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arXiv:math/9810073v2 [math.GT] 6 Apr 1999
FINITE TYPE INVARIANTS OF CLASSICAL AND VIRTUAL
KNOTS
MIKHAIL GOUSSAROV, MICHAEL POLYAK, AND OLEG VIRO
Abstract. We observe that any knot invariant extends to virtual knots. The
isotopy classification problem for virtual knots is reduced to an algebraic prob
lem formulated in terms of an algebra of arrow diagrams. We introduce a new
notion of finite type invariant and show that the restriction of any such invari
ant of degree n to classical knots is an invariant of degree ≤ n in the classical
sense. A universal invariant of degree ≤ n is defined via a Gauss diagram for
mula. This machinery is used to obtain explicit formulas for invariants of low
degrees. The same technique is also used to prove that any finite type invari
ant of classical knots is given by a Gauss diagram formula. We introduce the
notion of nequivalence of Gauss diagrams and announce virtual counterparts
of results concerning classical nequivalence.
1. Virtualization
Recently L. Kauffman introduced a notion of a virtual knot, extending the knot
theory in an unexpected direction. We show here that this extension motivates a
new approach to finite type invariants. This approach leads to new results both for
virtual and classical knots.
1.1. Diagrams and Gauss Diagrams. Knots (smooth simple closed curves in
R3) are usually presented by knot diagrams which are generic immersions of the
circle into the plane enhanced by information on overpasses and underpasses at
double points. A generic immersion of a circle into the plane is characterized by
its Gauss diagram, which consists of the circle together with the preimages of each
double point of the immersion connected by a chord. To incorporate the information
on overpasses and underpasses, the chords are oriented from the upper branch to the
lower one. Furthermore, each chord is equipped with the sign of the corresponding
double point (local writhe number). See Figure 1. The result is called a Gauss
diagram of the knot.
12
3
4
12
3
4
knot 41
+
+


++


Figure 1. A diagram of the figure eight knot and its correspond
ing Gauss diagram.
1991 Mathematics Subject Classification. 57M25.
1
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2M. GOUSSAROV, M. POLYAK, AND O. VIRO
A Gauss diagram is usually considered up to orientation preserving homeomor
phism of the underlying circle. The Gauss diagram defines (up to isotopy of S2) a
knot diagram on the sphere, i.e., a knot diagram embedded into S2via the embed
ding R2→ S2. Given a knot diagram on the sphere, a knot diagram on the plane
can be recovered modulo a finite ambiguity (this involves the choice of which con
nected component of the diagram contains the point at infinity), but the underlying
knot itself is recovered uniquely up to isotopy.
Thus Gauss diagrams can be considered as an alternative way to present knots.
Of course, they cannot compete with knot diagrams in creating a visual impression
of knots, but Gauss diagrams are simpler from the combinatorial point of view and
provide numerous advantages when we want to calculate knot invariants.
Unfortunately, not every picture which looks like a Gauss diagram is indeed a
Gauss diagram of some knot. Moreover, this is not easy to recognize. There is an
obvious algorithm [2] for checking this, which is just the result of attempting to
draw the corresponding knot diagram. However this requires a considerable amount
of effort.
1.2. Virtual Knots. The starting point for the present work is the idea that for
some purposes it is easier just to ignore the problem of whether a Gauss diagram
represents a knot, rather than trying to solve it. This gives rise to a generalization
of classical knot theory by replacing true knots with objects which generalize Gauss
diagrams of knots, but which are not necessarily associated to a knot. Of course,
these objects are to be considered up to an appropriate equivalence, which imitates
knot isotopy.
Although we had been led to this generalization by the internal logic of our
previous research on combinatorial formulae for Vassiliev knot invariants, as soon
as we formulated it, we recognized that we had rediscovered the theory of virtual
knots, which was announced last year by Louis Kauffman in several talks [4]. Our
main contribution to this newborn theory is to turn it into a useful tool for studying
classical knots.
A virtual knot diagram is a generic immersion of the circle into the plane, with
double points divided into real crossing points and virtual crossing points, with the
real crossing points enhanced by information on overpasses and underpasses (as for
classical knot diagrams). At a virtual crossing the branches are not divided into an
overpass and an underpass. The Gauss diagram of a virtual knot is constructed in
the same way as for a classical knot, but all virtual crossings are disregarded, see
Figure 2.
12
3
1
2
3
+
+

++

Figure 2. A diagram of a virtual knot with three real crossings
and one virtual crossing and its corresponding Gauss diagram.
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FINITE TYPE INVARIANTS OF CLASSICAL AND VIRTUAL KNOTS3
Virtuality of virtual knots is manifest through the fact that while the diagram of
a real knot is a picture describing a curve in R3, a virtual knot diagram apparently
does not pertain to any familiar 3dimensional geometric object. However we would
like to keep speaking about virtual knots in the same way we that speak about real
knots: a virtual knot is that thing presented by a virtual knot diagram. To resolve
this ambiguity, we introduce moves on virtual knot diagrams similar to the moves
of real knot diagrams which happen during an isotopy of a knot, and we will use
the term virtual knot to denote an equivalence class of virtual knot diagrams under
these moves. Two knot diagrams represent the same virtual knot, if one can be
obtained from the other by a sequence of these moves. This agrees with the tradition
of classical knot theory, where the term knot is often taken to refer to the isotopy
class of a knot.
1.3. Reidemeister Moves and Virtual Moves. As is wellknown, when a knot
changes by a generic isotopy, its diagram undergoes a sequence of Reidemeister
moves of one of the three types shown in Figure 3.
Figure 3. Reidemeister moves.
A diagram of a virtual knot can undergo the same Reidemeister moves, as well
as the moves shown in Figure 4. These additional moves are called virtual moves.
The first three of them are versions of the Reidemeister moves, but with virtual
crossings in place of crossings. The last one looks like the third Reidemeister move,
but involves two virtual crossings and one usual crossing.
Figure 4. Virtual moves.
Similar moves, but with two real crossings and one virtual crossing (shown in
Figure 5) are forbidden. If one allows these moves, this makes the theory trivial:
any virtual knot diagram can be unknotted by a sequence of moves shown in Figures
3, 4 and 5, see Section 5.3.
As mentioned above, a virtual knot is a class of virtual knot diagrams consisting
of diagrams which can be transformed into each other by sequences of Reidemeister
and virtual moves. A sequence of this kind is called a virtual isotopy.
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4 M. GOUSSAROV, M. POLYAK, AND O. VIRO
Figure 5. Forbidden moves.
