virtual clasper on long virtual knots
A -move is a family of local moves on knots and links, which gives a topological characterization of finite type invariants of knots. We extend the -move to (long) virtual knots by using the lower central series of the pure virtual braid, and call it an -move. We then prove that for long virtual knots an -equivalence generated by -moves is equal to -equivalence, which is an equivalence relation on (long) virtual knots defined by Goussarov-Polyak-Viro. Moreover we directly prove that two long virtual knots are not distinguished by any finite type invariants of degree if they are -equivalent, for any positive integer .
2010 Mathematics Subject Classification. 57M25, 57M27
The theory of finite type invariants of knots and links was introduced by Vassiliev  and Goussarov [4, 5] and developed by Birman-Lin . Goussarov [6, 7] and Habiro [9, 10] independently introduced theories of surgery along embedded graphs in 3-manifolds, called -graphs or variation axes by Goussarov, and claspers by Habiro. An -variation equivalence (called -equivalence in ) or -equivalence for links is generated by n-variations  or -moves , respectively. Goussarov proved in  that for string links and knots in , the n-variation (or -) equivalence coincides with the Goussarov-Ohyama -equivalence [4, 15]. Stanford proved in  that two links are not distinguished by any finite type invariant of degree if one is obtained from the other by inserting an element of the -th lower central series subgroup of the pure braid group. Goussarov  and Habiro [9, 10] independently proved that two knots are not distinguished by any finite type invariant of degree if and only if they are related by a finite sequence of -moves and ambient isotopies. Moreover Stanford  translated Habiro’s result for -moves into the pure braid setting.
On the other hand, a long virtual knot is defined by a (long) knot diagram with virtual crossings module Reidemeiser moves, introduced by Kauffman . Goussarov-Polyak-Viro  showed that the (long) virtual knot can be redefined as Gauss diagram and also gave the theory of finite type invariants on Gauss diagrams. They also defined an -equivalence on (long) virtual knots and notioned that the value of a finite type invariant of degree less than or equal to depended only the -equivalence class.
In this paper, we extend a -move to (long) virtual knots, called an -move, by using Stanford’s method. The -moves generate the -equivalence on (long) virtual knots. We prove that -equivalence coincides with -equivalence on long virtual knots. Moreover, we directly prove that, for any non-negative integer , if two long virtual knots are -equivalent, then they are not distinguished by any finite type invariants of degree . Their extensions and results are also establish on virtual string links.
The author thanks Professor Kazuo Habiro for a lot of comments, discussions and suggestions. The author also thanks Professor Vassily Manturov for comments and suggestions.
2. Gauss diagram
A Gauss diagram on the interval is an oriented interval with several oriented chords having disjoint end points and equipped with sign as in Figure 1. Here, we call the chord an arrow.
Reidemeister moves among Gauss diagrams are the following three moves in Figure 2: First Reidemeister move (RI) is in the top row. Second Reidemeister move (RII) is in the second row. Third Reidemeister move (RIII) is in the remain two rows.
Two Gauss diagrams and are said to be equivalent if and are related by Reidemeister moves. By we mean that and are equivalent. We define a long virtual knot to be the equivalence class of a Gauss diagram , which is denoted by . Similarly, the equivalence class of Gauss diagram on circle ( intervals) is virtual knot (-component virtual string, respectively).
3. Finite type invariant of virtual knot
Goussarov Polyak and Viro defined a finite type invariant for (long) virtual knots in . Similar way to classical knots, we can define Vassiliev-Goussarov filtration on -module generated by the set of (long) virtual knots.
Let be the set of long virtual knots. For each , let denote the set of equivalence classes of Gauss diagrams with dashed arrows equipped with sign with fixing dashed arrows. We construct a map as follows. Let be a Gauss diagram with dashed arrows. Let be the dashed arrows of . For in , let denote the Gauss diagram obtained from by replacing each dashed arrow with an arrow if and removing each dashed arrow if . We then define
Let be an invariant of with values in an abelian group . We extend it to by linearly. Then is said to be a finite type invariant of degree if vanishes for any long virtual knot with more than dashed arrows.
