariants.
Theorem 3 (Goussarov, Polyak, Viro, [3]). The map (IGPV)n :Z[K] → P→ Pn is universal in the sense
that if G is any abelian group, and v is a GPV finite-type invariant of order ≤ n, then there is a map
v′ :P →G such that the following diagram commutes:
n
Z[K] v //G
IGPV (cid:15)(cid:15) zzzzzvzIzG−zzP1==VOO(cid:31)(cid:31)(cid:31)v′
P //P
ϕn n
In particular, the vector space of rational valued invariants of type ≤ n is finite dimensional and can be
identified with HomZ(Pn,Q).
4 MICAHW.CHRISMAN
2. Proof of Main Theorem
2.1. A model of virtualization invariant knot invariants. Let C denote the set of signed chord dia-
grams. These are chord diagrams in the usual sense which have the additional structure of a sign at each
chord: ⊕ or ⊖. Let C denote the free abelian group generated by set of signed chord diagrams possessing
n
>n chords. The relations for Z[C] are as follows:
ε
ε −ε
∆RI: ε =0,∆RII: + + =0,
−ε
ε ε ε ε
ε ε
∆RIII: + + + =
ε
ε ε
ε ε ε
ε + ε + + ε
ε ε ε
Figure 3. ∆R relations
Let ∆R=h∆RI,∆RII,∆RIIIi. Define:
Z[C] Z[C]
V= , V =
∆R n C +∆R
n
Let VKdenote theset of virtual knot diagrams and Dthe set of Gauss diagrams. Definegˆ:Z[VK]→Z[D]
on generators to be the map which assigns to every virtual knot diagram its Gauss diagram. Define g :
Z[VK]/ker(gˆ) → Z[D] via Noether’s First Isomorphism Theorem. For D ∈ D, let D¯ ∈ C denote the chord
diagramobtainedfromDbyerasingthearrowheadofeveryarrowofD. Wewillalsorefertothisoperation
by themap Bar:Z[D]→Z[C]. DefineI :Z[D]→Z[C] on generators D∈Dby:
I(D)= D′ =I (D)
GPV
DX′⊂D
where thesum is over all Gauss diagrams D′ obtained from D bedeleting a subset of its arrows.
Proposition 4. For all v ∈ HomZ(V,Q), v ◦I is a virtual knot invariant that is invariant under the
virtualization move.
Proof. By the GPV theorem, it is sufficient to show that v(PI)=v(PII)=v(PIII)={0}. However, this is
clearly true since hPI,PII,PIIIi=∆R. For the second assertion, note that if K and K′ are obtained from
oneanotherbyasinglevirtualizationmove,thentheirGaussdiagramsdifferonlyinthedirectionofasingle
arrow. In that case, I(g(K))=I(g(K′)). (cid:3)
2.2. Universality of the model. In this section we establish the universality of I and I . The following
n
lemma is useful in this regard.
Lemma 5. Suppose that v∈HomZ(P,Q) is virtualization invariant. In other words,
v◦IGPV =v◦IGPV
(cid:16) (cid:17) (cid:18) (cid:19)
Then for all signed dashed arrow diagrams D, if D′ is obtained from D by changing the direction of one
arrow, then v(D)=v(D′).
ON THE GOUSSAROV-POLYAK-VIROFINITE-TYPE INVARIANTS AND THE VIRTUALIZATION MOVE 5
Proof. The proof is by induction on the numberof arrows of the dashed diagram D. If n=1, the result is
obvious in the case of knots. For long knots, we have:
ε ε
v◦I = v +v
GPV
v◦I (cid:16) ε (cid:17) = v(cid:16) ε (cid:17)+v(cid:0) (cid:1)
GPV
⇒v(cid:16) ε (cid:17) = v(cid:16) ε (cid:17) (cid:0) (cid:1)
(cid:16) (cid:17) (cid:16) (cid:17)
Suppose now that the theorem is true for n. Let D be a dashed arrow diagram with n+1 arrows, D′ a
diagram obtained from D by switching the direction of one arrow. Let B =i−1(D) and B′ =i−1(D′). Let
e denote thearrow whose direction is changed from D to D′.
v◦I (B) = v i(R)
GPV
!