Virtual moves do not affect Gauss diagrams. On the other hand, virtual moves
allow one to move the interior of any arc which does not pass through a real crossing
quite arbitrarily. Therefore we obtain the
1.A. Theorem. A Gauss diagram defines a virtual knot diagram up to virtual
moves.
This means that a virtual knot (modulo Reidemeister and virtual moves) is
equivalent to the corresponding Gauss diagram considered up to moves which are
the counterparts of Reidemeister moves for Gauss diagrams, see Figure 6. Since
in a Gauss diagram all the orientations and the cyclic ordering of the endpoints
of arrows are essential, each type of Reidemeister moves splits. In Figure 6, all
moves corresponding to the first and second Reidemeister moves are shown in the
top and middle rows, respectively. There are eight moves corresponding to the
third Reidemeister move, but we only show two of them in the bottom row. As
¨Ostlund [6] showed, the remaining six moves are unnecessary. That is, any sequence
of moves of a Gauss diagram can be replaced by a sequence of moves appearing
in Figure 6. Although in [6] this is proved for Gauss diagrams of knots, the same
proof works for virtual knots.
+
++
+






++
ε
ε
ε −ε
ε −ε
Figure 6. Moves of Gauss diagrams corresponding to Reidemeis
ter moves.
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FINITE TYPE INVARIANTS OF CLASSICAL AND VIRTUAL KNOTS5
1.4. Kauffman’s Results on Extending Knot Invariants to Virtual Knots.
Kauffman [4], [5] has proved that many knot invariants extend to invariants of
virtual knots. In particular, the notions of knot group, quandle and rack, and the
bracket polynomial, all extend in a straightforward manner. He also announced
that with the use of a ”virtual framing” there are extensions of all quantum link
invariants and the large collection of their corresponding Vassiliev invariants.
The extensions are done in a formal way, disregarding the original topological
nature of these invariants. For example, the knot group, which is defined for clas
sical knots as the fundamental group of the knot complement, is extended via a
formal construction of a Wirtinger presentation. This construction can be written
down in terms of a Gauss diagram as follows.
Let G be a Gauss diagram. If we cut the circle at each arrowhead (forgetting
arrowtails), the circle of G is divided into a set of arcs. To each of these arcs there
corresponds a generator of the group. Each arrow gives rise to a relation. Suppose
the sign of an arrow is ε, its tail lies on an arc labelled a, its head is the final point
of an arc labelled b and the initial point of an arc labelled c. Then we assign to this
arrow the relation c = a−εbaε. The resulting group is called the group of the Gauss
diagram. One can easily check that it is invariant under the Reidemeister moves
shown in Figure 6. Moreover, the group system1also extends. For the meridian,
take the generator corresponding to any of the arcs. To write down the longitude,
we go along the circle starting from this arc and write aε, when passing the head
of an arrow whose sign is ε and whose tail lies on the arc labelled a.
The notion of quandle [3] is extended in the same way as the knot group: the
generators remain the same, but each group relation c = a−εbaεis replaced with
the corresponding quandle relation c = a ⊲εb.
1.5. Knots Versus Virtual Knots. Any diagram of a classical knot can be con
sidered to be a virtual knot diagram. A virtual isotopy can turn it into a diagram
with virtual crossings, and then back again to a real knot diagram. Thus virtual
isotopy is a new relation among classical knots, which apriori could differ from
classical isotopy. However this is not the case.
1.B. Theorem (Virtual Isotopy Implies Isotopy). (See also Kauffman [4].) Vir
tually isotopic classical knots are isotopic.
Proof. The group system extends to virtual knots. Hence it is preserved under
virtual isotopy, and virtually isotopic knots have isomorphic group systems. Now
recall that the group system is a complete knot invariant: knots with isomorphic
group systems are isotopic.
Any invariant of virtual knots is obviously an invariant of classical knots. On
the other hand, by Theorem 1.B, any invariant of classical knots can be extended
to an invariant of virtual knots. Nevertheless, for some invariants it is not easy to
choose a natural extension. Even for the linking number, the extension to virtual
2component links is not unique (see Section 1.7 below). A similar situation occurs
for the degree 2 Vassiliev knot invariant considered in Section 3.2.
These examples are based on the same phenomenon. Unlike a classical knot, a
virtual knot cannot be turned upside down. A rotation of a classical knot by the
angle π around a horizontal line reverses all arrows of its Gauss diagram, while
1Recall that the group system of a knot is the knot group together with the class of subgroups
which are conjugate to the subgroup which is generated by a meridian and longitude.
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6 M. GOUSSAROV, M. POLYAK, AND O. VIRO
+
++
+
Figure 7. A virtual knot with different upper and lower groups.
their signs do not change. An application of this operation to a Gauss diagram of
a virtual knot gives rise to a virtual knot which may be non isotopic to the original
one. The composition of this operation with an invariant of virtual knots may be
another invariant.
A striking manifestation of this phenomenon comes from the knot group. Instead
of the upper Wirtinger presentation of a knot group, which was generalized to
virtual knots by Kauffman (see the preceding section), let us use the lower Wirtinger
presentation, i.e. compose Kauffman’s construction with arrows reversal. We will
call these groups the upper and the lower virtual knot groups, respectively. A
virtual knot with different upper and lower groups is shown in Figure 7. The upper
group of this knot is isomorphic to the group of the trefoil knot, while its lower
group is Z.
These examples may create an impression that virtual knot theory is more cum
bersome than the classical knot theory. However, this is not the case. Due to its
larger class of objects, the theory of virtual knots provides more flexibility. This
leads to significant simplification, especially in the theory of finite type invariants.
1.6. Long Knots. By a (classical) long knot we mean a smooth embedding R →
R3which coincides with the standard embedding outside a compact set.
An isotopy of long knots is a smooth isotopy in the class of embeddings above.
In the classical knot theory, long knots are introduced for purely technical reasons,
since adding the point at infinity turns a long knot into a knot in the sphere S3
and this construction establishes a onetoone correspondence between the isotopy
classes of long knots and the isotopy classes of knots.
Given a diagram of a long knot, the corresponding Gauss diagram is the line pa
rameterizing the knot, together with arcs connecting the preimages of each crossing.
As in the case of closed classical knots considered above, the arcs are oriented from
the upper branch to the lower one and equipped with signs which are equal to the
local writhe numbers of the corresponding crossing points. Each oriented signed
arc is called an arrow.
A virtual long knot diagram is a generic immersions R → R2with double points
divided into real and virtual crossing points, where real crossing points are enhanced
by information on overpasses and underpasses, as in a classical knot diagram. At
a virtual crossing the branches are not divided into an overpass and an underpass.
