OON GROUPS G n AND IMAGINARY GENERATORS
S.KIM AND V.O.MANTUROV
Abstract.
In the present paper, we construct a monomorphism from (Artin)pure braid group
P B n into a group, which is ‘bigger’ than P B n . Roughlyspeaking, this mapping is defined on words of braids by adding ‘new genera-tors’ between generators of P B n . By this mapping we can get a new invariantfor classical braids. As one of application of this invariant, we will show ex-amples, which are minimal words in P B n and the minimality can be shownby the invariant.MSC 57M25, 57M27. Introduction
In the papers [6, 8], the autors initiated the study of groups G kn , describingthe behaviour of dynamical systems of n particles governed by a certain generalposition codimension 1 property with respect to k -tuples of particles. The mainexamples coming from this theory are homomorphisms from the n -strand pure braidgroup to the groups G n and G n . It turns out that the standard Artin presentationcan also be described as a modification of such a standard presentation of (somesubgroup of) G n . Stadard presentations of groups G kn have a nice property: eachletter in each word of this presentation has lots of “invariants”: when applyingeach of the relations we either get a bijection between similar letters (in this casethe corresponding letters have the same invariants), or we get a cancellation of twoletters (in this case these two letters have the same values of the invariant). Thisleads to two obvious invariants.First,for a given word β from G kn and for each generator a m in β we get ‘indices” i a m ( i ) valued in the free product of ( k − n − k ) copies of the group Z for each i ∈ { , · · · n }\ m ; roughly speaking, we get an invariant by counting the number ofgenerators a m (cid:48) , which occur before a m in the word β , for each m (cid:48) ⊂ { , · · · n } , suchthat | m (cid:48) ∩ m | = k − i ∈ m (cid:48) .On the other hand, by using these indices one can construct a homomorphismfrom the group G kn to a free product of the groups Z . The non-triviality of theimage for this homomorphism can be easily checked and provides a sufficient con-diton for the initial braid to be non-trivial. In the present paper, by using a simpleintutive construction we show how to associatie with any word in Artin’s generatora certain word in a larger set of generators containing the initial word inside (The-orem 2.5). In other words, we get a phenomenon of imaginary generators whichallows one to read between letters : for a word in letters σ ij we can “see” the letters a ijk , placed between σ ij , so tat the equivalence of the initial words in σ ij leads tothe equivalence of the resulting words in σ ij , a ijk , see Definiton 2.3. In the algebraiclanguage this is described by means of an injection of a smaller group to a larger a r X i v : . [ m a t h . G T ] D ec S.KIM AND V.O.MANTUROV one; the compostion of this homomorphism with the obvious projection is the iden-tity homomorphism. This allows one to use the larger group (a small modificatonof the group G n ) as a modification of the small group (actualy, the braid group);which makes it possible to use G n for constructing invariants of crossings of a clas-sical braid: with each classical crossing we associate a set of invarians which do notchange under the third Reidemeister move; if we apply the second Reidemeistermove, the crossings which can cancel have the same invariants.This allows us to “foresee” that some two crossings of a classical braid can notbe cancelled. In Section 2, we shall define the group (cid:101) G n , which appears when wesplice “classical braid generators” into the group G n . In the Setion 3, we constructa new invariant of pure braids by using (cid:101) G n by means of a homomorphism from G n to the free product of some copies of the group Z (these homomorphisms lead toevident composite homomorphisms from (cid:101) G n to the free product mentioned above).In Secton 4, we get a sufficient condition for two adjacent generators b ij and b − ij not to cancel. In the present paper, we do not pretend to consturct necessary andsuficient condition for cancellability; we just give some striking examles when thiscancellability is impossible.In the end of the paper, we give a list of unsolved problems and topics for futurediscussion. 2. Homomorphisms from classical braids to (cid:101) G n Denote ¯ n := { , · · · , n } . Definition 2.1. [2]
The pure braids group
P B n is the group given by group pre-sentation generated by { b ij | ≤ i < j ≤ n } subject to the following relations: b rs b ij b − rs = b ij , if s < i or j < r,b − is b ij b is , if i < j = r < s,b − ij b − ir b ij b ir b ij , if i < j < r = s,b − is b − ir b is b ir b ij b − ir b − is b ir b is , if i < j < r < s. Definition 2.2.
