Simple loops on 2-bridge spheres in 2-bridge link complements
aa r X i v : . [ m a t h . G T ] J un SIMPLE LOOPS ON 2-BRIDGE SPHERES IN 2-BRIDGELINK COMPLEMENTS
DONGHI LEE AND MAKOTO SAKUMA
Abstract.
The purpose of this note is to announce complete answers tothe following questions. (1) For an essential simple loop on a 2-bridgesphere in a 2-bridge link complement, when is it null-homotopic in thelink complement? (2) For two distinct essential simple loops on a 2-bridgesphere in a 2-bridge link complement, when are they homotopic in the linkcomplement? We also announce applications of these results to charactervarieties and McShane’s identity. Introduction
Let K be a knot or a link in S and S a punctured sphere in the complement S − K obtained from a bridge sphere of K . Then the following naturalquestion arises. Question 1.1. (1)
Which essential simple loops on S are null-homotopic in S − K ? (2) For two distinct essential simple loops on S , when are they homotopicin S − K ? A refined version of the first question for 2-bridge spheres of 2-bridge linkswas proposed in the second author’s joint work with Ohtsuki and Riley [20,Question 9.1(2)], in relation with epimorphisms between 2-bridge links. Itmay be regarded as a special variation of a question raised by Minsky [9,Question 5.4] on essential simple loops on Heegaard surfaces of 3-manifolds.The purpose of this note is to announce a complete answer to Question 1.1for 2-bridge spheres of 2-bridge links established by the series of papers [11,12, 13, 14] and to explain its application to the study of character varietiesand McShane’s identity [15].
Mathematics Subject Classification.
Primary 57M25, 20F06The first author was supported by a 2-Year Research Grant of Pusan National University.The second author was supported by JSPS Grants-in-Aid 22340013 and 21654011. he key tool for solving the question is small cancellation theory, appliedto two-generator and one-relator presentations of 2-bridge link groups. Wenote that it has been proved by Weinbaum [32] and Appel and Schupp [5]that the word and conjugacy problems for prime alternating link groups aresolvable, by using small cancellation theory (see also [10] and references init). Moreover, it was shown by Sela [24] and Pr´eaux [21] that the word andconjugacy problems for any link group are solvable. A characteristic feature ofour work is that it gives complete answers to special (but also natural) wordand conjugacy problems for the link groups of 2-bridge links, which form aspecial (but also important) family of prime alternating links. (See [1, 4] forthe role of 2-bridge links in Kleinian group theory.)This note is organized as follows. In Sections 2, 3 and 4, we describe themain results, applications to character varieties and McShane’s identity. Theremaining sections are devoted to explanation of the idea of the proof of themain results. In Section 5, we describe the two-generator and one-relatorpresentation of the 2-bridge link group to which small cancellation theoryis applied, and give a natural decomposition of the relator, which plays akey role in the proof. In Section 6, we introduce a certain finite sequenceassociated with the relator and state its key properties. In Section 7, we recallsmall cancellation theory and present a characterization of the “pieces” of thesymmetrized subset arising from the relator. In Sections 8 and 9, we describeoutlines of the proofs of the main results.The authors would like to thank Norbert A’Campo, Hirotaka Akiyoshi,Brian Bowditch, Danny Calegari, Max Forester, Koji Fujiwara, Yair Min-sky, Toshihiro Nakanishi, Caroline Series and Ser Peow Tan for stimulatingconversations. 2. Main results