R⊂B
X
= v i(R) +v i(R)
   
ReX∈⊂RB  ReX∈/⊂RB 
   
v◦IGPV(B′) = v i(R)+v i(R′)
ReX′∈⊂RB′′  ReX′∈/⊂RB′′ 
   
Now, for e ∈ R, let R′ denote the diagram obtained from R by switching the direction of e. By applying
linearity of v and subtracting the two equations of interest, we obtain:
v(i(R))−v(i(R′))=0
ReX∈⊂RB
Now, for R⊂B, R has between 1 and n+1 arrows. Since R and R′ differ only in the direction of a single
arrow theinduction hypothesis implies that v(i(R))=v(i(R′)) for all R having ≤n arrows. Thus,
0=v(i(B))−v(i(B′))=v(D)−v(D′)
This establishes the lemma. (cid:3)
Theorem 6. The map I : Z[D] → V is universal in the sense that if v ∈ HomZ(P,Q) is virtualization
invariant, then there is a v′ ∈HomZ(V,Q) such that the following diagram commutes:
Z[D] I //V
(cid:31)
(cid:31)
IGPV v′
(cid:31)
(cid:15)(cid:15) (cid:15)(cid:15)
P //Q
v
Proof. Let C ∈ C and let C~ denote the signed arrow diagram obtained from C by directing the chords
of C. Define v′(C) = v(C~). Note that by Lemma 5, if C~′ is an arrow diagram with Bar(C~′) = C, then
v(C~)=v(C~′). Thus, v′ is well-defined on Z[C].
Tocompletetheproof,itisonlynecessarytoshowthatv′(r)=0forallr∈∆R. Foreachr∈∆RI,∆RII,
or ∆RIII, thereis a~r∈Bar−1(r) such that~r∈PI,PII, or PIII, respectively. For example, we have:
ε
ε −ε
r = + +
−ε
ε
ε −ε
~r = + +
−ε
6 MICAHW.CHRISMAN
Since v(PI)=v(PII)=v(PIII)=0, it follows that v′(r)=0. (cid:3)
Lemma 7. For v∈HomZ(Vn,Q), v◦I is a GPV finite-type invariant of order ≤n.
Proof. For D∈P, v◦I◦I−1 (D)=v(D). IfD has more than n dashed arrows, thenv(D)=0. (cid:3)
GPV
Letp :V→V denotethatnaturalprojection. DefineI :Z[D]→V tobethecomposition: I =p ◦I.
n n n n n n
Theorem 8. The map I : Z[D] → V is universal in the sense that for all v ∈ Hom(P ,Q) such that
n n n
v◦(IGPV)n is virtualization invariant, then there is a v′ ∈ HomZ(Vn,Q) such that the following diagram
commutes.
Z[D] In //V
(cid:31)n
(cid:31)
(IGPV)n v′
(cid:31)
(cid:15)(cid:15) (cid:15)(cid:15)
P //Q
n v
Proof. This follows immediately from Lemma 7 and Theorem 6. (cid:3)
ItisobviousthatconstantinvariantsareofGPVfinitetypeofeveryorder. Forvirtualknots,theyaregen-
erated bythecombinatorial formulah(cid:13),·i. Forlongvirtualknots,theyaregenerated bythecombinatorial
formula < ,·>. Denote both of thesegenerators by 1.
Lemma 9. For small orders, the following computations hold:
(1) For virtual knots, HomZ(V3,Q)=<1>
(2) For long virtual knots, HomZ(V2,Q)=<1>
Proof. Forthefirstassertion,wehavefrom[3]that1andthefollowingformulageneratetheGPVfinite-type
invariants of order ≤3.
3· − + + − − − + , ·
* + + + − +
Since thevalueon diagrams 3 and 5 is different,Lemma 5 implies our result.
For the second assertion, we have from [3] that 1 and the following two formulas generate the GPV
finite-typeinvariants of order ≤2.
, · , , ·
(cid:28) (cid:29) (cid:28) (cid:29)
In this case, Lemma 5 also implies our result. (cid:3)
2.3. Algebraic decomposition of HomZ(Pn,Q) and HomZ(Vn,Q). The hero of the decomposition is
Polyak’s algebra of arrow diagrams. Its precise relationship to the groups V and P is what allows us
n n
to obtain Theorem 1. While this relationship is already well-known (see [5]), it is prudent to describe it
carefully here.
DefineF~±, ~F to be the free abelian group generated by the set of signed arrow diagrams and unsigned
n n
arrowdiagramshavingexactlynarrows, respectively. DefineF±,F tobethefreeabelian groupgenerated
n n
by theset of signed chord diagrams and unsigned chord diagrams having exactly n chords, respectively.
Thedecomposition isthesame for theV and P . Therefore we definevariables which stand in place of
n n
either case.