Denote by the subgroup of generated by the set consisting of the element , where is in . It is easy to see that the ’s form a descending filtration of two-sided ideals of the monoid ring under the composition:
which we call the Vassiliev-Goussarov filtration on . Here for Gauss diagrams (or virtual knots) and ( and ), we denote by () their composition.
Later, we will redefine by using claspers.
Let be an abelian group and a positive integer. The following two conditions are equivalent. A map is an -valued finite type invariant of degree on and the map is a homomorphism of into which vanishes on
For , two long virtual knots and are said to be -equivalent if and are not distinguished by any finite type invariants of degree with values in any abelian group, equivalently, .
4. Definition of -equivalence
By using the pure virtual braid group, we introduce a new equivalence relation on Gauss diagrams, called -equivalence. Because the pure braid group is a subgroup of the pure virtual braid group (see [3, 13]), this is an extension of -equivalence. We then give properties of the set of -equivalence classes.
A pure virtual braid group on strands is a group represented by the following group representation.
Here, the element of the pure braid group is represented by a diagram as in Figure 3, where is correspondence with a horizontal arrow equipped with sign from the -th strand to the -th strand, and we determine that the orientation of the strand is from top to bottom. For example, the diagram in Figure 3 is correspondence with .
Let and . We denote the composition and tensor product of two elements of the pure virtual braid group as if and for any , , respectively. By we mean the -th lower central subgroup of the group , that is, and , which is the commutator of and , that is where .
Two Gauss diagrams and are related by an -move if there are a positive integer , an element in the -th lower central subgroup of the pure virtual braid group on strands and not in , and an embedding of strands such that , where is obtained from by attaching by an embedding of strands of in the interval of except for the end points of all arrows of as in Figure 4. By we mean that is obtained from by -move. In particular, we write if .
We call a pair for a clasper for . We define that a clasper is of degree if and , where is a positive integer, and denote the degree of the clasper by deg. Hereinafter, we omit the number of strands if it is not important. In particular, we call a pair a tree clasper for if is an -th commutator where and a forest clasper otherwise. Two claspers for are disjoint if the embeddings of all strands of claspers are disjoint in the interval of . For disjoint claspers and for , means or equivalently .
An -equivalence is an equivalence relation on Gauss diagrams generated by the -moves and Reidemeister moves. By we mean that and are -equivalent.
The -equivalence is an equivalence relation on Gauss diagrams.
First of all, we show the reflexive relation. For any and , the identity element and for any embedding . Therefore . Secondly, we show the symmetric relation. Let where . Then there is an embedding of such that . Since the Gauss diagram is up to a sequence of second Reidemeister moves, we have that . Finally, the case of transitive relation is obvious. ∎
If , then an -move is achieved by an -move. Therefore -equivalence implies -equivalence.
By the property of the lower central series, . ∎
Two Gauss diagrams and are -equivalent if and only if there exists a clasper of degree such that equals to up to a sequence of Reidemeister moves.
A necessary condition is obvious. To prove a sufficient condition, we will show the following three statements (1), (2) and (3). (1) If is obtained from by a first (second or third, respectively) Reidemeister move and then an -move ( or , respectively) (), then there is an -move ( or , respectively) and a sequence of Reidemeister moves such that is obtained from by the -move ( or , respectively) and then the sequence of Reidemeister moves. (2) If is obtained from by an -move and then another -move , then there is an -move () and two sequences of second Reidemeister moves such that is obtained from by one sequence of the second Reidemeister moves and then the -move and then the other sequence of the second Reidemeister moves. (3) For any clasper of degree more than or equal to for , there exists a clasper of degree for such that , because any is represented by the product of elements in . By (1), (2) and (3), if and are -equivalent, there is an -move and a sequence of Reidemeister moves such that is obtained from by the -move and then the sequence of Reidemeister moves.
We show (1). We consider the case of the first Reidemeister move RI. In Figure 5, these Gauss diagrams are identical except in a local place of RI represented by this figure. By gray line we mean a clasper. Given a clasper , we can move the ends of chords of clasper out the arrow derived from RI by a sequence of second Reidemeister moves. We denote the obtained clasper by (See Figure 5). Moreover similar considerations apply to the other first Reidemeister move.