The Gauss diagram of a virtual long knot is constructed in the same way as for
a classical long knot, but all the virtual crossings are disregarded. A virtual long
knot is a class of diagrams which can be transformed into each other by sequences
of Reidemeister and virtual moves (shown in Figures 3 and 4).
Surprisingly, virtual long knots differ from virtual knots. That is, there is no
onetoone correspondence between the virtual isotopy classes of virtual long knots
and virtual knots. Addition of a point at infinity of the plane of the diagram turns
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FINITE TYPE INVARIANTS OF CLASSICAL AND VIRTUAL KNOTS7
a diagram of a virtual long knot into a virtual knot diagram on S2. Removal of
a point from the complement of the diagram yields a virtual knot diagram on R2.
One can easily prove that its virtual isotopy class does not depend on the choice of
this point. Thus we have a natural map from the set of virtual long knots to virtual
knots. This map is surjective but not injective. The simplest pair of virtual long
knots which are not virtually isotopic, but give rise to isotopic virtual closed knots
is shown in Figure 8. These virtual long knots are distinguished by the invariant
v2,2defined in Section 3.2 below. The upper diagram can be transformed into the
Figure 8.
lower diagram by moving the underpassing arc of the leftmost crossing through
the point at infinity. If these were classical knots, these transformation could be
replaced by moving the same arc under the rest of the diagram by a sequence of
Reidemeister moves. In our case this is impossible, since we cannot apply the move
of Figure 5.
As a result, the theories for long and closed virtual knots are quite different,
although their restrictions to usual knots coincide.
1.7. Links. For links, the basic notions of the virtual theory are introduced in a
straightforward way. The only change is that the underlying circle of a Gauss dia
gram is replaced with several circles. Simple examples show that in many respects
it is richer than the classical one and sometimes looks surprising. For instance,
for 2component links there are two independent versions of the linking number.
The invariant lk1/2may be computed as a sum of signs of real crossings where
the first component passes over the second one. Similarly, one can define lk2/1by
exchanging the components in the definition of lk1/2above.
String links are related to links like long knots are to knots. A classical n
component string link is a smooth embedding of a disjoint union of n copies of R
into R3which coincides with the standard embedding outside a compact set. Here,
by the standard embedding, we mean the one given by the formula t ?→ (t,k,0),
with t ∈ R and k = 1,...,n. All the basic notions of the virtual theory extend
naturally to string links. A Gauss diagram in this case consists of n parallel lines
and signed arrows with end points on these lines.
2. Finite Type Invariants
2.1. Crossing Virtualization Versus Crossing Change. In the realm of vir
tual knots there is an elementary operation which does not exist for classical knots.
A real crossing can be turned into a virtual one. In terms of Gauss diagrams it
looks even simpler: we erase an arrow. This operation simplifies the knot in the
sense that after applying it a sufficient number of times we eventually get to the
unknot.
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8 M. GOUSSAROV, M. POLYAK, AND O. VIRO
In the classical knot theory an operation with this property is widely used. This
is the crossing change. However, it is more complicated in several ways. First,
in order to turn a knot into the unknot, one must apply this move according to
a certain pattern, say making the diagram descending or ascending, whereas vir
tualizing crossings leads to the unknot automatically. Second, the unknotting by
crossing changes involves a choice: even if we have chosen to proceed towards a de
scending diagram, we still have to choose a point at which the descent begins. The
result considered as a diagram depends on this choice. Unknotting by virtualization
eliminates these technically unpleasant problems. Third, crossing changes do not
diminish the number of crossings, while each virtualization diminishes the number
of real crossings. Finally, virtualization is more elementary than crossing chang
ing, since a crossing change can be presented as the composition of one crossing
virtualization and the inverse of another.
A more general operation defined on Gauss diagrams of virtual knots is passage
to subdiagrams. Here D′is a subdiagram of D if all the arrows of D′belong to D.
In this case we write D′⊂ D.
2.2. Classical Finite Type Invariants. The standard theory of finite type in
variants is based on crossing change as the basic modification.
Recall that a function ν defined on the set of knot isotopy types and taking
values in an abelian group G is said to be a finite type invariant of degree ≤ n, if
for any knot diagram D and n + 1 crossing points d1, d2, ... , dn+1of D
?
σ
(−1)σν(Dσ) = 0.
(1)
Here σ = {σ1,...,σn+1} runs over (n+1)tuples of zeros and ones, σ is the number
of ones in σ, and Dσis the diagram obtained from D by switching all crossings di
with σi= 1.
This description can be simplified by extending a knot invariant to knots with
double points (called also singular knots).
crossing point one moves the upper branch downwards through the lower branch.
The knot with a double point is identified with the formal difference between the two
knots obtained by resolving the double point in two ways. This can be formulated
as the following formal relation:
A double point appears when at a
=−(2)
Double points are depicted with thick points, so as to distinguish them from virtual
crossings.
Any knot invariant extends to formal linear combinations of knot diagrams by
linearity. Under the identification in (2), the alternating sum in the left hand side
of equality (1) becomes the value of ν on a knot with n + 1 double points. Thus a
knot invariant has degree at most n if its extension vanishes on every singular knot
having at least n + 1 double points.
2.3. A New Notion of Finite Type Invariant. The counterpart in the virtual
theory of the notion of finite type invariant can be described as follows. We intro
duce a new kind of crossing, which is called semivirtual. At a semivirtual crossing
there are still over and underpasses. In a diagram a semivirtual crossing is shown
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FINITE TYPE INVARIANTS OF CLASSICAL AND VIRTUAL KNOTS9
as a real one, but surrounded by a small circle. Semivirtual crossings are related
to the other types of crossings by the following formal relation:
=− (3)
In a Gauss diagram a semivirtual crossing is presented by a dashed arrow. The
relation (3) becomes
ε
= (4)
ε
−
Let D be a virtual knot diagram and {d1,...,dn} be an ntuple of its real
crossings points. For an ntuple σ = {σ1,...,σn} of zeros and ones, define Dσ to
be the diagram, obtained from D, by switching all the crossings di, with σi= 1, to
virtual crossings. Denote by σ the number of ones in σ. The formal alternating
sum
?
is called a diagram with n semivirtual crossings. We depict the corresponding
alternating sum of Gauss diagrams by the Gauss diagram of D with all the arrows
associated to {d1,...,dn} being dashed. This agrees with the convention (4) on
semivirtual crossings.