The group G n is the group given by group presentation generatedby { a { ijk } | { i, j, k } ⊂ ¯ n, |{ i, j, k }| = 3 } subject to the following relations: (1) a { ijk } = 1 for { i, j, k } ⊂ ¯ n , (2) a { ijk } a { stu } = a { stu } a { ijk } , for |{ i, j, k } ∩ { s, t, u }| < , (3) a { ijk } a { ijl } a { ikl } a { jkl } = a { jkl } a { ikl } a { ijl } a { ijk } for dictinct i, j, k, l .We denote a ijk := a { ijk } . Note that a ijk = a jik = · · · , but b ij (cid:54) = b ji . In [8] V.O.Manturov and I.N.Nikonovworked on that braids on n strands can be considered as dynamical systems, inwhich n distinct points move on the plane. Roughly speaking, the image of a braidin the group G n can be obtained by reading every moment, in which three points( i, j, k ) are placed on the same line. More precisely, the homomorphism φ n from P B n to G n is defined by(2.1) φ n ( b ij ) = ( c i,i +1 ) − ( c i,i +2 ) − · · · ( c i,j − ) − ( c i,j ) c i,j − · · · c i,i +2 c i,i +1 , for each generator b ij of the group P B n , i, j ∈ ¯ n, i < j , where(2.2) c i,k = n (cid:89) l = k +1 a ikl k − (cid:89) l =1 a ikl , for k ∈ { i + 1 , · · · , j } . Definition 2.3.
The group (cid:101) G n is given by group presentation generated by { a { ijk } | { i, j, k } ⊂ ¯ n, |{ i, j, k }| = 3 } and { σ ij | i, j ∈ { , . . . , n } , |{ i, j }| = 2 } subject to the followingrelations: (a) a { ijk } = 1 for { i, j, k } ⊂ { , · · · , n } , |{ i, j, k }| = 3 , (b) a { ijk } a { stu } = a { stu } a { ijk } , if |{ i, j, k } ∩ { s, t, u }| < , (c) a { ijk } a { ijl } a { ikl } a { jkl } = a { jkl } a { ikl } a { ijl } a { ijk } for dictinct i, j, k, l , (d) σ ij σ kl = σ kl σ ij for dictinct i, j, k, l , (e) σ ij a { stu } = a { stu } σ ij , if |{ i, j } ∩ { s, t, u }| < , (f) a { ijk } σ ij σ ik σ jk = σ jk σ ik σ ij a { ijk } for dictinct i, j, k , (g) σ ij a { ijk } σ ik σ jk = σ jk σ ik a { ijk } σ ij for dictinct i, j, k , (h) σ ij σ ik a { ijk } σ jk = σ jk a { ijk } σ ik σ ij for dictinct i, j, k , (i) σ ij σ ik σ jk a { ijk } = a { ijk } σ jk σ ik σ ij for dictinct i, j, k .We denote a ijk ∼ a { ijk } ; notice that σ ij (cid:54) = σ jk . Following [8], braids can be presented by dynamical systems, in which pointsmove. But we consider braids on n strands as n moving points with one additionalfixed (infinite) point. Let us define a mapping from P B n to (cid:101) G n . Now we considerpure braids as moving n points on upper semi-disk. As the above, mapping from P B n to (cid:101) G n will be defined by “reading” moments when some three points are onthe same line, which is analogous to the construction of mapping from P B n to G n .Let n enumerated points P = { p , · · · , p n } be placed on semicircle { z ∈ C | | z | =1 , Imz ≥ } in numerated order with respect to the courter-clockwise orientation.Let us place one more (infinite) point p ∞ in the center of semicircle. When thepoints move, if three points { p i , p j , p k } ⊂ P are on the same line, we write thegenerator a ijk . If points p j , p i , p ∞ are on the same directed line from p ∞ in thisorder and the point p i passes the directed line from left to right, then we write thegenerator σ ij , see Fig. 1. For i, j ∈ ¯ n, i < j define c i,j = ( n (cid:89) k = j +1 a ijk ) σ − ij ( j − (cid:89) k =1 a ijk ) , (2.3) ¯ c i,j = ( n (cid:89) k = j +1 a ijk ) σ ij ( j − (cid:89) k =1 a ijk ) , (2.4) c j,i = ( n (cid:89) k = j +1 a ijk ) σ − ji ( j − (cid:89) k =1 a ijk ) , (2.5) ¯ c j,i = ( n (cid:89) k = j +1 a ijk ) σ ji ( j − (cid:89) k =1 a ijk ) . (2.6)Then the mapping (cid:101) φ from P B n to (cid:101) G n is defined by(2.7) (cid:101) φ ( b ij ) = c − i,i +1 c − i,i +2 · · · c − i,j − ¯ c i,j ¯ c j,i c i,j − · · · c i,i +2 c i,i +1 . S.KIM AND V.O.MANTUROV i j k a ijk ij ij σ i j ji σ -1 ij ij σ i j ji σ -1-1 Figure 1.