For a rational number r ∈ ˆ Q := Q ∪ {∞} , let K ( r ) be the 2-bridge linkof slope r , which is defined as the sum ( S , K ( r )) = ( B , t ( ∞ )) ∪ ( B , t ( r ))of rational tangles of slope ∞ and r (see Figure 1). The common boundary ∂ ( B , t ( ∞ )) = ∂ ( B , t ( r )) of the rational tangles is identified with the Con-way sphere ( S , P ) := ( R , Z ) /H , where H is the group of isometries of theEuclidean plane R generated by the π -rotations around the points in the lat-tice Z . Let S be the 4-punctured sphere S − P in the link complement S − K ( r ). Any essential simple loop in S , up to isotopy, is obtained as theimage of a line of slope s ∈ ˆ Q in R − Z by the covering projection onto S .The (unoriented) essential simple loop in S so obtained is denoted by α s . We lso denote by α s the conjugacy class of an element of π ( S ) represented by(a suitably oriented) α s . Then the link group G ( K ( r )) := π ( S − K ( r )) isidentified with π ( S ) / hh α ∞ , α r ii , where hh·ii denotes the normal closure. S Figure 1. ( S , K ( r )) = ( B , t ( ∞ )) ∪ ( B , t ( r )) with r = 1 / B , t ( r )) and ( B , t ( ∞ )), respectively, are the inside andthe outside of the bridge sphere S .Let D be the Farey tessellation , whose ideal vertex set is identified withˆ Q . For each r ∈ ˆ Q , let Γ r be the group of automorphisms of D generatedby reflections in the edges of D with an endpoint r , and let ˆΓ r be the groupgenerated by Γ r and Γ ∞ . Then the region, R , bounded by a pair of Fareyedges with an endpoint ∞ and a pair of Farey edges with an endpoint r formsa fundamental domain of the action of ˆΓ r on H (see Figure 2). Let I and I be the closed intervals in ˆ R obtained as the intersection with ˆ R of the closureof R . Suppose that r is a rational number with 0 < r <
1. (We may alwaysassume this except when we treat the trivial knot and the trivial 2-componentlink.) Write r = 1 m + 1 m + . . . + 1 m k =: [ m , m , . . . , m k ] , here k ≥
1, ( m , . . . , m k ) ∈ ( Z + ) k , and m k ≥
2. Then the above intervalsare given by I = [0 , r ] and I = [ r , r = ( [ m , m , . . . , m k − ] if k is odd,[ m , m , . . . , m k − , m k −
1] if k is even, r = ( [ m , m , . . . , m k − , m k −
1] if k is odd,[ m , m , . . . , m k − ] if k is even. Figure 2.
A fundamental domain of ˆΓ r in the Farey tessella-tion (the shaded domain) for r = 5 /
17 = [3 , , r in the study of 2-bridge link groups. Proposition 2.1. (1)
If two elements s and s ′ of ˆ Q belong to the same orbit ˆΓ r -orbit, then the unoriented loops α s and α s ′ are homotopic in S − K ( r ) . (2) For any s ∈ ˆ Q , there is a unique rational number s ∈ I ∪ I ∪ {∞ , r } such that s is contained in the ˆΓ r -orbit of s . In particular, α s is homotopic to α s in S − K ( r ) . Thus if s ∈ {∞ , r } , then α s is null-homotopic in S − K ( r ) . Thus the following question naturally arises (see [20, Question 9.1(2)]).
Question 2.2. (1)
Does the converse to Proposition 2.1(2) hold? Namely, isit true that α s is null-homotopic in S − K ( r ) if and only if s belongs to the ˆΓ r -orbit of ∞ or r ? (2) For two distinct rational numbers s, s ′ ∈ I ∪ I , when are the unorientedloops α s and α s ′ homotopic in S − K ( r ) ? he following theorem proved in [11] gives a complete answer to Ques-tion 2.2(1). Theorem 2.3.
The loop α s is null-homotopic in S − K ( r ) if and only if s belongs to the ˆΓ r -orbit of ∞ or r . In other words, if s ∈ I ∪ I , then α s is notnull-homotopic in S − K ( r ) . This has the following application to the study of epimorphisms between2-bridge link groups (see [11, Section 2] for precise meaning).
Corollary 2.4.
There is an upper-meridian-pair-preserving epimorphism from G ( K ( s )) to G ( K ( r )) if and only if s or s + 1 belongs to the ˆΓ r -orbit of r or ∞ . The following theorem proved in [12, 13, 14] gives a complete answer toQuestion 2.2(2).
Theorem 2.5.
Suppose that r is a rational number such that < r ≤ / .For distinct s, s ′ ∈ I ∪ I , the unoriented loops α s and α s ′ are homotopic in S − K ( r ) if and only if one of the following holds. (1) r = 1 /p , where p ≥ is an integer, and s = q /p and s ′ = q /p satisfy q = q and q / ( p + p ) = 1 /p , where ( p i , q i ) is a pair of relatively primepositive integers. (2) r = 3 / , namely K ( r ) is the Whitehead link, and the set { s, s ′ } equalseither { / , / } or { / , / } . The proof of Theorem 2.5 reveals the structure of the normalizer of anelement of G ( K ( r )) represented by α s . This enables us to show the following. Theorem 2.6.