Denote by K the set of virtual knots. Let ν : K → G be an invariant of virtual
knots with values in an abelian group G. Extend it to Z[K] by linearity. We say
that ν is an invariant of finite type, if for some n ∈ N it vanishes for any virtual
knot K with more than n semivirtual crossings. The minimal such n is called the
degree of ν.
Note that (3) and (2) imply
σ
(−1)σDσ
=− (5)
It follows that for any finite type invariant of the virtual theory, its restriction to
classical knots is a finite type invariant (of at most the same degree) in the classical
sense.
The definition of finite type invariants extends to virtual links in a natural way.
A particularly simple example is given by the invariants lk1/2and lk2/1considered
in Section 1.7. These invariants of 2component virtual links have degree one.
2.4. The Algebra of Arrow Diagrams. An arrow diagram (on a circle) is an
abstract diagram, which consists of an oriented circle with pairs of distinct points
connected by dashed arrows. Each arrow is equipped with a sign. The algebra of
arrow diagrams A is the free abelian group generated by all arrow diagrams. We
call A an algebra, because there is indeed a natural multiplication in A making it
into an associative algebra.2The algebra A⊗Q is isomorphic to the one introduced
in [7]. However in this paper we will not make use of the multiplicative structure
in A.
2The product of arrow diagrams A1, A2is the sum (with appropriate multiplicities, cf. [6]) of
all diagrams each of which is the union of subdiagrams isomorphic to A1 and A2.
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10 M. GOUSSAROV, M. POLYAK, AND O. VIRO
Denote the set of all Gauss diagrams by D. Starting from any Gauss diagram
we get an arrow diagram just by making all its arrows dashed. The extension of
this map to Z[D] defines a natural isomorphism i : Z[D] → A.
There is another important map I : D → A, assigning to a Gauss diagram D
the sum of all its subdiagrams and then making each of them dashed:
I(D) =
?
D′⊂D
i(D′).
Thus the map I can described by the following symbolic formula:
I :
ε
?→
ε
+
.
(6)
The reason for using the same dashed arrows both for semivirtual crossings in
Z[D] and for arrows in A becomes clear if one compares formulas (6) and (4).
Extend I to Z[D] by linearity.
2.A. Proposition. I : Z[D] → A is an isomorphism. The inverse map I−1: A →
Z[D] is defined on the generators of A by
I−1(A) =
?
A′⊂A
(−1)A−A′i−1(A′),
where A − A′ is the number of arrows of A which do not belong to A′.
A Gauss diagram is called semivirtual if each of its arrows is dashed.
2.B. Corollary. Semivirtual diagrams form a basis of Z[D].
2.C. Remark. We can now explain an additional reason for the dual use of dashed
arrows: I maps each semivirtual Gauss diagram to the arrow diagram with the
same arrows.This observation extends to diagrams containing both solid and
dashed arrows. Consider such a diagram D as an element of Z[D]. Then each
diagram appearing in I(D) ∈ A contains all the dashed arrows of D.
Thus we see that I can be interpreted as a presentation of a Gauss diagram by
a linear combination of semivirtual diagrams.
2.5. The Polyak Algebra. The Polyak algebra is the quotient of A by the fol
lowing relations:
ε
= 0(7)
ε
−ε
+
ε
+
−ε
= 0(8)
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FINITE TYPE INVARIANTS OF CLASSICAL AND VIRTUAL KNOTS11
(9)
ε
ε
ε
+
ε
ε
+
ε
ε
+
ε
ε
=
ε
ε
ε
+
ε
ε
+
ε
ε
+
ε
ε
Here we follow the common convention that the unshown parts of all diagrams
involved in each of the relations coincide. The embeddings of the shown parts into
the whole diagrams should preserve the orientations in (9).
The same relations define analogous algebras for long knots, links and string
links.
The quotient of A by the relations (7)  (9) is an algebra, since the relations
generate an ideal of A, but we shall not go into further detail on this point. Denote
this algebra by P. This algebra is closely related to the algebra A introduced in
[8].
The isomorphism I induces an isomorphism I : Z[K] → P of quotient algebras.
Indeed, the equivalence relation induced in D by the Reidemeister moves shown
in Figure 3, can be rewritten in terms of diagrams with semivirtual crossings as
follows
= 0 (10)
++= 0(11)
(12)
+++=
+++
Note that the map I turns (10) – (12) into (7) – (9). Thus for any Gauss diagram
D of a virtual knot K, I(D) defines a Pvalued invariant of K. Moreover, since the
Gauss diagram determines K, the invariant I(D) distinguishes virtual knots. Thus
we obtain the following theorem.
2.D. Theorem. Let D be any diagram of a virtual knot K. The formula K ?→
I(D) ∈ P defines a complete invariant of virtual knots.
2.6. The Truncated Algebras Pnand the Universal Finite Type Invariant.
Define the truncated algebra Pnby putting A = 0 for any diagram A ∈ P with more
than n arrows. Denote by In: K → Pn the composition of I : K → P with the
projection P → Pn.
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12 M. GOUSSAROV, M. POLYAK, AND O. VIRO
Let P be an abelian group and p : K → P be a Pvalued invariant of virtual
knots. We call p a universal invariant of degree n, if for every abelian group G and
every invariant ν : K → G of degree at most n factors through p, i.e. there exists a
map π : P → G, such that ν = π ◦ p.
2.E. Theorem. The map In: K → Pndefines a universal invariant of degree n.
Proof. Since, by Remark 2.C, I preserves all dashed arrows, Inmaps a knot with
more than n semivirtual crossings to zero. Therefore Inis an invariant of degree
at most n.
Let ν : K → G be an invariant of degree at most n. We have to prove that
ν ◦ I−1: P → G factors through Pn. Observe that, by Remark 2.C, for any arrow
diagram A its image I−1(A) ∈ Z[D] can be identified with the same diagram A,
but considered as a semivirtual Gauss diagram. Since the degree of ν is at most
n, ν vanishes on each diagram with more than n semivirtual crossings. Therefore
ν◦I−1vanishes on each arrow diagram with more than n arrows and ν◦I−1factors
through Pn.
2.F. Corollary. The space of Qvalued invariants of degree at most n is finite
dimensional, of dimension equal to rk(Pn). It can be identified with the dual space
P∗
nof Qvalued linear functions on Pn.
3. Gauss Diagram Formulas for Finite Type Invariants
3.1. Gauss Diagram Formulas. Since the algebra A has a distinguished basis,
consisting of arrow diagrams, there is a natural orthonormal scalar product (·,·)
on A. Namely, on the generators of A we put (D1,D2) to be 1, if D1= D2, and
0 otherwise and then extend (·,·) bilinearly. This allows us to define the pairing
?·,·? : A × D → Z in the following way. For any D ∈ D and A ∈ A put
?A,D? = (A,I(D)) = (A,
?