Generators related for each cases
Example 2.4.
Let n = 6 , i = 2 , j = 4 and points are placed on semi-circle shownin Fig. 2. By reading every triple points, we obtain the following word in the group (cid:101) G : (cid:101) φ ( b ) = a σ a a a a a σ a a a a σ a a a a a σ − a , see Fig. 3. By definition, we obtain (2.8) (cid:101) φ ( b ) = c − ¯ c ¯ c c . The sets of codimension 1 is related to generators, and the sets of codimension 2are related to relations. They are analogous to those from [8] in the case of (n+1)points. In this work, if a point passes to triple point, which contains the infinitepoint ( ∞ ), then it is possible to give an additional information and write one ofgenerators not of a ij ∞ , but of σ ij , σ − ij , σ ji or σ − ji . This is the main idea to provethe theorem given below, but this theorem differs from the main theorem in [8],roughly speaking, the image has more generators and relations than the image in[8]. Theorem 2.5.
The mapping (cid:101) φ from P B n to (cid:101) G n , defined by (2.7), is a homomor-phism. In other words, if pure braids β, β (cid:48) are equivalent in P B n , then (cid:101) φ ( β ) = (cid:101) φ ( β (cid:48) ) in (cid:101) G n . σ σ σ σ Figure 2.
A generator b in P B and its dynamical system Proof.
Let z k = e πi ( n − k ) / ( n − , k = 1 , · · · , n, be points on semicircle C = { z = a + bi ∈ C | | z | = 1 , b ≥ } . Pure braids can be considered as dynamical systems, inwhich points move on the plane such that the position of points in the start and inthe end are same as asserted in the above. Now we clearly formulate the image ofeach generator in P B n . For i < j a generator b ij can be considers as the followingdynamical system (see Fig. 4):(1) The point z i moves along the semi-circle C passes beside z i +1 , z i +2 , · · · , z j − to the point z j .(2) The point z i turn around z j in the counter-clockwise orientation.(3) The point z i comes back to the initial position beside z j − , · · · , z i +1 .When three points are placed on the same straight line, we write one of generatorsof (cid:101) G n with respect to the rules, which are shown in Fig. 1. In the end of the process,we obtain a word (cid:101) φ ( b ij ) = c − i,i +1 c − i,i +2 · · · c − i,j − ¯ c i,j ¯ c j,i c i,j − · · · c i,i +2 c i,i +1 . Now we will show that if two pure braids β and β (cid:48) are equivalent in P B n , then (cid:101) φ ( β )and (cid:101) φ ( β (cid:48) ) are equivalent in (cid:101) G n . Firstly, we introduce the following two definitions:a dynamical system is nice, if the following holds: P1:
If points are placed on the same straight line, then the number of thepoints is less than or equal to 3.
P2:
For each moment there is at most one triple of points { p i , p j , p k } , whichare collinear. If such a triple of point happens, we call it the critical momentof type { i, j, k } . P3:
The number of critical moments is finite.A dynamical system D is called stable , if it satisfies the followings: C1:
Every dynamical system D (cid:48) in the neighborhood U ( D ) of D (that is, D (cid:48) is obtained from D by transforming D ) is nice, S.KIM AND V.O.MANTUROV φ’(b ) = c -1 c c c σ a a a a a a σ a a a a σ a a σ a a a a -1 Figure 3.
The diagram of the image of b along (cid:101) φ C2:
For each dynamical system D (cid:48) in U ( D ), (cid:101) φ ( D ) = (cid:101) φ ( D (cid:48) ).Without loss of generality we may assume that the pure braids have the formsof nice stable dynamical system.Let { β t } t ∈ I be a isotopy such that β = β and β = β (cid:48) . Without loss of generality,we may assume that { β t } t ∈ I satisfies the followings:(1) For every t ∈ ( t , t ) ⊂ I , if β t is stable and nice, then the set of criticalmoments of β t changes continuously.(2) In I there are finite s ∈ I such that β s is not nice or not stable. In otherwords, β s satisfies one of the followings; A: Suppose that there are three points { z i , z j , z k } ⊂ { z · · · z n } , suchthat the points { z i , z j , z k } are on the same line in the moment t andthey are still on the same line in the first approximation centered at t . For some (cid:15) >
0, the word (cid:101) φ ( β s − (cid:15) ) has the form of F a ijk a ijk B , but (cid:101) φ ( β s + (cid:15) ) has the form of F B (see. Fig. 6). That is, when β t passes the a i(k-1)k a a ik(k+1) a ikn σ ik Figure 4.
