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arXiv:0904.2862v1 [math.GT] 18 Apr 2009

Cobordisms of Free Knots and Gauss Words
D. P. Ilyutko V. O. Manturov ,

Abstract We investigate cob ordisms of free knots. Free knots and links are also called homotopy classes of Gauss words and phrases. We define a new strong invariant of free knots which allows to detect free knots not cob ordant to the trivial one.

1

Introduction

The aim of the present work is to consider cobordisms of free knots. Free knots and links [Ma] (also called homotopy classes of Gauss words and phrases, see [Tu, Gib]) are a substantial simplification of homotopy classes of curves on 2-surfaces, and at the same time, a simplification of virtual knots and links, introduced by Kauffman in [Ka]. A conjecture due to Turaev [Tu] stating that all free knots are trivial was solved very recently indep endently by Manturov [Ma] and Gibson [Gib]. We consider cobordism classes of free knots and give a strong invariant which allows to detect free knots not cob ordant to the trivial one.

2

Basic Definitions

We shall encode free knots and links by using framed graphs. Throughout the pap er by a four-valent graph we mean the following generalization: a finite 1-complex with each connected comp onent b eing homeomorphic either to a circle or to a four-valent graph; by a vertex of a four-valent graph we mean only vertices of those comp onents which are homeomorphic to four-valent graphs, and by edges we mean b oth edges of four-valent graphs and circular comp onents (the latter will b e called cyclic edges). We call a four-valent graph framed if for every vertex of it a way of splitting of the four incident half-edges into two pairs is indicated. Half-edges b elonging to the same pair are


Partially supported by grants of RF President NSh н 660.2008.1, RFFI 07н01н00648, RNP 2.1.1.7988.

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to b e called opposite. We shall also use the term opposite with resp ect to edges containing opp osite half-edges. By an isomorphism of framed 4-graphs we always mean a framing-preserving homeomorphism. By a unicursal component of a framed four-valent graph we mean either its connected comp onent homeomorphic to the circle or an equivalence class of edges of some noncircular comp onent, where the equivalence is generated by the opp osite relation. Let K b e a framed four-valent graph, and let X b e a vertex of X . Let (a, c) and (b, d) b e the two pairs of opp osite half-edges of the graph K at X . By smoothings of K at X we call two framed four-valent graphs obtained from K by deleting X and reconnecting the adjacent (non-opp osite) edges: for one graph we connect the pairs (a, b) and (c, d), and for the other graph we connect (a, d) and (b, c). Consider a chord diagram1 C . Then the corresp onding framed four-valent graph G(C ) with a unique unicursal comp onent is constructed as follows. With the chord diagram having no chords one associates the graph G0 consisting of a unique circular comp onent. Otherwise the edges of the graph are in one-to-one corresp ondence with arcs of the chord diagrams, and vertices are in one-to-one corresp ondence with chords. Those arcs which are incident to the same chord end, corresp ond to formally opp osite half-edges. The first Reidemeister move for four-valent framed graphs is an addition/removal of a loop, see Fig. 1. The second Reidemeister move is an addition/removal of a bigon, formed by a pair of edges, which are adjacent (not opp osite) to each other at each of the two vertices, see Fig. 2. The third Reidemeister move is shown in Fig. 3. Definition 2.1. A free link is an equivalence class of framed four-valent graphs modulo Reidemeister moves. Obviously, the numb er of comp onents of a framed four-valent graph does not change under Reidemeister moves, thus, one can talk ab out the numb er of comp onents of a framed links. A free knot is a 1-comp onent free link. Free knots can b e treated as equivalence classes of corresp onding Gauss diagrams (chord diagrams) modulo corresp onding moves on Gauss diagrams (which mimic the moves on four-valent framed graphs). Analogously, for free-links one can introduce Gauss diagrams on several circles which stand for different comp onents of the link, and chord ends may b elong to different circles. Here a chord diagram (Gauss diagram) on one circle is a split collection of unordered 0 0 pairs of distinct p oints on a circle S1 § § § Sk S 1 . The circle S 1 is here called the cycle
Usually in literature one considers oriented and non-oriented singular knots and corresponding oriented and non-oriented chord diagrams. In the present paper, we restrict ourselves to the non-orientable case. Many theorems and constructions can be easily extended to the orientable case.
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2


Figure 1: The First Reidemeister Move and Its Chord Diagram Version

Figure 2: The Second Reidemeister Move and Its Chord Diagram Version

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Figure 3: The Third Reidemeister Move and Its Chord Diagram Version
of the chord diagram. 0 Every pair of p oints Si is called a chord and p oints from this pair are called chord 0 0 0 ends. We call two chords Si , Sj linked if the chord ends Si b elong to different connected 1 \S 0 . comp onents of S j A chord is even if the numb er of chords linked with it, is even, and odd otherwise. By an even symmetric configuration C on a chord diagram D we mean a set of pairwise disjoint segments Ci on the cycle of the chord diagram which p ossess the following prop erties: 1) the ends of the segments do not coincide with chord ends; 2) the numb er of chords inside any segment is finite; 3) the numb er of endp oints of chords inside any segment is even; 4) every chord having one endp oint in C has the other endp oint in C . 5) Consider the involution i of the cycle which fixes all p oints outside the segments Ci and reflects all segments along the radii. This involution naturally defines new p oints to b e chord ends, and thus defines the new chord diagram i(D ). We require that the configuration C is symmetric, i. e., chord diagrams D and i(D ) are equal. Note that if a chord has two endp oints in different segments Ci and Cj then it is not self-symmetric. By an elementary cobordism we mean a transformation of a chord diagram deleting all chords b elonging to the even symmetric configuration, as well as the inverse transformation.