D′⊂D
i(D′)).
(13)
Informally speaking, we count subdiagrams of D with weights, where the weight of
a diagram D′is the coefficient of i(D′) in A.
In the case of Gauss diagrams corresponding to usual knots, this pairing (in
a slightly different form) was introduced in [7] as a tool for writing down Gauss
diagram formulas for knot invariants. We will use it below for the same purpose in
the framework of virtual knots.
Using equation (13) and Theorem 2.D it is easy to see that any Zvalued invariant
of finite type of virtual knots can be obtained by a Gauss diagram formula ?A,·? :
K → Z for some A ∈ A. The maximal number of arrows of the diagrams in
the linear combination giving A is an upper bound for the degree. However, in
general the expression ?A,D? depends on the choice of the Gauss diagram D of a
virtual knot. In our earlier work [7] we did not present any systematic method for
producing arrow polynomials A which give invariants, and we posed the following
question: “Which arrow polynomials define knot invariants...?” We can now answer
this question in the framework of the virtual theory: A defines an invariant of degree
at most n if and only if all diagrams in A have at most n arrows and A satisfies the
equations (A,R) = 0, where R runs over the left hand sides of the relations R = 0
defining Pnin A, see Section 2.5.
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FINITE TYPE INVARIANTS OF CLASSICAL AND VIRTUAL KNOTS13
The general method for producing all invariants of degree n requires a com
putation of the algebra Pn. Some simple observations allow one to reduce this
computation. First, by a repeated use of (11), one can eliminate arrows with the
negative sign. Second, since all the diagrams with more than n arrows vanish, for
diagrams with n arrows equations (11) and (12) become simpler and contain only 2
and 6 terms respectively, all of them with exactly n arrows. The simplified version
of (11) implies the following rule for elimination of negative arrows in diagrams
with exactly n arrows: if two such diagrams differ only by the signs of k arrows,
they differ in Pn by multiplication by (−1)k. This allows one to drop the signs
of arrows in diagrams with n arrows, using the convention that a diagram with k
negative signs of arrows is counted with coefficient (−1)k. The simplified version
of (12) involves only diagrams with exactly n arrows and looks as follows:
(14)
++=
++
This 6Tequation (introduced earlier in [8]) is an oriented version of the wellknown
4Trelation for chord diagrams. The 4Trelation can be recovered from (14) by
repeated use of the following formula which follows from (5):
=+
.
(15)
3.2. Invariants of Small Degree. Computation of Pnfor small n can be done
by hand and leads to some interesting results. Similarly to the case of classical
knots, there are no invariants of degree one. More surprisingly, the algebra P2is
also trivial, so there are no invariants of degree two! However, for long knots the
corresponding algebra is 2dimensional, so there are two independent invariants v2,1
and v2,2of degree 2. These invariants are given by
v2,1(·) =
?
, ·
?
;
v2,2(·) =
?
, ·
?
(16)
This illustrates a curious feature of the theory of virtual knots which was discussed
in Section 1.6. For classical knots, there is onetoone correspondence between the
isotopy classes of knots and long knots, hence any invariant of long knots is an
invariant of closed knots. We now see that for virtual knots this is no longer true.
Another interesting feature of this theory is that many invariants, which coincide
for usual knots (due to the existence of certain symmetries), are different on the
larger class of virtual knots. The invariants v2,1and v2,2provide a good illustration.
In degree three there is only one invariant, given by
?
3−++−−−+
+
−−
+
,·
?
.
It vanishes on real knots. Similarly to degree two, for long virtual knots there are
several invariants of degree three, which give the same degree three invariant of real
Page 14
14 M. GOUSSAROV, M. POLYAK, AND O. VIRO
knots. Here is an example of such an invariant:
+
+++++
++++
+

+ ++ + ++


, .
3.3. The Case of Classical Knots. Our work in this direction started with a
search for combinatorial formulas for finite type invariants of classical knots. The
first results were summarized in the paper [7] of the second and third authors.
There a class of combinatorial formulas similar to (13) was introduced, and numer
ous special formulas of this sort were found. In [7] we posed the following question:
“Can any Vassiliev invariant be calculated as a function of arrow polynomials eval
uated on the knot diagram?” In the terminology used above, an arrow polynomial
evaluated on the knot diagram is an expression of the type given in (13).
This question has been answered in the affirmative by the first author. The
result is formulated as follows.
3.A. Theorem (Goussarov). Let G be an abelian group and let ν be a Gvalued
invariant of degree n of long (real) knots. Then there exists a function π : A → G
such that ν = π ◦I and such that π vanishes on any arrow diagram with more than
n arrows.
3.B. Corollary. Any integervalued finite type invariant of degree n of long knots
can be presented as ?A,·?, where A is a linear combination of arrow diagrams on a
line with at most n arrows.
The next section is devoted to the proof of this theorem.
To a large extent, the present paper was motivated by an analysis of the proof
of 3.A, which originally was rather cumbersome. The main difficulties in this proof
were caused by the necessity of requiring all the numerous Gauss diagrams involved
to be realizable. The desire to get rid of this restriction motivated our interest in vir
tual knots. Indeed, for virtual knots the problem stated in [7] is solved by Theorem
2.E above. The universal invariant of Theorem 2.E is essentially?
Unfortunately, for classical knots the new technique does not give a universal
invariant. However it gives powerful and simple machinery to generate Gauss dia
gram formulas for any invariant which can be extended to a finitetype invariant of
virtual knots. Hoping for the best, we conjecture
A∈Pn?A,D?A,
so any Gvalued invariant can be presented by a Gauss diagram formula.
3.C. Conjecture. Every finitetype invariant of classical knots can be extended
to a finitetype invariant of long virtual knots.
This may require the consideration of virtual framing. The extension given by
Kauffman [4] [5] of numerous invariants to virtual knots strongly supports this
conjecture.
The main open problem concerning finite type invariants of classical knots is
whether such invariants distinguish nonisotopic knots. The positive solution of
this problem would follow from the positive solution of the corresponding problem
Page 15
FINITE TYPE INVARIANTS OF CLASSICAL AND VIRTUAL KNOTS15
for virtual knots. By Theorems 2.D and 2.E, the latter can be reformulated in
purely algebraic terms as the question whether the natural map
P → lim
←−Pn
is injective.