The dynamical systems associating with b ij and c ik moment β s , (cid:101) φ ( β s + (cid:15) ) is obtained from (cid:101) φ ( β s − (cid:15) ) by the relation (a) inDefinition 2.3. B: Suppose that four points are on the same line in the moment β s (see. Fig. 7). If they have not the infinite point, then for some (cid:15) > (cid:101) φ ( β s − (cid:15) ) contains a product of a ijk , a ijl , a ikl , a jkl in some order, (cid:101) φ ( β s + (cid:15) ) have the product of a ijk , a ijl , a ikl , a jkl in the reverse order.When t is changed, (cid:101) φ ( β s + (cid:15) ) is obtained from (cid:101) φ ( β s − (cid:15) ) by the relation(c) in Definition 2.3.If one of the four points is the infinite point, then for some (cid:15) >
0, theword (cid:101) φ ( β s − (cid:15) ) has a product of a ijk , σ ij , σ ik , σ jk in some order, andthe word (cid:101) φ ( β s + (cid:15) ) has the product of a ijk , σ ij , σ ik , σ jk in the reverseorder. When β t passes the moment β s , the word (cid:101) φ ( β s + (cid:15) ) is obtainedfrom the word (cid:101) φ ( β s − (cid:15) ) by the relations (f),(g),(h),(i) of relations ofthe group (cid:101) G n in Definition 2.3. C: Suppose that two sets m and m (cid:48) of three points on the lines l and l (cid:48) respectively in the moment s , such that | m ∩ m (cid:48) | < (cid:101) φ ( β t ) is changed according to one of the relations (b),(d),(e) ofrelations of the group (cid:101) G n in Definition 2.3.As the above, we can rewrite every type of deformations (codimension 2), whichcorrespond to general position isotopies between two braids. Passing those momentswhich are either not good or not stable, the word is deformed by one of relationsof the group (cid:101) G n and the proof is completed. (cid:3) Homomorphisms from (cid:101) G n to G n +1 Define the homomorphism pr : (cid:101) G n → G n by S.KIM AND V.O.MANTUROV i k-1 kk-2 nk+11 i k-1 kk-2 nk+11 a i(k-1)k a i(k-2)k a σ ik a iknik(k+1) a Figure 5.
The diagram of c − ij (3.1) pr ( a ijk ) = (cid:26) , if ∞ ∈ { i, j, k } ,a ijk , if ∞ (cid:54)∈ { i, j, k } . The proposition below follows from the definition of pr : Proposition 3.1. pr ◦ (cid:101) φ = φ . i j k k ki j i j Figure 6.
Case A kji l k j i k j i
Figure 7.
Case B kji l m kji l kji l
Figure 8.
Case CNow we define homomorphism i from G n to (cid:101) G n by i ( a ijk ) = a ijk . Then i ◦ pr = Id G n .Besides, we define the homorphism π from (cid:102) G n to G n +1 by π ( a ijk ) = a ijk and π ( σ ij ) = a ij ( n +1) . In [8] V.O.Manturov and I.M.Nikonov studied homomorphismsfrom G kn to the free product of copies of groups Z . This, in turn, leads to the homomorphism w ijk : G n → F n is constructed, where F n = (cid:104){ σ | σ : { , , · · · n }\{ i, j, k } → Z × Z } | { σ = 1 }(cid:105) ∼ = Z ∗ n − . For each generator a ijk in β = F a ijk B ∈ G kn , let us define the mapping i a ijk :¯ n \{ i, j, k } → Z × Z by i c ( l ) = ( N jkl + N ijl , N ikl + N ijl ) ∈ Z × Z , for l ∈ ¯ n \{ i, j, k } , where N jkl is the number of occurencies of a ikl in F . We call i a ijk “index” of a ijk in β . Let { c , · · · , c m } be the ordered set of all a ijk in β suchthat if β = T l c l B l , then c s ∈ T l for s < l . Define w ijk : G n → F n by w ( i,j,k ) ( β ) = i c i c · · · i c m . Note that if generators on the right hand side have indices, then we can defineindices for generators on the left hand side. In other words, if we get a homo-morphism x : G → H for generators of the group G and H , and if indices i forgenerators in the group H are defined, · · · a · · · → · · · x ( a ) · · · then the indices j for generators in G can be defined by j ( a ) := i ( x ( a )) . That is, for a generator b in (cid:102) G n , the index j b can be defined by means of G n +1 asfollow: j b ( l ) := i π ( b ) ( l ) , for each l ∈ n + 1 \{ i, j, k } . Analogously we define the homomorphism from (cid:102) G n tothe free product of copies of groups Z as follow: (cid:101) w ijk := w ( i,j,k ) ◦ π : (cid:102) G n → G n +1 → Z ∗ n +1) − . Example 3.2.