Let r be a rational number such that < r ≤ / . Suppose K ( r ) is hyperbolic, i.e., r = q/p and q
6≡ ± p ) , and let s be a rationalnumber contained in I ∪ I . (1) The loop α s is peripheral if and only if one of the following holds. (i) r = 2 / and s = 1 / or s = 3 / . (ii) r = n/ (2 n + 1) for some integer n ≥ , and s = ( n + 1) / (2 n + 1) . (iii) r = 2 / (2 n + 1) for some integers n ≥ , and s = 1 / (2 n + 1) . (2) The conjugacy class α s is primitive in G ( K ( r )) with the following ex-ceptions. (i) r = 2 / and s = 2 / or / . In this case α s is the third power of someprimitive element in G ( K ( r )) . (ii) r = 3 / and s = 2 / . In this case α s is the second power of someprimitive element in G ( K ( r )) . iii) r = 2 / and s = 3 / . In this case α s is the second power of someprimitive element in G ( K ( r )) . At the end of this section, we describe a relation of Theorem 2.3 with thequestion raised by Minsky in [9, Question 5.4]. Let M = H + ∪ S H − be a Hee-gaard splitting of a 3-manifold M . Let Γ ± := M CG ( H ± ) be the mapping classgroup of H ± , and let Γ ± be the kernel of the map M CG ( H ± ) → Out( π ( H ± )).Identify Γ ± with a subgroup of M CG ( S ), and consider the subgroup h Γ , Γ − i of M CG ( S ). Now let ∆ ± be the set of (isotopy classes of) simple loops in S which bound a disk in H ± . Let Z be the set of essential simple loops in S which are null-homotopic in M . Note that Z contains ∆ ± and invariant under h Γ , Γ − i . In particular, the orbit h Γ , Γ − i (∆ + ∪ ∆ − ) is a subset of Z . ThenMinsky posed the following question. Question 2.7.
When is Z equal to the orbit h Γ , Γ − i (∆ + ∪ ∆ − ) ? The above question makes sense not only for Heegaard splittings but alsobridge decompositions of knots and links. In particular, for 2-bridge links, thegroups Γ ∞ and Γ r in our setting correspond to the groups Γ and Γ − , andhence the group ˆΓ r corresponds to the group h Γ , Γ − i . To make this precise,recall the bridge decomposition ( S , K ( r )) = ( B , t ( ∞ )) ∪ ( B , t ( r )), and let ˜Γ + (resp. ˜Γ − ) be the mapping class group of the pair ( B , t ( ∞ )) (resp. ( B , t ( r ))),and let ˜Γ ± be the kernel of the natural map ˜Γ + → Out( π ( B − t ( ∞ ))) (resp.˜Γ − → Out( π ( B − t ( r )))). Identify ˜Γ ± with a subgroup of the mapping classgroup M CG ( S ) of the 4-times punctured sphere S . Recall that the Fareytessellation D is identified with the curve complex of S and there is a naturalepimorphism from M CG ( S ) to the automorphism group Aut ( D ) of D , whosekernel is isomorphic to ( Z / Z ) . Then the groups Γ ∞ and Γ r , respectively,are identified with the images of ˜Γ and ˜Γ − by this epimorphism. Moreover,the sets { α ∞ } and { α r } , respectively, correspond to the sets ∆ + and ∆ − .Theorem 2.3 says that the set Z of simple loops in S which are null-homotopicin S − K ( r ) is equal to the orbit h Γ ∞ , Γ r i (∆ + ∪ ∆ − ). Thus Theorem 2.3 maybe regarded as an answer to the special variation of Question 2.7.3. Application to character varieties
In this section and the next section, we assume r = q/p , where p and q arerelatively prime positive integers such that q
6≡ ± p ). This is equivalentto the condition that K ( r ) is hyperbolic, namely the link complement S − K ( r ) admits a complete hyperbolic structure of finite volume. Let ρ r be the SL(2 , C )-representation of π ( S ) obtained as the composition π ( S ) → π ( S ) / hh α ∞ , α r ii ∼ = π ( S − K ( r )) → Isom + ( H ) ∼ = PSL(2 , C ) , where the last homomorphism is the holonomy representation associated withthe complete hyperbolic structure.Now, let T be the once-punctured torus obtained as the quotient ( R − Z ) / Z , and let O be the orbifold ( R − Z ) / ˆ H where ˆ H is the group gener-ated by π -rotations around the points in ( Z ) . Note that O is the orbifoldwith underlying space a once-punctured sphere and with three cone pointsof cone angle π . The surfaces T and S , respectively, are Z / Z -covering and( Z / Z ) -covering of O , and hence their fundamental groups are identified withsubgroups of the orbifold fundamental group π ( O ) of indices 2 and 4, respec-tively. The PSL(2 , C )-representation ρ r of π ( S ) extends, in a unique way, tothat of π ( O ) (see [4, Proposition 2.2]), and so we obtain, in a unique way, aPSL(2 , C )-representation of π ( T ) by restriction. We continue to denote it by ρ r . Note that ρ r : π ( T ) → PSL(2 , C ) is type-preserving , i.e., it satisfies thefollowing conditions.