Burnside groups and n -moves for links
aa r X i v : . [ m a t h . G T ] J a n BURNSIDE GROUPS AND n -MOVES FOR LINKS HARUKO A. MIYAZAWA, KODAI WADA AND AKIRA YASUHARA
Abstract.
Let n be a positive integer. M. K. D¸abkowski and J. H. Przytyckiintroduced the n th Burnside group of links which is preserved by n -moves,and proved that for any odd prime p there exist links which are not equivalentto trivial links up to p -moves by using their p th Burnside groups. This givescounterexamples for the Montesinos-Nakanishi 3-move conjecture. In general,it is hard to distinguish p th Burnside groups of a given link and a trivial link.We give a necessary condition for which p th Burnside groups are isomorphicto those of trivial links. The necessary condition gives us an efficient wayto distinguish p th Burnside groups of a given link and a trivial link. As anapplication, we show that there exist links, each of which is not equivalent toa trivial link up to p -moves for any odd prime p . Introduction
Let n be a positive integer. An n -move on a link is a local change as illustratedin Figure 1.1. Two links are n -move equivalent if they are transformed into eachother by a finite sequence of n -moves. Note that if n is odd then an n -move maychange the number of components of a link. Since a 2-move is generated by crossingchanges and vice versa, we can consider an n -move as a generalization of a crossingchange. Any link can be transformed into a trivial link by a finite sequence ofcrossing changes. Therefore, it is natural to ask whether or not any link is n -moveequivalent to a trivial link. In 1980s, Yasutaka Nakanishi proved that all links with10 or less crossings and Montesinos links are 3-move equivalent to trivial links, andhe conjectured that any link is 3-move equivalent to a trivial link (see [8, Problem1.59 (1)]). This conjecture is called the Montesinos-Nakanishi -move conjecture ,and have been shown to be true for several classes of links, for example, all linkswith 12 or less crossings, closed 4-braids and 3-bridge links [1, 9, 11]. n n -move Figure 1.1.
After 20 years, in [2, 3] M. K. D¸abkowski and J. H. Przytycki introduced the n th Burnside group of a link as an n -move equivalent invariant, and proved that forany odd prime p there exist links which are not p -move equivalent to trivial linksby using their p th Burnside groups. More precisely, they proved that the closure ofthe 5-braid ( σ σ σ σ ) and the 2-parallel of the Borromean rings are not 3-moveequivalent to trivial links [2], and that the closure of the 4-braid ( σ σ ) is not p -move equivalent to a trivial link for any prime number p ≥ Mathematics Subject Classification.
Primary 57M25, 57M27; Secondary 20F50.
Key words and phrases.
Link; Burnside group; Magnus expansion; Montesinos-Nakanishi 3-move conjecture; Fox coloring; virtual link; welded link.This work was supported by JSPS KAKENHI Grant Numbers JP17J08186, JP17K05264.
It is easy to see that the p th Burnside group is preserved by p -moves. While the p th Burnside group is a powerful invariant, it is hard to distinguish p th Burnsidegroups of given links in general. Hence to find a way to distinguish given Burnsidegroups is very important. In this paper, we give a necessary condition for which p thBurnside groups of links are isomorphic to those of trivial links (Lemma 2.1). Thenecessary condition gives us an efficient way to distinguish p th Burnside groups ofa given link and a trivial link. As a consequence, we have a new obstruction totrivializing links by p -moves (Theorem 4.1). In fact, by using Theorem 4.1, we showthat there exist links, each of which is not p -move equivalent to a trivial link forany odd prime p (Theorem 4.3). Our method is naturally extended to both virtual and welded links. We prove that there exists a welded link which is not p -moveequivalent to a trivial link for any odd prime p (Remark 4.5).2. Free Burnside groups
Throughout this paper, for a group G let γ q G denote the q th term of the lowercentral series of G , that is, γ G = G and γ q +1 G = [ γ q G, G ] ( q = 1 , , . . . ).Let F m = h x , . . . , x m i be the free group of rank m . We set F ( m, n ) = F m /W n ,where W n is the normal subgroup of F m generated by W n = { w n | w ∈ F m } fora positive integer n . The group F ( m, n ) is called the m generator free Burnsidegroup of exponent n . Let F q ( m, n ) denote the quotient group F ( m, n ) /γ q F ( m, n ).We remark that F ( m, n ) is not always finite but F q ( m, n ) is a finite group for all q , see for example [12, Chapter 2]. Lemma 2.1.