For a braid b ∈ P B , π ◦ (cid:101) φ ( b ) = π ( a σ a a a σ a a a a σ a a a a a σ − a )= a a a a a a a a a a a a a a a a a a . Then i a = (cid:18) (cid:19) and i a = (cid:18) (cid:19) , where i c = ( i c (1) , i c (3) , i c (5) , i c (6)) . We define indices j for generators σ and σ with respect to i . j σ := i a = (cid:18) (cid:19) and j σ := i a = (cid:18) (cid:19) . Non-cancellable generators b ij and b − ij We will use the group (cid:101) G n to know whether two classical crossing cannot becanceled or not. Example 4.1.
Let β = [ b , b ] in P B . π ◦ (cid:101) φ ( β ) = π ( σ a σ a σ σ a σ a σ − a σ − σ − a σ − σ − )= a a a a a a a a a a a a a a a a in G . Then (cid:101) w ( (cid:101) φ ( β )) = (1 , , , , , , (cid:54) = 1 in F , where i a (4) = ( N + N , N + N ) mod , and β is non trivialbraid in P B . Note that the braid β is Brunnian, and w ijk ◦ φ ( β ) = 1 for Brunnianbraids β , that is, w ( φ ( β )) = 1 (Theorem 3.4 from [5] ). Theorem 4.2.
For pure braids in the form of β = Ab ij Bb − ij C ∈ P B n , where A, B, C ∈ P B n , if the total number of b ik and b jk in B is odd, and there are no b ij and b − ij in B , then b ij and b − ij cannot be canceled by relations of P B n .Proof. Without loss of generality, assume that i < j < k . Note that (cid:101) w ijk ◦ (cid:101) φ ( β ) = (cid:101) w ijk ◦ (cid:101) φ ( f ijk ( β )), where f : P B n → P B n is the endomorphism defined by(4.1) f ijk ( b st ) = (cid:26) , if |{ s, t } ∩ { i, j, k }| (cid:54) = 2 ,b st , if |{ s, t } ∩ { i, j, k }| = 2 . The homomorphism (cid:101) w ijk is valued on the free product of copies of Z and itfollows that (cid:101) w ijk ◦ (cid:101) φ ( f ijk ( B )) = 1 for two generators b ij and b − ij to be can-celled. By the assumption we have f ijk ( B ) = c · · · c k for k ≡ c i ∈ { b ik , b − ik , b jk , b − jk } . Now we calculate (cid:101) w ijk ◦ (cid:101) φ ( b ij ) = w ijk ◦ π ◦ (cid:101) φ ( b ij ), where i a ijk ( n + 1) = ( a ij ( n +1) + a jk ( n +1) , a ik ( n +1) + a jk ( n +1) ) ∈ Z × Z . By the definitionit follows that π ◦ (cid:101) φ ( b ij ) = π ( c − i,i +1 · · · c − i,j − ¯ c i,j ¯ c j,i c i,j − · · · c i,j − )= c − i,i +1 · · · c − i,j − c i,j c i,j c i,j − · · · c i,j − . In the words c i,j , c i,k , c j,k there are generators a ijk , a ij ( n +1) , a ik ( n +1) , a jk ( n +1) ,but they are not in c s,t for { s, t }∩{ i, j, k } <
2. Therefore c s,t for { s, t }∩{ i, j, k } < i a ijk ( n + 1), and we focus on w ijk ( c i,j c i,j ). Then w ijk ( c i,j c i,j ) = a ij ( j +1) · · · a ijk · · · a ij ( n +1) a ij · · · a ij ( i − a ij ( j +1) · · · a ijk · · · a ij ( n +1) a ij · · · a ij ( i − , and there are exactly two a ijk , denote them by c and c . From simple calculations,we obtain that w ijk ◦ π ◦ (cid:101) φ ( b ij ) = i c i c , where i c ( n + 1) = (0 , , i c ( n + 1) = (1 , . Analogously, we can get that (cid:101) w ijk ◦ (cid:101) φ ( b ik ) = i c i c i c i c , where i c ( n + 1) = (1 , , i c ( n + 1) = (1 , , i c ( n + 1) = (1 , , i c ( n + 1) = (1 , , and (cid:101) w ijk ◦ (cid:101) φ ( b jk ) = i c i c , where i c ( n + 1) = (1 , , i c ( n + 1) = (0 , . That is, the images of the homomorphism w ijk ◦ π ◦ (cid:101) φ of { b ij , b ik , b jk } have differentvalues. It is easy to show that if w ijk ◦ π ◦ (cid:101) φ ( P ) = 1 for a product P of { b ij , b ik , b jk } ,then P is the product of words ( b ij b ik b jk ) ± and ( b jk b ik b ij ) ± up to relations in P B n . But k ≡ b ij in B , (cid:101) w ijk ◦ (cid:101) φ ( f ijk ( B )) = (cid:101) w ijk ◦ (cid:101) φ ( c · · · c k ) (cid:54) = 1 , and hence b ij and b − ij cannot be cancelled. (cid:3) Remark 4.3.