(1) ρ r is irreducible, i.e., its image does not have a common fixed point on ∂ H .(2) ρ r maps a peripheral element of π ( T ) to a parabolic transformation.By extending the concept of a geometrically infinite end of a Kleinian group,Bowditch [7] introduced the notion of the end invariants of a type-preservingPSL(2 , C )-representation of π ( T ). Tan, Wong and Zhang [30] (cf. [26])extended this notion (with slight modification) to an arbitrary PSL(2 , C )-representation of π ( T ). (To be precise, [30] treats SL(2 , C )-representations.However, the arguments work for PSL(2 , C )-representations.)To recall the notion of end invariants, let C be the set of free homotopyclasses of essential simple loops on T . Then C is identified with ˆ Q , the vertexset of the Farey tessellation D by the following rule. For each s ∈ ˆ Q , let β s be the essential simple loop on T obtained as the image of a line of slope s in R − Z . Then the correspondence s β s gives the desired identification ˆ Q ∼ = C . The projective lamination space PL is then identified with ˆ R := R ∪ {∞} and contains C as the dense subset of rational points. Definition 3.1.
Let ρ be a PSL(2 , C )-representation of π ( T ).(1) An element X ∈ PL is an end invariant of ρ if there exists a sequenceof distinct elements X n ∈ C such that X n → X and such that {| tr ρ ( X n ) |} n isbounded from above.(2) E ( ρ ) denotes the set of end invariants of ρ . n the above definition, it should be noted that | tr ρ ( X n ) | is well-definedthough tr ρ ( X n ) is defined only up to sign. Note also that the condition that {| tr ρ ( X n ) |} n is bounded from above is equivalent to the condition that thehyperbolic translation lengths of the isometries ρ ( X n ) of H are bounded fromabove.Tan, Wong and Zhang [26, 30] showed that E ( ρ ) is a closed subset of PL and proved various interesting properties of E ( ρ ), including a characteriza-tion of those representations ρ with E ( ρ ) = ∅ or PL , generalizing a resultof Bowditch [7]. They also proposed an interesting conjecture [30, Conjec-ture 1.8] concerning possible homeomorphism types of E ( ρ ). The following isa modified version of the conjecture of which Tan [25] informed the authors. Conjecture 3.2.
Suppose E ( ρ ) has at least two accumulation points. Theneither E ( ρ ) = PL or a Cantor set of PL .They constructed a family of representations ρ which have Cantor sets as E ( ρ ), and proved the following supporting evidence to the conjecture. Theorem 3.3.
Let ρ : π ( T ) → SL(2 , C ) be discrete in the sense that the set { tr( ρ ( X )) | X ∈ C} is discrete in C . Then if E ( ρ ) has at least three elements,then E ( ρ ) is either a Cantor set of PL or all of PL . The above theorem implies that the end invariants E ( ρ r ) of the representa-tion ρ r induced by the holonomy representation of a hyperbolic 2-bridge link K ( r ) is a Cantor set. But it does not give us the exact description of E ( ρ r ).By using the main results stated in Section 2, we can explicitly determine theend invariants E ( ρ r ). To state the theorem, recall that the limit set Λ(ˆΓ r )of the group ˆΓ r is the set of accumulation points in the closure of H of theˆΓ r -orbit of a point in H . Theorem 3.4.
For a hyperbolic -bridge link K ( r ) , the set E ( ρ r ) is equal tothe limit set Λ(ˆΓ r ) of the group ˆΓ r . We would like to propose the following conjecture.
Conjecture 3.5.