Similar way to RI, in the case of RII and RIII, we give claspers and as in Figure 6 and 7, which are one of RII and RIII. Here, in Figure 5 for simplicity we draw only one strand is embedding in each interval between endpoints of arrows derived from RIII.
We show (2). Given a Gauss diagram and a clasper for , we can transform to by a sequence of the second Reidemeister moves. Let and . There is an embedding of in such that . Moreover, is up to a sequence of the second Reidemeister moves. We set . ∎
It is obvious that Proposition 4.6 is equivalent to the following statement. There exists the union of disjoint claspers of degree such that equals to up to a sequence of the Reidemeister moves.
In , Meilhan and Yasuhara also extended the concept of the clasper to welded knots, which is a quotient of virtual knot.
Let . Let be a Gauss diagram and a clasper of degree for . Then for any Gauss diagram which is equivalent to there is a clasper of degree for such that is equivalent to .
Since and , we have that . It is from Proposition 4.6 that there is a clasper of degree for such that . ∎
A Gauss diagram is -trivial if is -equivalent to the trivial Gauss diagram .
The next proposition is well-known fact of group theory.
Let be a group. Let and be elements in the -th and -th lower central subgroup of , respectively. Then the commutator of and is in -th lower central subgroup of .
Let be a Gauss diagram. Let , . Let be a clasper of degree and a clasper of degree for , where they are disjoint. Let be the -th strand of and the -th one of . Suppose that there is no end point of arrows and no embedding of another strands of claspers on the interval between embeddings and . Then, these embeddings may replace each other up to -equivalence as in Figure 8. Let and be claspers of degree and obtained from and by replacing and as in Figure 8. Then, there exists a clasper of degree such that is equivalent to .
We call the transformation between two claspers a sliding.
For and , we construct and its embedding . Let be an element of pure virtual braid from by adding strands before 1st strand of and strands after -th strand of . Let be an element of from by adding strands between -th and -th strand of and, strands between -th and -th strand of . Then the -th strand of and -th strand of are the same order in and .
If both of the orientations of and are compatible or not with the orientation of the interval of , the product has a natural embedding induced by and . Let . Then by Proposition 4.12. Since , we may replace and each other and leave other embeddings up to -equivalence. If only one of the orientations of and is compatible with that of the interval of D, to adjust and the orientation we set , where is the mirror image of for a horizontal line and its embedding is induced by and . ∎
Let . Let be an -trivial Gauss diagram and be an -trivial one. Then the Gauss diagram is -equivalent to .
By assumption, there are two claspers and with and such that and . Then by Lemma 4.13 we have . ∎
For any -trivial Gauss diagram , there is an -trivial Gauss diagram such that both and are -trivial.
By assumption, there is a clasper with such that . We define . Then by Lemma 4.13 we have . ∎
The set of equivalence classes of Gauss diagrams has a monoid structure under the composition. For , let denote the submonoid of consisting of the equivalence classes of Gauss diagrams which are -trivial. There is a descending filtration of monoids
For , denotes the quotient of by -equivalence. It is easy to see that the monoid structure on induces that of . There is a filtration on of finite length
For , the monoid is an abelian group.
Let . We then have as follows.
(1) The monoid is a group.
(2) for with .
(3) The group is nilpotent.
(1) We fix and prove it by induction on . If , it is obvious. Assume that is a group for some with . We then have a shot exact sequence of monoids:
Here, and are groups by the assumption of induction and Lemma 4.17.
Therefore is also a group.
(2) It is from Proposition 4.14, and are commute up to -equivalent. Here . Therefore .
(3) From (2), it is easy to check. ∎
5. -equivalence and -equivalence
In this section, we prove that -equivalence coincides with -equivalence defined by Goussarov-Polyak-Viro .
 Let . A Gauss diagram on strands is said to be -trivial if the Gauss diagram satisfies the following condition. There exist non-empty disjoint subsets of the set of arrows of such that for any non-empty subfamily of the set the Gauss diagram obtained from by removing all arrows which belongs to the subfamily is trivial up to a sequence of second Reidemeister moves.