4. Proof of Goussarov’s Theorem
4.1. Scheme of the Proof. The standard method for calculation of a Vassiliev
invariant ν goes as follows (see [11]). One picks a set of singular knots which span
(using relation (2)) the free abelian group generated by all nonsingular knots. The
invariant ν is determined by its actuality table, i.e. its values on this set. Given
a knot diagram, one unknots it, making it descending by a sequence of crossing
changes. Under each crossing change the invariant jumps. The jump is equal to
the value of ν on the knot with a double point by (2). Then each of these singular
knots is deformed to a knot with a single double point from the actuality table by
an isotopy and a sequence of crossing changes. The jumps of ν correspond to knots
with two double points. They are again deformed to the knots from the actuality
table. The process eventually stops when the number of double points exceeds the
degree of ν (by definition of the degree).
In the proof of 3.A, both the actuality table and the procedure of expansion
described above are made canonical. This is done by generalizing the notion of a
descending diagram to singular knots.3
More importantly, this is done in terms of Gauss diagrams, so that the notion of
descending diagram and the procedure of expansion extend to virtual knots. For a
real descending knot the isotopy class and hence the value of ν is determined by the
part of the Gauss diagram encoding the double points. For a virtual descending
diagram the isotopy type is not determined by this part of the Gauss diagram.
Nevertheless we extend ν to virtual descending diagrams literally in the same way.
We do not know whether the result is an invariant of virtual knots.4However, for
our purposes, this formal extension turns out to be sufficient.
Next we use the isomorphism I−1: A → Z[D] (see Proposition 2.A) to define
π : A → G as ν ◦ R. Some special properties of the extended map ν : Z[D] → G
are then used to prove that π vanishes on diagrams with more than n chords.
4.2. Descending Singular Diagrams. On a diagram of a long virtual singular
knot each double point is naturally equipped with a sign. Indeed, the branches at
a double point are ordered and the sign is the intersection number of the branches
(taken in this order). On a Gauss diagram of a long singular knot, each double
point is shown by a dashed chord equipped with the above sign. A diagram D′is
called a subdiagram of a diagram D if D′consists of all the chords and some arrows
of D.
Recall that a diagram of a real long knot is descending if going along the knot
in the positive direction we pass each crossing first going over and then under. In
terms of Gauss diagrams it means that all the arrows are directed to the right.
We now extend this notion to virtual long knots with double points. We still
require that all the arrows are directed to the right. There is also an additional
3A similar notion called almost monotone diagram was considered by BarNatan [1], but the
procedure of expansion in [1] involves some choices.
4A positive answer to this question would imply Conjecture 3.C.
Page 16
16 M. GOUSSAROV, M. POLYAK, AND O. VIRO
condition: there is no chord whose left endpoint has an endpoint of an arrow as
immediate left neighbor. In other words, the situations shown in Figure 9 are
forbidden.
Figure 9.
A real long knot with a Gauss diagram of this type can be presented by a diagram
such that
• all the double points are in the left halfplane,
• all the crossings are in the right halfplane,
• the intersection of the diagram with the left halfplane is an embedded tree,
• the intersection with the right halfplane is an ordered collection of arcs; each
of them is descending and lies below all the previous ones.
See Figure 10.
+++
  
Figure 10. A descending long real knot and its Gauss diagram.
4.A. Lemma. Let D1 and D2 be Gauss diagrams of real descending long knots
and let ν be an invariant of long knots. If the chord parts of D1and D2coincide
then ν(D1) = ν(D2).
Proof. One can see that the isotopy class of a real descending long knot is deter
mined by the chord part of its Gauss diagram. Indeed, the chord part determines
the tree in the left halfplane (recall that the chords have signs, which define the
embedding locally). The rule for connecting the endpoints of the tree by arcs in the
right halfplane is determined by the mutual position of the chords in the Gauss
diagram. Since the diagram is descending, the connection by the arcs is unique up
to isotopy.
Page 17
FINITE TYPE INVARIANTS OF CLASSICAL AND VIRTUAL KNOTS17
4.3. Reduction to Descending Diagrams. There is an algorithm for expressing
the Gauss diagram of a long knot with double points as a linear combination of
descending diagrams. This algorithm consists of steps of two types. At each step,
one inspects the Gauss diagram from the left to the right looking for the first
fragment where the diagram fails to be descending. Such a fragment may either be
a bad arrow or a bad chord. An arrow is bad if it is directed to the left. A chord
is bad if an immediate left neighbor of its left endpoint is an endpoint of an arrow,
as in Figure 9.
In the case of a bad arrow the step of the algorithm is the replacement of the
diagram with the sum of two diagrams according to the formula
+
In terms of Gauss diagrams this replacement is as follows:
ε −ε−ε
+ ε
In the case of a bad chord the step of the algorithm is the pulling of the crossing
over or under the appropriate branch by isotopy:
In terms of Gauss diagrams, this corresponds to one of the transformations shown
in Figure 11. The different cases in Figure 11 correspond to different orientations
and possible orderings of the three arcs.
Since we deal with an invariant of degree n, the diagrams with more than n
chords are disregarded. Thus, when one applies a step of the algorithm to a bad
arrow in a diagram with n chords, the summand with n + 1 chords disappears.
Denote by Dnthe free abelian group generated by Gauss diagrams of virtual long
singular knots with at most n chords (note that Z[D] = D0⊂ Dn). We will think of
a step of our algorithm as of an operator acting on Dn. Denote this operator by P.
By the definition of P, for any descending Gauss diagram D we have P(D) = D.
4.B. Lemma. For any diagram D ∈ Dnthere exists m such that Pm(D) is a sum
of descending diagrams.
Proof. Let l(D) be the number of chords of D which have one of the endpoints to
the left of the first bad fragment. As is easy to see, l(D′) ≥ l(D) for each diagram
in the expansion of P(D). However the number of such chords in a nondescending
diagram is at most n. Therefore it suffices to prove that the diagram cannot change
infinitely many times in subsequent iterations of P without changing l.
Consider the number of arrowheads on the ray to the left of the left endpoint of
the (l(D) + 1)th chord. For any diagram involved in the expansion of P(D) this
number is not greater than that for D. If it is the same for one of these diagrams,
Page 18
18M. GOUSSAROV, M. POLYAK, AND O. VIRO
−εεεε
−εεε
−εε ε ε
ε
ε −εε
ε
εε −ε
ε
ε ε −ε
−ε
−ε
ε
ε
ε
−ε −ε
εε
ε
ε
−ε−ε
Figure 11.
then it has less arrowtails on the same ray. This can happen only finitely many
times.