Let ρ : π ( T ) → PSL(2 , C ) be a type-preserving represen-tation such that E ( ρ ) = Λ(ˆΓ r ). Then ρ is conjugate to the representation ρ r . 4. Application to McShane’s identity
In his Ph.D. thesis [17], McShane proved the following surprising theorem. heorem 4.1. Let ρ : π ( T ) → PSL(2 , R ) be a type-preserving fuchsian rep-resentation. Then X s ∈ ˆ Q
11 + e l ρ ( β s ) = 12In the above identity, l ρ ( β s ) denotes the translation length of the orientation-preserving isometry ρ ( β s ) of the hyperbolic plane. This identity has been gen-eralized to cusped hyperbolic surfaces by McShane himself [18], to hyperbolicsurfaces with cusps and geodesic boundary by Mirzakhani [19], and to hyper-bolic surfaces with cusps, geodesic boundary and conical singularities by Tan,Wong and Zhang [27]. A wonderful application to the Weil-Petersson vol-ume of the moduli spaces of bordered hyperbolic surface was found by Mirza-khani [19]. Bowditch [7] (cf. [6]) showed that the identity in Theorem 4.1is also valid for all quasifuchsian representations of π ( T ), where l ρ ( β s ) isregarded as the complex translation length of the orientation-preserving isom-etry ρ ( β s ) of the hyperbolic 3-space. Moreover, he gave a nice variation ofthe identity for hyperbolic once-punctured torus bundles, which describes thecusp shape in terms of the complex translation lengths of essential simple loopson the fiber torus [8]. Other 3-dimensional variations have been obtained by[2, 3, 26, 27, 28, 29, 30, 31].As an application of the main results stated in Section 2, we can obtain yetanother 3-dimensional variation of McShane’s identity, which describes thecusp shape of a hyperbolic 2-bridge link in terms of the complex translationlengths of essential simple loops on the bridge sphere. This proves a conjectureproposed by the first author in [23].To describe the result, note that each cusp of the hyperbolic manifold S − K ( r ) carries a Euclidean structure, well-defined up to similarity, and hence itis identified with the quotient of C (with the natural Euclidean metric) by thelattice Z ⊕ Z λ , generated by the translations [ ζ ζ + 1] and [ ζ ζ + λ ]corresponding to the meridian and (suitably chosen) longitude respectively.This λ does not depend on the choice of the cusp, because when K ( r ) is atwo-component link there is an isometry of S − K ( r ) interchanging the twocusps. We call λ the modulus of the cusp and denote it by λ ( K ( r )). Theorem 4.2.
For a hyperbolic -bridge link K ( r ) with r = q/p , the followingidentity holds: X s ∈ int I
11 + e l ρr ( β s ) + 2 X s ∈ int I
11 + e l ρr ( β s ) + X s ∈ ∂I ∪ ∂I
11 + e l ρr ( β s ) = − . urther the modulus λ ( K ( r )) of the cusp torus of the cusped hyperbolic mani-fold S − K ( r ) with respect to a suitable choice of a longitude is given by thefollowing formula: λ ( K ( r )) = ( P s ∈ int I e lρr ( βs ) + 4 P s ∈ ∂I e lρr ( βs ) if p is odd, P s ∈ int I e lρr ( βs ) + 2 P s ∈ ∂I e lρr ( βs ) if p is even. The main results stated in Section 2 are used to establish the absoluteconvergence of the infinite series.5.
Presentations of 2-bridge link groups
In the remainder of this note, p and q denote relatively prime positive in-tegers such that 1 ≤ q ≤ p and r = q/p . Theorems 2.3 and 2.5 are provedby applying the small cancellation theory to a two-generator and one-relatorpresentation of the link group G ( K ( r )). To recall the presentation, let a and b ,respectively, be the elements of π ( B − t ( ∞ ) , x ) represented by the orientedloops µ and µ based on x as illustrated in Figure 3. Then π ( B − t ( ∞ ) , x )is identified with the free group F ( a, b ). Note that µ i intersects the disk, δ i , in B bounded by a component of t ( ∞ ) and the essential arc, γ i , on ∂ ( B , t ( ∞ )) = ( S , P ) of slope 1 /
0, in Figure 3. γ δ γ δ x μ μ Figure 3. π ( B − t ( ∞ ) , x ) = F ( a, b ), where a and b arerepresented by µ and µ , respectively.To obtain an element, u r , of F ( a, b ) represented by the simple loop α r (witha suitable choice of an orientation and a path joining α r to the base point x ), note that the inverse image of γ (resp. γ ) in R − Z is the union ofthe single arrowed (resp. double arrowed) vertical edges in Figure 4. Let a abb a ab bb a z z Figure 4.