Chord Case Arrow Case
B V P
n n n
B C A
n n n
R ∆R ∆P
F± F± ~F±
n n n
F F F~
n n n
In this section, we investigate thefollowing short exact sequence:
0 // BBnn++1+RR //Bn+1 πn //Bn //0
ON THE GOUSSAROV-POLYAK-VIROFINITE-TYPE INVARIANTS AND THE VIRTUALIZATION MOVE 7
and (more importantly), its dual:
0 //HomZ(Bn,Q) πn∗ //HomZ(Bn+1,Q) //HomZ BBnn++1+RR,Q
(cid:16) (cid:17)
In the next section, we will show that the rightmost module in the dual sequence vanishes and hence π∗
n
is an isomorphism. For this, we use some intermediate groups which are isomorphic to the groups in the
grading of Polyak’s algebra of arrow diagrams. Therelations for theintermediate groups are given below.
1~T : =0, N~S : ε + −ε =0
n n
6~T±n : ε ε + ε ε + ε ε = εε + ε ε + ε ε
6~T : + + = + +
n
Using the same convention as above, define 1T± = Bar(∆PI ), NS = Bar(N~S ), 6T± = Bar(6~T±), and
n n n n n n
6T =Bar(6~T ). Define6T =6T or6~T ,6T± =6T± or6~T±,1T± =1T± or∆PI ,andNS =NS or
n n n n n n n n n n n n n
N~S .
n
Theorem 10. There is an injection:
B +R F
HomZ n−1 ,Q ֒→HomZ n ,Q
B +R h1T ,6T i
(cid:18) n (cid:19) (cid:18) n n (cid:19)
Proof. First we rewrite Bn−1+R using Noether’s Second Isomorphism Theorem.
Bn+R
B +R F±+(B +R)
n−1 = n n
B +R B +R
n n
F±
∼= n
F±∩(B +R)
n n
Lemma 11. 6T±,1T±,NS ⊂F±∩(B +R)
n n n n n
Proof. It is ce(cid:10)rtainly true that(cid:11)LHS ⊂ F±. We will show that LHS ⊂ B +R for the chord diagram case
n n
only:
ε ε ε ε ε
± ε
6Tn : + ε + ε − ε − ε − ε ε
ε ε ε ε
ε ε
= + + +
ε ε ε
ε ε ε
− ε ε − ε − ε − ε ε ∈∆R
(cid:27)
ε ε
ε
+ − ε + ε ε ∈Cn
(cid:18) (cid:19)(cid:27)
ε −ε ε ε −ε ε
NS : + = + + −
n
−ε −ε
∈∆R ∈Cn
This completes theproof of the lemma. (cid:3)
| {z } | {z }
Therefore thereis an exact sequence:
h6T±n,1FTn±±n,NSni // Fn±∩(FBn±n+R) //0
8 MICAHW.CHRISMAN
DefineΞ:F± →F asin[5]. ForF ∈F±,letm(F)denotethenumberof⊖signsappearinginthediagram.
n n n
Denote by |F| ∈ F the diagram obtained by deleting all the signs of F. Then Ξ is defined on generators
n
by:
Ξ(F)=(−1)m(F)|F|
Extend Ξ toall of F± using linearity. It is clear that:
n
Ξ( 6T± ) ⊂ h6T i
n n
Ξ((cid:10)1T±n(cid:11)) ⊂ h1Tni
Ξ(hNS i) = {0}
(cid:10) n(cid:11)
ThesurjectionF± →F →F /h1T ,6T ihaskernel 6T±,1T±,NS . Takingthedualoftheaboveshort
n n n n n n n n
exact sequencegives the injection:
(cid:10) (cid:11)
B +R F± F
HomZ n−1 ,Q ֒→HomZ n ,Q ∼=HomZ n ,Q
(cid:18) Bn+R (cid:19) 6T±n,1T±n,NSn ! (cid:18)h1Tn,6Tni (cid:19)
(cid:3)
(cid:10) (cid:11)
Asweshallshortlysee,itisconvenienttochangefromZ-modulestovectorspacesoverQ. Thefollowing
computation shows that this does not affect our result.
F F
HomZ n ,Q ∼= HomZ n ,HomZ(Q,Q)
h1T ,6T i h1T ,6T i
(cid:18) n n (cid:19) (cid:18) n n (cid:19)
F
∼= HomZ n ⊗Q,Q
h1T ,6T i
(cid:18) n n (cid:19)
2.4. Proof of Theorem 1. To provethetheorem, we must also consider thefour term relation in F :
n
4T : − = −
n
Defineµ :F →~F to be:
n n n
µ = +
n
!