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We say that two Gauss diagrams are cobordant if one can b e obtained from the other by a sequence of elementary cob ordisms and third Reidemeister moves. Remark 2.1. The first two Reidemeister moves are partial cases of elementary cob ordisms, unlike the third Reidemeister move. Since the first two Reidemeister moves are partial cases of elementary cob ordisms, it makes sense to talk ab out cobordism classes of free knots, not only cobordism classes of Gauss diagrams. Remark 2.2. The definition of cob ordisms given ab ove agrees with the definition of word cobordism (nanoword cob ordism) [Tu]. With each (cyclic) double occurrence word in some given alphab et one associates a Gauss diagram, and words which are cob ordant in Turaev's sense yield cob ordant diagrams. However, nanowords usually corresp ond to Gauss diagrams with some decoration (labeling ) of chords, and the notion of elementary cob ordism usually requires some conditions imp osed on chords b elonging to the symmetric configuration. In this sense, cob ordism classes of free knots are the simplest variant of cob ordism classes of nanowords. Nevertheless, we show that there are free knots which are not cob ordant to zero. The main result of the present pap er is to prove the existence of (in fact, infinitely many) free knots which are not cob ordant to zero. To prove this problem, we shall construct a cob ordism invariant for free knots.

3

The Mapping and Its Iterations

Let K b e a framed four-valent graph. Each unicursal comp onent Ki of K can b e treated as a four-valent framed graph with a Gauss diagram Di . Thus, some vertices of the graph K can b e represented by chords of one of Di 's (namely, those vertices lying on one unicursal comp onent). Among these, let us choose even vertices (in the sense of the Gauss diagram Di ), and at each even vertex X of K , we consider the smoothing KX (one of the two p ossible) for which the numb er of unicursal comp onents is greater than that of K by 1. Now let G b e the set of Z2 -linear combinations of equivalence classes of framed fourvalent graphs modulo second and third Reidemeister moves. Set (K ) = KX G , where the sum is taken over all crossings X of K . Then the following statement holds. Statement 3.1. The map is a wel l defined map from G to G .
X

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Proof. Indeed, assume K is obtained from K by a third Reidemeister move. Consider the three crossings a1 , a2 , a3 of K involved in this move, and the corresp onding crossings a , a2 , a of K . By construction, ai lies in one unicursal comp onent of K if and only if a 3 1 i lies in one unicursal comp onent of K . Moreover, a is even if and only if ai is even. It is now easy to see that whenever ai is even, the smoothing Kai gives the same impact to G as that of Ka (the corresp onding framed 4-valent graphs are either isomorphic or i differ by a second Reidemeister move). Now, if K and K differ by a second Reidemeister move and K has two more crossings a, b in comparison with K , then it obviously follows that either b oth crossings a, b are even or none of them is even. In the first case, the summands in (K ) are in one-to-one corresp ondence with those in (K ) and the corresp onding diagrams in each pair differ by a second Reidemeister move. If b oth a and b are odd, then it is obvious that the smoothings at these crossings give equal impact to K , and since we are working over Z2 , they cancel each other. Consequently, if we take k times, the resulting map k is also invariant. So, if K and K are two framed four-valent graphs which are obtained from each other by a third Reidemeister move, then for every p ositive integer k we have k (K ) = k (K ).

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The map

Let L b e a framed four-valent graph with k unicursal comp onents. With L we associate a graph (L) (not necessarily four-valent, but without loops and multiple edges) and a numb er I (L) according to the following rule. The graph (L) will have k vertices which are in one-to-one corresp ondence with unicursal comp onents of L. Two vertices are connected by an edge if and only if the corresp onding comp onents share an odd numb er of p oints. The following statement is evident. Statement 4.1. If two framed four-valent graphs L and L are homotopic then (L) = (L ). Now, we define the numb er j (L) from the graph (L) in the following way. If (L) is not connected we set j (L) = 0, otherwise j (L) is set to b e the numb er of edges of (L).

5

The invariant

Fix a natural numb er n. Let G b e a linear space generated over Z2 by formal vectors {ai }, i N.

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Let K b e a framed four-valent graph with one unicursal comp onent, and let n (K ) b e the corresp onding linear combination of four-valent graphs. For a four-valent framed graph L, we set I (L) = aj (L) if j (L) = 0 and I (L) = 0 otherwise. We extend this map to Z2 нlinear combinations of framed four-valent graphs by linearity. Now, set I(n) (K ) = I (n (K )). The main result of the pap er is the following Theorem 5.1. If K and K are cobordant then I
(n)

(K ) = I

(n)

(K ).