The line of slope 2 / u / = bab − a − and u / = a ˆ u / b ˆ u − / = abab − a − baba − b − . Since the inverse im-age of γ (resp. γ ) in R − Z is the union of the single arrowed(resp. double arrowed) vertical edges, a positive intersectionwith a single arrowed (resp. double arrowed) edge correspondsto a (resp. b ). L ( r ) be the line in R of slope r passing through the origin, and let L + ( r )be the line in R − Z obtained by slightly modifying L ( r ) near each of thelattice points in L ( r ) so that L + ( r ) takes an upper circuitous route around it,as illustrated in Figure 4. Pick a base point z from the intersection of L + ( r )with the second quadrant, and consider the sub-line-segment of L + ( r ) boundedby z and z := z + (2 p, q ). Then the image of the sub-line-segment in S ishomotopic to the loop α s . Let u r be the word in { a, b } obtained by reading theintersection of the line-segment with the vertical lattice lines (= the inverseimages of γ and γ ) as in Figure 4. Then u r ∈ F ( a, b ) ∼ = π ( B − t ( ∞ )) isrepresented by the simple loop α r , and we obtain the following two-generatorone-relator presentation. G ( K ( r )) = π ( S − K ( r )) ∼ = π ( B − t ( ∞ )) / hh α r ii∼ = F ( a, b ) / hh u r ii ∼ = h a, b | u r i . To describe the explicit formula for u r , set ǫ i = ( − ⌊ iq/p ⌋ where ⌊ x ⌋ is thegreatest integer not exceeding x . Then we have the following (cf. [22, Propo-sition 1]). Let ǫ i = ( − ⌊ iq/p ⌋ , where ⌊ x ⌋ is the greatest integer not exceeding x .
1) If p is odd, then u q/p = a ˆ u q/p b ( − q ˆ u − q/p , where ˆ u q/p = b ǫ a ǫ · · · b ǫ p − a ǫ p − .(2) If p is even, then u q/p = a ˆ u q/p a − ˆ u − q/p , where ˆ u q/p = b ǫ a ǫ · · · a ǫ p − b ǫ p − .In the above formula, ˆ u q/p is obtained from the open interval of L ( r ) boundedby (0 ,
0) and ( p, q ). z z z z z v v v v (0,0) ( , ) p q ( , ) p q ( , ) p q ( , ) p q , p q + ( p + q ) Figure 5.
The decomposition of the relator u r = v v v v We now describe a natural decomposition of the word u r , which plays akey role in the proof of the main results. Let r i = q i /p i ( i = 1 ,
2) be therational number introduced in Section 2. Then ( p, q ) = ( p + p , q + q ) andthe parallelogram in R spanned by (0 , p , q ), ( p , q ) and ( p, q ) does notcontain lattice points in its interior. Consider the infinite broken line, L b ( r ),obtained by joining the lattice points . . . , (0 , , ( p , q ) , ( p, q ) , ( p + p , q + q ) , (2 p, q ) , . . . which is invariant by the translation ( x, y ) ( x + p, y + q ). Let L + b ( r ) be thetopological line obtained by slightly modifying L b ( r ) near each of the latticepoints in L b ( r ) so that L + b ( r ) takes an upper or lower circuitous route aroundit according as the lattice point is of the form d ( p, q ) or d ( p, q ) + ( p , q ) forsome d ∈ Z , as illustrated in Figure 5. We may assume the base points z and z in L + ( r ) also lie in L + b ( r ). Then the sub-arcs of L + ( r ) and L + b ( r ) ounded by z and z are homotopic in R − Z by a homotopy fixing the endpoints. Moreover, the word u r is also obtained by reading the intersection ofthe sub-path of L + b ( r ) with the vertical lattice lines. Pick a point z ∈ L + b ( r )whose x -coordinate is p + (small positive number), and set z := z + ( p, q )and z := z + ( p, q ). Let L + b,i ( r ) be the sub-path of L + b ( r ) bounded by z i − and z i ( i = 1 , , , v i , of u r corresponding to L + b,i ( r ).Then we have the decomposition u r = v v v v , where the lengths of the subwords v i are given by | v | = | v | = p + 1 and | v | = | v | = p −