Let G be a group with a presentation h x , . . . , x m R, W n i , where R is a set of words. If G/γ q G and F q ( m, n ) are isomorphic, then R ⊂ W n × γ q F m for any q .Proof. First we note that F q ( m, n ) ∼ = h x , . . . , x m | W n , γ q F m i and G/γ q G ∼ = h x , . . . , x m | R, W n , γ q F m i . Consider a sequence of two natural projections ψ and φ : F m ψ −→ h x , . . . , x m | W n , γ q F m i φ −→ h x , . . . , x m | R, W n , γ q F m i . Then for any r ∈ R , φ ( ψ ( r )) vanishes because ψ ( r ) = rW n × γ q F m . So ψ ( r ) ∈ ker φ .On the other hand, we have that | F q ( m, n ) | = |h x , . . . , x m | W n , γ q F m i| = | G/γ q G | × | ker φ | . Since F q ( m, n ) is finite, | ker φ | = 1. Therefore we have that r ∈ W n × γ q F m . (cid:3) Burnside groups of links
Let L be a link in the 3-sphere S and D an unoriented diagram of L . In [4, 6,7, 13], a group Π (2) D of D is defined as follows: Each arc of D yields a generator,and each crossing of D gives a relation yx − yz − , where x and z correspond to theunderpasses and y corresponds to the overpass at the crossing, see Figure 3.1. Thegroup Π (2) D is an invariant of L . We call it the associated core group of L and denoteit by Π (2) L . Remark 3.1.
M. Wada [13] proved that Π (2) L is isomorphic to the free product ofthe fundamental group of the double branched cover M (2) L of S branched along L and the infinite cyclic group Z . That is, Π (2) L ∼ = π ( M (2) L ) ∗ Z . Moreover, D¸abkowski URNSIDE GROUPS AND n -MOVES FOR LINKS 3 x yzyx − yz − Figure 3.1.
Relation of the associated core groupand Przytycki [2, 3] pointed out that for a diagram D of L , π ( M (2) L ) is obtainedfrom the group Π (2) D of D by putting any fixed generator x = 1.In [2, 3], D¸abkowski and Przytycki introduced n -move equivalence invariants of L by using Π (2) L and π ( M (2) L ) as follows. Definition 3.2 ([2, 3]) . Suppose that Π (2) D = h x , . . . , x m | R i . Then π ( M (2) L ) ∼ = h x , . . . , x m | R, x m i . The unreduced n th Burnside group b B L ( n ) of L is defined as h x , . . . , x m | R, W n i . The n th Burnside group B L ( n ) of L is defined as h x , . . . , x m | R, x m , W n i . Proposition 3.3 ([2, 3]) . b B L ( n ) and B L ( n ) are preserved by n -moves. We will focus on the unreduced n th Burnside group b B L ( n ) from now on. Let b B qL ( n ) denote the quotient group b B L ( n ) /γ q b B L ( n ), which is a finite group for all q .Then the proposition above immediately implies the following corollary. Corollary 3.4. b B qL ( n ) and | b B qL ( n ) | are preserved by n -moves for any q . Remark 3.5.
Let Z n denote the cyclic group Z /n Z of order n . Let L be a linkand D a diagram of L . A map f : { arcs of D } → Z n is a Fox n -coloring of D if f satisfies f ( x ) + f ( z ) = 2 f ( y ) for each crossing of D , where x and z correspondto the underpasses and y corresponds to the overpass at the crossing. The set ofFox n -colorings of D forms an abelian group and is an invariant of L . It is knownthat the order of the abelian group is equal to | b B L ( n ) | (see [10, Proposition 4.5]).Moreover, if L is the m -component trivial link, then b B L ( n ) ∼ = Z mn .4. Obstruction to trivializing links by p -moves Let p be a prime number. The Magnus Z p -expansion E p is a homomorphismfrom F m into the formal power series ring in non-commutative variables X , . . . , X m with Z p coefficients defined by E p ( x i ) = 1+ X i and E p ( x − i ) = 1 − X i + X i − X i + · · · ( i = 1 , . . . , m ). Then we have the following theorem. Theorem 4.1.