Let β be a braid on n strands and let two strands i and j be fixed.Let c , c be classical crossings σ ij , σ − ij between i and j strands. Is it possible for c and c in β to be cancelled? To answer the question, we need to use indices forcrossings c , c .As asserted in the previous section, the indices are defined by the homomorphismfrom P B n to (cid:101) G n . That is, to obtain the “local” information (indices for crossings)we use “the global” information (homomorphisms (cid:101) w ijk ). As Theorem 4.2 indices(obstruction to reducing) work in groups G kn .In the simplest case, G n , “an algorithm of descending” takes place, in otherwords, if a word β , which is obtained from G n and is not minimal, then there mustbe two “cancellable” crossings, i.e. β = Xa ij Y a ij Z , and β can be converted into β (cid:48) = X ˜ Y a ij a ij Z by relations of G n , without change of the length of the word, and inthe final word β (cid:48) there are two crossings a ij , which are directly reduced. Step by step,we obtain a word of minimal length. Moreover, if two words β (cid:48) and β (cid:48)(cid:48) obtainedfrom a word β have the minimal length, then there is a sequence S of relationsbetween β (cid:48) and β (cid:48)(cid:48) such that every word, which is obtained by a subsequence of S from β (cid:48) , does not have reducing crossings. It means that Diamond lemma holds forthe standard group presentation of G n . In [7] this is proved by means of the Coxetergroups.In the case of groups G n the Diamond lemma does not take place: there is anelement, which has two different representatives of the minimal length in the group G . For example, let words a a a a a a a a and a a a a a a a a in the group G . They are equivalent by the following sequence of deformations a a a a a a = a a a a a a a a = a a a a a a a a = a a a a a a . To the words a a a a a a and a a a a a a any relationsfrom Definition 2.2 cannot be applied except for relations a ijk = 1 , that is, they haveminimal lengths. But, without relations a ijk = 1 , the word a a a a a a ,cannot be deformed to the word a a a a a a .It is well-known that in Artin presentation of the classical braids group, therepresentative of the minimal length is not unique. For example, it is related tohandle reductions, which play important role in the algorithm Dehornoy [3] .It is interested to study connections between handle reductions for classical braidsand the phenomenon in G as the above. References [1] J.Birman, K.H.Ko, S.J.Lee,
A new approach to the word and conjugacy problems in the braidgroups,
Advances in Mathematics 139, 32235 (1998).[2] D.C.Cohen, M.Falk, R.Randell,
Pure braid groups are not residually free , ConfigurationSpaces(Geometry, Combinatorics and Topology), 2012, pages 213-230.[3] P. Dehornoy,
A Fast Method for Comparing Braids,
Advances in Mathematics Volume 125,Issue 2, 10 February 1997, Pages 200-235.[4] R.Fenn, R.Rim´anyi, C.Rourke,
The braid-permutation group , Topology, Volueme 36, Issue 1,January 1997, Pages 123-135.[5] S.Kim, V.O.Manturov,
On groups Gnk, braids and Brunnian braids , arXiv:1606.03563v2[math.GT] 14 Jun 2016. [6] V.O.Manturov, Non-Reidemeister knot theory and its applications in dynamical systems, ge-ometry, and topology, arXiv:1501.05208v1 [math.GT] 21 Jan 2015.[7] V.O.Manturov,