Two Gauss diagrams and are related by -variation if is obtained from by attaching an -trivial Gauss diagram on several strands to segments of without endpoints of any arrow. Two Gauss diagrams are said to be -equivalent if they are related by -variations and Reidemeister moves.
For any , -equivalence and -equivalence on long virtual knots are equal.
It is obvious that if two Gauss diagrams are -equivalent, then they are -equivalent. Therefore it suffices to prove that if Gauss diagrams and are related by an -variation then they are -equivalent.
Let be an -trivial Gauss diagram such that is obtained from by attaching . Let be disjoint sets of arrows of satisfying the condition in Definition 5.1. By the property of -triviality, coinsids with the Gauss diagram obtained from by removing all arrows in up to second Reidemeister move. Therefore, by the method of (1) in proof of Proposition 4.6 it is sufficient to consider the case that all arrows of belong to . Let be the set of tree claspers of degree 1 corresponding to the arrows in .
We define a weight for a clasper , which is a subset of , and denote it by . We consider as a set of tree claspers each clasper of which assigns as weight if the clasper corresponds with an arrow of . Let be a finite subset of . Then denote the subset of each clasper of which has a subset of as weight, and denote the subset of each clasper of which has as weight. Let . We then can regard as . Moreover, by the property of -triviality if is a proper subset of , then up to a sequence of second Reidemeister moves. We show the following claim, which proves the theorem.
Suppose that and are as above. Then there exists a set of tree claspers of degree with weight such that is equivalent to .
Let us first prove the case that is equivalent to the trivial Gauss diagram . Then by Lemma 4.9, it is sufficient to show that the case that is trivial. To prove this claim, we prove the following statement depending on a positive integer .
(A) There exists a set of tree claspers () for such that for each deg() where means the number of a set, and for every subset of .
We prove it by induction on for . For , we can set . Under the assumption of the claim, assuming the statement (A) to hold for , we will prove it for . Let be a set of tree claspers for satisfying (A) for . We take a subset of such that and is not empty. Then we shift all tree clasper in to the ahead with fixing claspers in by sliding of claspers (Lemma 4.13) until all end points of all clasper in are completely to the ahead of those in . We denote the obtained set of tree claspers for by . Here we define the weight of new tree claspers obtained by sliding in Lemma 4.13 as follows. If two claspers have the weight and , then the new tree clasper has the weight . We remark that for every and deg() for any new tree .
Let be a set of tree claspers for the Gauss diagram . We consider a subset of . If , then it is clear that . If not, then the new tree claspers do not contain in . Hence . By the assumption of claim, is equivalent to . Therefore the Gauss diagrams and are equivalent for every .
Repeating this procedure for such that , we obtain a set of tree claspers ’s with for a Gauss diagram which is equivalent to . By Lemma 4.9, we obtain a set of tree claspers preserving above condition for , which is the required set satisfying (A) for . This proves the claim for the case that .
Next we prove the case that is not equivalent to the trivial one. Since the set of -equivalence classes has a group structure, there is an inverse of up to -equivalence. Then is -equivalent to . It is -equivalent to , since and are -equivalent. Because is . By Proposition 4.6, there exists a clasper of degree such that . Therefore it follows from the case that there exists a set of claspers of degree such that . Hence is -equivalent to . ∎
6. -equivalence and -equivalence
Goussarov-Polyak-Viro  mentioned that the value of a finite type invariant of degree less than or equal to depends only on the -equivalence classes. Therefore it follows from Theorem 5.2 that -equivalence implies -equivalence, indirectly. In this section, we give this relation directly, by redefining the two-sided ideal of the monoid ring by using claspers.
Let . A scheme of size , , for a Gauss diagram is the set of disjoint claspers for . Denote an element of by
where runs over all subsets of . The degree of a scheme is the sum of the degree of its elements, denoted by deg().
Let be a Gauss diagram and a scheme of size for of degree . Then for any Gauss diagram which is equivalent to there is a scheme of size for of degree such that is equal to in .
It is easy to check. ∎
Let , be integers with . Let