Let L be a link with Π (2) L ∼ = h x , . . . , x m | R i and b B L ( p ) ∼ = Z mp . If L is p -move equivalent to a trivial link, then for any r ∈ R , E p ( r ) = 1 + X ( i ,...,i p ) c ( i , . . . , i p ) X i · · · X i p + d ( p + 1) for some c ( i , . . . , i p ) ∈ Z p such that c ( i , . . . , i p ) = c ( i σ (1) , . . . , i σ ( p ) ) for any per-mutation σ of { , . . . , p } , where ( i , . . . , i p ) runs over { , . . . , m } p and d ( k ) denotesthe terms of degree ≥ k .Proof. If L is p -move equivalent to a trivial link T , then b B T ( p ) ∼ = b B L ( p ) ∼ = Z mp .By Remark 3.5 the number of components of T is m , and hence b B qL ( p ) ∼ = F q ( m, p )for any positive integer q . Thus we have that R ⊂ W p × γ q F m by Lemma 2.1.In particular, for any r ∈ R , we have that r = ( Q j u pj ) v for some Q j u pj ∈ W p and v ∈ γ q F m because W p is a verbal subgroup. HARUKO A. MIYAZAWA, KODAI WADA AND AKIRA YASUHARA
Now we may assume that q ≥ p +1. Hence we have that E p ( v ) = 1+ d ( p +1). Foreach j , E p ( u j ) can be written in the form 1 + P mi =1 a ji X i + d (2) for some a ji ∈ Z p .Then we have that E p ( u pj ) = 1 + m X i =1 a ji X i + d (2) ! p = 1 + X ( i ,...,i p ) a ji · · · a ji p X i · · · X i p + d ( p + 1) . Thus we have that E p Y j u pj = Y j X ( i ,...,i p ) a ji · · · a ji p X i · · · X i p + d ( p + 1) = 1 + X ( i ,...,i p ) c ( i , . . . , i p ) X i · · · X i p + d ( p + 1)for some c ( i , . . . , i p ) ∈ Z p such that c ( i , . . . , i p ) = c ( i σ (1) , . . . , i σ ( p ) ) for any per-mutation σ of { , . . . , p } . Therefore E p ( r ) is the desired form. (cid:3) Even though 4 is not prime, we can show the following theorem by a similarway to the proof of Theorem 4.1. We note that 4-moves preserve the number ofcomponents of a link.
Theorem 4.2.
Let L be an m -component link with Π (2) L ∼ = h x , . . . , x m | R i . If L is -move equivalent to a trivial link, then for any r ∈ R , E ( r ) = 1 + X ( i ,i ,i ,i ) c ( i , i , i , i ) X i X i X i X i + d (5) for some c ( i , i , i , i ) ∈ Z such that c ( i , i , i , i ) = c ( i σ (1) , i σ (2) , i σ (3) , i σ (4) ) forany permutation σ of { , , , } , where ( i , i , i , i ) runs over { , . . . , m } . By applying Theorem 4.1, we have the following theorem.
Theorem 4.3.
The closure of the -braid ( σ σ σ σ ) and the -parallel of theBorromean rings are not p -move equivalent to trivial links for any odd prime p . Remark 4.4.
D¸abkowski and Przytycki proved Theorem 4.3 for p = 3 [2, Theo-rem 6]. In their proof, the condition that p = 3 is essential, and hence it seemshard to show Theorem 4.3 by using their arguments. Proof of Theorem 4.3.