4.4. The Extension of ν and the Construction of π. Denote by Dre
subgroup of Dngenerated by Gauss diagrams of real long singular knots. Any finite
type invariant of classical knots of degree at most n extends to Dre
n the
nby linearity.
4.C. Obvious Lemma. The operator P : Dn→ Dn preserves Dre
tion of P to Dre
npreserves any invariant of degree at most n.
n. The restric
We now extend an invariant ν of degree at most n to virtual descending diagrams.
For any such diagram D there exists a descending diagram Dreof a real knot with
the same double points, i.e., the same chord part of the Gauss diagram. Drecan be
obtained by turning all the virtual crossings of D into appropriate real ones. Put
ν(D) = ν(Dre). By Lemma 4.A, ν(Dre) does not depend on the choice of Dre.
Next we extend ν to all virtual diagrams. By Lemma 4.B, for any diagram
D ∈ Dn there exists m such that Pm(D) is a sum of descending diagrams and
hence ν(Pm(D)) is already defined. Put ν(D) = ν(Pm(D)). Lemma 4.C implies
that on Drethis agrees with the initial definition of ν. Since Pm+1(D) = Pm(D)
we get:
4.D. Obvious Lemma. The operator P : Dn → Dn preserves ν, i.e. ν ◦ P =
ν.
Page 19
FINITE TYPE INVARIANTS OF CLASSICAL AND VIRTUAL KNOTS 19
We are now in a position to construct the map π : A → G of Theorem 3.A.
Define π : A → G as the composition
A
I−1
− − − − → Z[D] ⊂ Dn
ν
− − − − → G,
where I−1is the isomorphism of Proposition 2.A and ν is the extension of the
original finite type invariant to Dn. Then for any diagram D of a long knot
ν(D) = π(I(D)) =
?
D′⊂D
π(i(D′)).
In order to prove Theorem 3.A, we must show that π(A) = 0 for any arrow
diagram A with more than n arrows. The rest of this section is devoted to the
proof of this fact .
4.5. The Analogues of Dn and P for Arrow Diagrams. The algebra A of
arrow diagrams on the line is generated by diagrams consisting of the line and
dashed arrows (oriented signed arcs). Consider now diagrams which in addition to
arrows also contain dashed chords, i.e. unoriented signed arcs. Denote by Anthe
free abelian group generated by such diagrams with at most n chords.
The maps i,I : Z[D] → A defined in Section 2.4 on Gauss diagrams without
chords extend to isomorphisms i,I : Dn→ An. The chord parts of the diagrams
remain intact under both i and I, while the arrows are dealt with as in Section 2.4.
We now define an operator Q : An → An, which is an analogue of P.
diagram A ∈ An is called descending, if i−1(A) is descending. Put Q(A) = A
if A is descending. Otherwise, find the leftmost bad fragment of A (the notion
of a bad fragment is borrowed from Dn via i). If it is a bad arrow, we define
Q(A) = iPi−1(A). If it is a bad chord, put Q(A) =?A′where the sum runs over
in Figure 11, all the chords and at least one more arrow. In other words, we sum
up all seven subdiagrams of iPi−1(A) which contain all the arrows and chords also
belonging to A plus at least one more arrow.
A
all the subdiagrams of iPi−1(A), each of which contains all the arrows not shown
4.E. Remark. Observe that in both cases, we sum up all subdiagrams of diagrams
in iPi−1(A) which are not subdiagrams of A, but contain all arrows of A except
for the arrow involved into the bad fragment. The arrows of A which are not in
the leftmost bad fragment play a passive role in the construction of Q: if A′is a
subdiagram of A obtined by removing arrows which are not in the leftmost bad
fragment, then Q(A′) is obtained from Q(A) by removing the same arrows from
each of the summands.
4.F. Obvious Lemma. For any diagram A ∈ An, the total number of arrows and
chords in each diagram appearing in Q(A) is at least the total number of arrows
and chords in A.
4.G. Lemma. For any diagram A ∈ An, there exists m such that Qm(A) is a sum
of descending diagrams.
The proof of this Lemma is completely analogous to the proof of Lemma 4.B.
4.H. Lemma. For any nondescending diagram D ∈ Dn, there is a splitting
I(D) = U + V with U,V ∈ Ansuch that
I(P(D)) = Q(U) + V
(17)
Page 20
20 M. GOUSSAROV, M. POLYAK, AND O. VIRO
and such that U = i(D) + U′, where U′is a sum of diagrams each of which has
fewer arrows than D.
Proof. Let U be the sum of all the subdiagrams of i(D) which include the first bad
fragment of i(D). These subdiagrams contain the same bad fragment as the whole
diagram i(D). As follows from Remark 4.E, Q(U) is the sum of all subdiagrams
of diagrams in iP(D) which are not subdiagrams of i(D). Then V is the sum of
the subdiagrams of i(D) which do not contain the arrow from the bad fragment (in
the case of a bad chord, this is the arrow shown on the left hand side of Figure 11)
and these subdiagrams of i(D) remain unchanged, when one applies P to D. Thus
I(P(D)) = Q(U) + V .
4.I. Lemma. The operator Q : An→ Anpreserves π, i.e. π ◦ Q = π.
Proof. Let A ∈ Anbe a diagram and D = i−1(A). Let us prove that π(Q(A)) =
π(A) by induction on the number of arrows in A. If this number equals 0, then
A is descending and Q(A) = A by definition of Q. Suppose inductively that the
statement is correct for any diagram whose number of arrows is less then the number
of arrows in A and let us prove the statement for A. Apply π to (17):
π ◦ Q(U) + π(V ) = π ◦ I ◦ P(D) = ν ◦ P(D)
By Lemma 4.D and the definition of π
ν ◦ P(D) = ν(D) = π ◦ I(D) = π(U) + π(V ).
Thus π ◦ Q(U) = π(U). By the induction assumption, π ◦ Q(U′) = π(U′), where
U′= U−A (as in Lemma 4.H), and we obtain the desired equality π(Q(A)) = π(A).
This completes the induction step.
4.J. Lemma. Let A ∈ Anbe a descending diagram such that the total number of
arrows and chords in A is greater than n. Then π(A) = 0.
Proof. Let D = i−1(A). By the definition of π and Proposition 2.A,
π(A) = ν ◦ R(A) =
?
D′⊂D
(−1)D−D′ν(D′).
Since any subdiagram D′of D is descending and has the same chord part, ν(D′) =
ν(D) by the construction of ν. Therefore
π(A) =
??
D′⊂D
(−1)D−D′
?
ν(D).