1. This decomposition plays a key role in the followingsection. 6.
Sequences associated with the simple loop α r We begin with the following observation.(1) The word u r is reduced , i.e., it does not contain xx − or x − x for any x ∈ { a, b } . It is also cyclically reduced , i.e., all its cyclic permutationsare reduced.(2) The word u r is alternating , i.e., a ± and b ± appear in u r alternately,to be precise, neither a ± nor b ± appears in u r . It is also cyclicallyalternating , i.e., all its cyclic permutations are alternating.This observation implies that the word u r is determined by the S -sequencedefined below and the initial letter (with exponent). Definition 6.1. (1) Let w be a nonempty reduced word in { a, b } . Decompose w into w ≡ w w · · · w t , where, for each i = 1 , . . . , t −
1, all letters in w i have positive (resp. negative)exponents, and all letters in w i +1 have negative (resp. positive) exponents.(Here the symbol ≡ means that the two words are not only equal as elementsof the free group but also visibly equal, i.e., equal without cancellation.) Thenthe sequence of positive integers S ( w ) := ( | w | , | w | , . . . , | w t | ) is called the S -sequence of v .(2) Let ( w ) be a nonempty reduced cyclic word in { a, b } represented by aword w . Decompose ( w ) into( w ) ≡ ( w w · · · w t ) , where all letters in w i have positive (resp. negative) exponents, and all lettersin w i +1 have negative (resp. positive) exponents (taking subindices modulo t ). hen the cyclic sequence of positive integers CS ( w ) := (( | w | , | w | , . . . , | w t | )) iscalled the cyclic S -sequence of ( w ). Here the double parentheses denote thatthe sequence is considered modulo cyclic permutations.In the above definition, by a cyclic word , we mean the set of all cyclicpermutations of a cyclically reduced word. By ( v ), we denote the cyclic wordassociated with a cyclically reduced word v . Definition 6.2.
For a rational number r with 0 < r ≤
1, let u r be the wordin { a, b } defined in Section 5. Then the symbol S ( r ) (resp. CS ( r )) denotesthe S -sequence S ( u r ) of u r (resp. cyclic S -sequence CS ( u r ) of ( u r )), which iscalled the S-sequence of slope r (resp. the cyclic S-sequence of slope r ).We can easily observe the following. S ( r ) = S ( u r ) = ( S ( v ) , S ( v ) , S ( v ) , S ( v )) ,CS ( r ) = CS ( u r ) = (( S ( v ) , S ( v ) , S ( v ) , S ( v ))) , where u r = v v v v is the natural decomposition of u r obtained at the end ofthe last section. It is also not difficult to observe S ( v ) = S ( v ) and S ( v ) = S ( v ). By setting S := S ( v ) = S ( v ) and S := S ( v ) = S ( v ), we have thefollowing key propositions. Proposition 6.3.
The decomposition S ( r ) = ( S , S , S , S ) satisfies the fol-lowing. (1) Each S i is symmetric, i.e., the sequence obtained from S i by reversingthe order is equal to S i . (Here, S is empty if k = 1 .) (2) Each S i occurs only twice in the cyclic sequence CS ( r ) . (3) Set m := ⌊ q/p ⌋ . Then S ( r ) consists of only m and m + 1 , and S beginsand ends with m + 1 , whereas S begins and ends with m . Proposition 6.4.
Let S ( r ) = ( S , S , S , S ) be as in Proposition 6.3. For arational number s with < s ≤ , suppose that the cyclic S -sequence CS ( s ) contains both S and S as a subsequence. Then s / ∈ I ∪ I . Small cancellation conditions for 2-bridge link groups
A subset R of the free group F ( a, b ) is called symmetrized , if all elements of R are cyclically reduced and, for each w ∈ R , all cyclic permutations of w and w − also belong to R . Definition 7.1.
Suppose that R is a symmetrized subset of F ( a, b ). A nonemptyword v is called a piece if there exist distinct w , w ∈ R such that w ≡ vc nd w ≡ vc . Small cancellation conditions C ( p ) and T ( q ), where p and q areintegers such that p ≥ q ≥
3, are defined as follows (see [16]).(1) Condition C ( p ): If w ∈ R is a product of n pieces, then n ≥ p .(2) Condition T ( q ): For w , . . . , w n ∈ R with no successive elements w i , w i +1 an inverse pair ( i mod n ), if n < q , then at least one of the products w w , . . . , w n − w n , w n w is freely reduced without cancellation.The following proposition enables us to apply small cancellation theory tothe group presentation h a, b | u r i of G ( K ( r )). Proposition 7.2.
Let r be a rational number such that < r < , and let R be the symmetrized subset of F ( a, b ) generated by the single relator u r of thegroup presentation G ( K ( r )) = h a, b | u r i . Then R satisfies C (4) and T (4) . This proposition follows from the following characterization of pieces, whichin turn is proved by using Proposition 6.3.