Let γ be the 5-braid ( σ σ σ σ ) described by a diagram inFigure 4.1. We put labels x i ( i = 1 , , , ,
5) on initial arcs of the diagram. Progressfrom left to right, then the arcs are labeled by using relations of the associated coregroup. Thus we obtain labels Q i of terminal arcs of γ as follows (see [2, Lemma5]): Q i = x x − x x − x x − x x − x x − x i x − x x − x x − x x − x x − x . Let γ be the closure of γ . Since we have relations Q i x − i for Π (2) γ , Π (2) γ has thepresentation h x , x , x , x , x | r , r , r , r , r i , where r i = Q i x − i . We note that b B γ ( p ) ∼ = Z p for any odd prime p . On the other hand, by computing E p ( r ), thenthe coefficient of X X X is 0 and that of X X X is 2 in E p ( r ). Theorem 4.1implies that γ is not p -move equivalent to a trivial link.Let γ ′ be the 6-braid described by a diagram in Figure 4.2. We put labels x i on initial arcs, y i on terminal arcs, and Q i on arcs of the diagram as illustrated inFigure 4.2 ( i = 1 , , , , , URNSIDE GROUPS AND n -MOVES FOR LINKS 5 x x x x x Q Q Q Q Q Figure 4.1. γ = ( σ σ σ σ ) x x x x x x y y y y y y Q Q Q Q Q Q Figure 4.2. γ ′ whose closure is the 2-parallel of the Bor-romean rings L BR labels Q i are expressed as follows: Q i = x x − x x − x x − x i x − x x − x x − x = y y − y y − y y − y y − y y − y i y − y y − y y − y y − y y − y ( i = 1 , ,x x − x i x − x = Q Q − y i Q − Q = x x − x x − x x − y i x − x x − x x − x ( i = 3 , ,x x − x i x − x = y y − y y − y y − y i y − y y − y y − y ( i = 5 , . Since the closure of γ ′ is the 2-parallel of the Borromean rings L BR , Π (2) L BR hasthe presentation h x , x , x , x , x , x | r , r , r , r , r , r i , where r i = ( x x − x x − x x − x i x − x x − x x − x ) − × x x − x x − x x − x x − x x − x i x − x x − x x − x x − x x − x ( i = 1 , , ( x x − x i x − x ) − x x − x x − x x − x i x − x x − x x − x ( i = 3 , , ( x x − x i x − x ) − x x − x x − x x − x i x − x x − x x − x ( i = 5 , . We note that b B L BR ( p ) ∼ = Z p for any odd prime p . On the other hand, by computing E p ( r ), then the coefficient of X X X is 1 and that of X X X is 0 in E p ( r ).Theorem 4.1 implies that L BR is not p -move equivalent to a trivial link. (cid:3) Remark 4.5.
For a welded link L , we can similarly define the associated core group Π (2) L and the unreduced n th Burnside group b B L ( n ) of L . We note that Theorems 4.1and 4.2 hold for welded links. Hence, we can show that there exists a welded linkwhich is not p -move equivalent to a trivial link for any odd prime p as follows. Let b be the welded 4-braid described by a virtual diagram in Figure 4.3. We put labels x i and Q i ( i = 1 , , ,
4) on initial and terminal arcs of the diagram, respectively. Byusing relations of the associated core group, the labels Q i are expressed as follows: Q i = (cid:26) x x − x x − x x − x x − x x − x x − x if i = 3 ,x i otherwise . Let b be the closure of b , then Π (2) b ∼ = h x , x , x , x | Q x − i . We note that b B b ( p ) ∼ = Z p for any odd prime p . On the other hand, by computing E p ( Q x − ), we HARUKO A. MIYAZAWA, KODAI WADA AND AKIRA YASUHARA x x x x Q Q Q Q Figure 4.3.
Welded 4-braid b have that the coefficient of X X X is 1 and that of X X X is 0 in E p ( Q x − ).Therefore, we have that b is not p -move equivalent to a trivial link by Theorem 4.1. Remark 4.6.
All of the three links γ, L BR and b above are not 4-move equivalentto trivial links by Theorem 4.2. Because terms of degree 3 survive in E ( r ) forsome relation r of Π (2) L ( L = γ, L BR , b ). References [1] Q. Chen,
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E-mail address : [email protected] Faculty of Education and Integrated Arts and Sciences, Waseda University, 1-6-1Nishi-Waseda, Shinjuku-ku, Tokyo, 169-8050, Japan
E-mail address : [email protected] Department of Mathematics, Tsuda University, 2-1-1 Tsuda-Machi, Kodaira, Tokyo,187-8577, Japan
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