As one can easily check by induction on the number of arrows in A, the sum in
parentheses is equal to 1 if A has no arrows and is 0 otherwise. Since all the
diagrams in Anhave at most n chords and the total number of arrows and chords
in A is greater than n, it has at least one arrow. Hence π(A) = 0.
4.K. Lemma. Let A ∈ Anbe a diagram such that the total number of arrows and
chords in A is greater than n. Then π(A) = 0.
Proof. Let m be the number which exists for A by Lemma 4.G. By Lemma 4.I,
π(A) = π(Qm(A)). By Lemma 4.F, the expansion of Qm(A) contains only descend
ing diagrams with the total number of chords and arrows greater than n. Then by
Lemma 4.J, π(A) = 0.
Page 21
FINITE TYPE INVARIANTS OF CLASSICAL AND VIRTUAL KNOTS 21
This concludes the proof of Theorem 3.A.
5. nEquivalence
5.1. nTrivial Gauss Diagrams. Let D be a Gauss diagram and let its arrows be
colored with n colors. Consider all subdiagrams which can be obtained from D be
removing all arrows colored with one or several colors. If all arrows of each of these
diagrams can be removed by the second Reidemeister moves, then the coloring is
said to be destroying.
A Gauss diagram, based on a union of several disjoint segments, is called ntrivial
if it admits a destroying coloring with n + 1 colors.
The property of ntriviality does not change if one reverses the orientation of
some of the segments. It also does not change if one simultaneously reverses the
orientations or the signs of all arrows connecting two segments.
5.2. nVariations. On a Gauss diagram D, choose several segments which do not
contain an endpoint of any arrow. Adjoin to D arrows of an (n − 1)trivial Gauss
diagram based on the chosen segments. This transformation of D is called an
nvariation.
It is easy to see that an addition of any number of arrows is a 1variation. On a
virtual knot diagram an addition of an arrow can be realized as follows:
.
In Figure 12 we show the simplest 2variations. To get the corresponding destroy
ing coloring, one colors all arrows connecting the first two strings with one color
and the other arrows with the other color. It is easy to see that these 2variations
do not change lki/j.
−δ
δ
ε
−ε
−ε
ε
−δ
δ
Figure 12.
On a virtual knot diagram these 2variations can be realized as
Observe that these modifications coincide, up to isotopy, with the forbidden moves
of Figure 5.
Some obvious properties of nvariations are:
1. An nvariation is a kvariation for any k < n.
2. Composition of several nvariations is an nvariation.
Page 22
22 M. GOUSSAROV, M. POLYAK, AND O. VIRO
A less obvious property is: the result of an isotopy followed by an nvariation
can be presented as the result of other nvariations followed by an isotopy.
The following proposition is a key property of nvariations.
5.A. Proposition. After any nvariation, one can apply another nvariation such
that the final result is the initial diagram, up to a sequence of second Reidemeister
moves.
5.3. nEquivalence. Two Gauss diagrams are said to be nequivalent if they can
be transformed to each other by a sequence of isotopies and (n+1)variations. For
example, the Gauss diagrams shown in Figure 12 are 1equivalent. Moreover, one
can prove that any two 1equivalent Gauss diagrams can be transformed to each
other by a sequence of isotopies and the 2variations of Figure 12. A 1equivalence
class of string links is completely determined by the invariants lki/j. Therefore any
two closed virtual knots are 1equivalent and can be transformed to each other by
a sequence of the Reidemeister moves and the forbidden moves of Figure 5.
The transition from the use of Gauss diagrams to that of nequivalence classes
yields better results when the set of Gauss diagrams is equipped with a natural
multiplication. For example, in the case of virtual string links (and, in particular,
long knots) nequivalence classes form a group.
In the cases of virtual knots and (closed) links the set of nequivalence classes
has a more complicated algebraic structure. As in the case of classical links, the
following trick works. Consider a virtual string link with 2n strings. It can be
turned into a closed one by adding n arcs from above and below (this generalizes
the plat presentation of a link with braids replaced by string links). This gives rise to
a map from the group of nequivalence classes of virtual string links to the set of n
equivalence classes of closed virtual links. This map is a double coset factorization.
From the left we quotient out by string links which become nequivalent to the
trivial one by adding only arcs from below, and from the right, similarly with arcs
from above. Both sets of string links give rise to subgroups which are not normal
in general.
The value of a finite type invariant of degree ≤ n depends only on the n
equivalence class. Usually, in a nongroup situation, invariants of degree ≤ n do
not separate all the nequivalence classes. For instance, virtual closed knots do not
admit an invariant of degree 2.
References
[1] D.BarNatan, Polynomial invariants are polynomial, Harvard University Preprint, De
cember 1994.
[2] G.Cairns and D.M.Elton, The planarity problem for signed Gauss words, J. Knot The
ory and Its Ramifications 2 (1993) 359–367.
[3] D.Joyce, A classifying invariant of knots,the knot quandle, J. Pure Appl. Algebra, 23
(1982), 3765.
[4] L.Kauffman, Virtual Knots, talks at MSRI Meeting in January 1997 and AMS Meeting
at University of Maryland, College Park in March 1997.
[5] L.Kauffman, Private communication, 4.08.1998.
[6] O.P.¨Ostlund, Preprint Uppsala University, 1997.
[7] M.Polyak and O.Viro, Gauss diagram formulas for Vassiliev invariants, International
Math. Research Notices, (1994), No. 11, 445–453
[8] M.Polyak, Arrow diagrams and link homotopy invariants, Preprint
[9] K.Reidemeister, Zur dreidimensionalen Topologie, Abh. Math. Sem. Univ. Hamburg, 9¯
(1933), 189–194.
Page 23
FINITE TYPE INVARIANTS OF CLASSICAL AND VIRTUAL KNOTS 23
[10] J.Singer, Threedimensional manifolds and their Heegaard diagrams, Trans. Amer.
Math. Soc. 35 (1933), 88–111.
[11] V.A.Vassiliev, Cohomology of knot spaces, Theory of singularities and its applications
(Providence) (V.I.Arnold, ed.) Amer. Math. Soc., Providence, 1990.
M. Goussarov, POMI, Fontanka 27, St. Petersburg, 191011, Russia
Email address: goussar@pdmi.ras.ru
M. Polyak, School of Mathematics, TelAviv University, 69978 TelAviv, Israel
Email address: polyak@math.tau.ac.il
O. Viro, Department of Mathematics, Uppsala University S751 06 Uppsala, Sweden;
POMI, Fontanka 27, St.Petersburg, 191011, Russia.
Email address: oleg@math.uu.se
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