Proposition 7.3. (1)
A subword w of the cyclic word ( u ± r ) is a piece if andonly if S ( w ) does not contain S as a subsequence and does not contain S in its interior, i.e., S ( w ) does not contain a subsequence ( ℓ , S , ℓ ) for some ℓ , ℓ ∈ Z + . (2) For a subword w of the cyclic word ( u ± r ) , w is not a product of two piecesif and only if S ( w ) either contains ( S , S ) as a proper initial subsequence orcontains ( S , S ) as a proper terminal subsequence. Outline of the proof of Theorem 2.3
Let R be the symmetrized subset of F ( a, b ) generated by the single relator u r of the group presentation G ( K ( r )) = h a, b | u r i . Suppose on the contrary that α s is null-homotopic in S − K ( r ), i.e., u s = 1 in G ( K ( r )), for some s ∈ I ∪ I .Then there is a van Kampen diagram M over G ( K ( r )) = h a, b | R i such thatthe boundary label is u s . Here M is a simply connected 2-dimensional complexembedded in R , together with a function φ assigning to each oriented edge e of M , as a label , a reduced word φ ( e ) in { a, b } such that the following hold.(1) If e is an oriented edge of M and e − is the oppositely oriented edge,then φ ( e − ) = φ ( e ) − .(2) For any boundary cycle δ of any face of M , φ ( δ ) is a cyclically reducedword representing an element of R . (If α = e , . . . , e n is a path in M ,we define φ ( α ) ≡ φ ( e ) · · · φ ( e n ).)We may assume M is reduced , namely it satisfies the following condition: Let D and D be faces (not necessarily distinct) of M with an edge e ⊆ ∂D ∩ ∂D , nd let eδ and δ e − be boundary cycles of D and D , respectively. Set φ ( δ ) = f and φ ( δ ) = f . Then we have f = f − . Moreover, we may assumethe following conditions:(1) d M ( v ) ≥ v ∈ M − ∂M .(2) For every edge e of ∂M , the label φ ( e ) is a piece.(3) For a path e , . . . , e n in ∂M of length n ≥ e i ∩ e i +1 has degree 2 for i = 1 , , . . . , n − φ ( e ) φ ( e ) · · · φ ( e n ) cannotbe expressed as a product of less than n pieces.Since R satisfies the conditions C (4) and T (4) by Proposition 7.2, M is a[4 , d M ( v ) ≥ v ∈ M − ∂M ;(2) d M ( D ) ≥ D ∈ M .Here, d M ( v ), the degree of v , denotes the number of oriented edges in M having v as initial vertex, and d M ( D ), the degree of D , denotes the number of orientededges in a boundary cycle of D .Now, for simplicity, assume that M is homeomorphic to a disk. (In general,we need to consider an extremal disk of M .) Then by the Curvature Formulaof Lyndon and Schupp (see [16, Corollary V.3.4]), we have X v ∈ ∂M (3 − d M ( v )) ≥ . By using this formula, we see that there are three edges e , e and e in ∂M such that e ∩ e = { v } and e ∩ e = { v } , where d M ( v i ) = 2 for each i = 1 ,
2. Since φ ( e ) φ ( e ) φ ( e ) is not expressed as a product of two pieces, wesee by Proposition 7.3 that the boundary label of M contains a subword, w ,with S ( w ) = ( S , S , ℓ ) or ( ℓ, S , S ). This in turn implies that the S -sequenceof the boundary label contains both S and S as subsequences. Hence, byProposition 6.4, we have s I ∪ I , a contradiction.9. Outline of the proof of Theorem 2.5
Suppose, for two distinct s, s ′ ∈ I ∪ I , the unoriented loops α s and α s ′ are homotopic in S − K ( r ). Then there is a reduced annular R -diagram M such that u s is an outer boundary label and u ± s ′ is an inner boundary label of M . Again we can see that M is a [4 , X v ∈ ∂M (3 − d M ( v )) ≥ . y using this formula, we obtain the following very strong structure theoremfor M , which plays key roles throughout the series of papers [12, 13, 14]. Theorem 9.1.
Figure 6(a) illustrates the only possible type of the outer bound-ary layer of M , while Figure 6(b) illustrates the only possible type of whole M .(The number of faces per layer and the number of layers are variable.) In the above theorem, the outer boundary layer of the annular map M isthe submap of M consisting of all faces D such that the intersection of ∂D with the outer boundary of M contains an edge, together with the edges andvertices contained in ∂D . (a) (b) Figure 6.
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E-mail address : [email protected] Department of Mathematics, Graduate School of Science, Hiroshima Uni-versity, Higashi-Hiroshima, 739-8526, Japan
E-mail address : [email protected]@math.sci.hiroshima-u.ac.jp