Here, the sum is over all such resolutions. In fact, if F ∈ F , then µ (F) is a sum of 2n arrow diagrams.
n n
The map µ is also called theaverage map. The following theorem is dueto Polyak.
n
Theorem 12 (Polyak [5]). The average map µ :F →F~ satisfies the following properties.
n n n
(1) µ 4T ⊂ 6~T
n n n
(2) µ (cid:0)(cid:10)1T (cid:11)(cid:1)⊂D1~T E
n n n
(3) The(cid:0)(cid:10)avera(cid:11)g(cid:1)e mDap deEscends to a well-defined map:
F F~
µ : n ⊗Q→ n ⊗Q
n
1Tn,4Tn 1~Tn,6~Tn
(cid:10) (cid:11) D E
Defineµ′(D)= 1 ·µ (D). The following diagrams describe this map.
n 2n n
F F
µ′ : n ⊗Q→ n ⊗Q
n
1T ,4T 1T ,6T
n n n n
Fn ⊗Q(cid:10) (cid:11) 2n·µ′n (cid:10) // (cid:11) Fn ⊗Q
h1Tn,4Tni h1Tn,6~Tni
NNNNµNnNNNNNN&& ppppppBpaprppp88
F~n ⊗Q
h1~Tn,6~Tni
ON THE GOUSSAROV-POLYAK-VIROFINITE-TYPE INVARIANTS AND THE VIRTUALIZATION MOVE 9
In fact, for all D ∈F , µ′(D)=D. Hence, µ′ is a surjection and Fn ⊗Q is a homomorphic image
n n n h1Tn,6Tni
of Fn ⊗Q. Since µ′ 6T = {0}, diagrams in F satisfy relations 1T , 6T , and 4T in the
h1Tn,4Tni n n n n n n
homomorphic image Fn (cid:0)(cid:10)⊗Q(cid:11).(cid:1)Now, note that we may write 6T in thefollowing form:
h1Tn,6Tni n
− − − + − =0
(cid:18) (cid:19)
4Tn 2Tn
Therefore,|the two-term relation (see{[z6]) is also satisfied in theh}om|omorphic im{zage Fn } ⊗Q.
h1Tn,6Tni
2T : =
n
Lemma 13. If the two term relation is satisfied, then all chord diagrams with n chords are equivalent to
the diagram given below:
Virtual Knot Case Long Virtual Knot Case
nchords
nchords | {z }
Since these diagrams all vani|sh in{zthe p}resence of 1Tn, it follows that h1TnF,n6Tni ⊗Q ∼= {0}. Hence
the dual HomZ h1TnF,n6Tni ⊗Q,Q ∼= {0} and we conclude that πn∗ is an isomorphism. This implies
that HomZ(Vn,Q(cid:18)) ∼= HomZ(Vn+1,(cid:19)Q) for all n. In the case of virtual knots, we have by Lemma 9 that
HomZ(V3,Q)=<1>. In the case of long virtual knots, HomZ(V2,Q)=<1>. This completes the proof of
Theorem 1.
Corollary 14 (Kauffman[4],Chrisman[1]). Ifv isthen-th Birmancoefficient ofthe Jones-Kauffman poly-
n
nomial, then v is of Kauffman finite-type ≤n, but not of GPV finite-type ≤m for any m.
n
Proof. The invariants v are invariant underthevirtualization move[4]. (cid:3)
n
References
[1] MicahChrisman.Twistlatticesandthejones-kauffmanpolynomialforlongvirtualknots.Journal ofKnot Theoryand
ItsRamifications,toappear.
[2] Roger Fenn, Louis H. Kauffman, and Vassily O. Manturov. Virtual knot theory—unsolved problems. Fund. Math.,
188:293–323,2005.
[3] Mikhail Goussarov, Michael Polyak, and Oleg Viro. Finite-type invariants of classical and virtual knots. Topology,
39(5):1045–1068,2000.
[4] LouisH.Kauffman.Virtualknottheory.EuropeanJ.Combin.,20(7):663–690,1999.
[5] MichaelPolyak.Onthealgebraofarrowdiagrams.Lett.Math.Phys.,51(4):275–291,2000.
[6] J. Mostovoy S. Chmutov, S. Dushin. CDBook:Introduction to Vassiliev Knot Invariants. http://www.math.ohio-
state.edu/chmutov/preprints/.