The proof of this theorem follows from two statements. By virtue of Statement 4.1, the mapping I(n) (§) is invariant with resp ect to the third Reidemeister move (since so is n ). Moreover, the following statement holds Statement 5.1. If K2 is obtained from K1 by an elementary cobordisms (removal of an even symmetric configuration ), then I(n) (K1 ) = I(n) (K2 ) Having proved Statement 5.1, we shall get Theorem 5.1. Indeed, it will follow that I(n) is invariant under all elementary cob ordisms and third Reidemeister moves, hence, under arbitrary cob ordisms. We will prove Statement 5.1 in the last section of our pap er

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6

An example

Example. Consider the free knot K represented by the Gauss diagram shown in Fig 4. Let us calculate I(3) (K ). We have: ( 2 ( = + )= + ) + ( + + + ) + ( + + , ) ,

) = (

3 (

) = (

) + (

) + (

)

+(

) + (

) + (

)

=

+

+

+

+

+

=

+

.

Thus, applying 3 we get I(3) (K ) = a4 . Thus, by Theorem 5.1, the cob ordism class of K is non-trivial. Obviously, there are infinitely many such examples. We shall construct them and discuss the cobordism group of free knots as well as further invariants in a separate publication

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Sketch of the Proof of Statement 5.1

Let K2 b e the Gauss diagram obtained from a Gauss diagram K1 by deleting an even ~ symmetric configuration K . The chords of K1 b elong to three sets: 1. the set of those chords which corresp onding to chords of K2 , we denote them for b oth K1 and K2 by j 's. ~ 2. the set of those chords j which are fixed under the involution i on K ,

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Figure 4: The Gauss diagram of a free knot not cobordant to the unknot
3. the set of pairs of chords k and k = i(k ) which are obtained from each other by п the involution i (here k = k ). п Now, for every framed four-valent graph L, n (L) is a sum of some subsequent smoothings (p1 ...pk ) (L), where all pi 's are vertices of L where pi occurs to b e even after smoothing all p1 , . . . , pi-1 . Now, n (K1 ) naturally splits into 3 typ es of summands: 1. Those where all pi are some j . These smoothings are in one-to one corresp ondence with smoothings of K2 . We claim that the corresp onding elements I ((p1 ...pk ) (K1 )) and I ((p1 ...pk ) (K2 )) are equal. 2. Those where at least one of pi is j , and neither 's nor 's occur among pi . We п (p1 ...pk ) (K )) is zero. claim that each of these summands I ( 1 3. Those summands where at least one of pi 's is j or j . п пп These summands are naturally paired: the elements I ((p1 ...pk ) (K1 )) and I ((p1 ...pk ) (K1 )) are equal. Let us first prove 1. Since segments of our even symmetric configurations have no common p oints with chords j , we see that after smoothing along any of 's, every segment will completely b elong to one circle. This means that the corresp onding graphs ((p1 ...pk ) (K1 )) and ((p1 ...pk ) (K2 )) are isomorphic. Indeed, the vertices corresp onding to do not change the graph at all, since every vertex corresp onding to is an intersection of one comp onent of ... with itself. Moreover, the vertices Xi and Xi corresp onding to i and i b elong to the same pair п of comp onent, so their impact to the graph cancels. To prove 2, let us take one pj = k , and consider the segment Ci of K1 where k lies. Without loss of generality, we may assume that k is the innermost chord in K1 amongst those chords pl we use for smoothings. Now, our summand looks like ...k ... . Smoothing K1 along k cuts a free knot (comp o-

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nent) which has ends only in the segment Ci . It is obvious that this unicursal comp onent will b e split in the sense of the graph : it will have even intersection with any other unicursal comp onent (since whenever it shares a vertex X of some other comp onent, this vertex necessarily corresp onds either to some (or to some ) and the corresp onding п п (resp., ) will b e shared by the same two comp onents). The proof of 3 follows from one basic fact: the number of components of a 1-manifold obtained from a chord diagram by smoothing some of its chords can be defined from the intersection graph of this chord diagram, see [Sob]. We shall give a rigorous proof of 3 in a separate publication.

The authors express their gratitude to V.P.Ilyutko for implementing a computer program calculating the invariant I.

References
[Gib] Gibson, A., Homotopy Invariants of Gauss Words, ArXiv:Math.GT/0902.0062. [Ka] L. H. Kauffman, Virtual knot theory, Eur. J. Combinatorics. 1999. V. 20, N. 7, pp. 662н690. [Ma] Manturov, V.O., On Free Knots, ArXiv:Math.GT/0901.2214 [Sob] E. Sob oleva, Vassiliev Knot Invariants Coming from Lie Algebras and 4-Invariants (2001), Journal of Knot Theory and Its Ramifications, 10 (1), pp. 161н169. [Tu] Turaev, V.G., Cobordisms of Words, Arxiv:math.CO/0511513 , v.2.

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