RATIONAL TANGLES AND THE MODULAR GROUP

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1 MOSCOW MATHEMATICAL JOURNAL Volume, Number, April June 0, Pages 9 6 RATIONAL TANGLES AND THE MODULAR GROUP FRANCESCA AICARDI Dedicated to V.I. Arnold, with endless gratitude Abstract. There is a natural way to define an isomorphism between the group of transformations of isotopy classes of rational tangles and the modular group. This isomorphism allows to give a simple proof of the Conway theorem, stating the one-to-one correspondence between isotopy classes of rational tangles and rational numbers. Two other simple ways to define this isomorphism, one of which suggested by Arnold, are also shown. 000 Math. Subj. Class. 57M7, 0F6. Key words and phrases. Tangles, rational tangles, modular group, continued fractions, braids group, spherical braids group. Introduction A tangle with four ends (here called simply tangle) is an embedding of two closed segments in a ball such that their endpoints are four distinct points of the bounding sphere and the image of the interior of the segments lie at the interior of the ball. Figure. Two tangles Received June 6, 0. 9 c 0 Independent University of Moscow

2 0 F. AICARDI Two tangles are said isotopic if one can be deformed continuously to the other in the set of tangles with fixed endpoints. A tangle is said rational if can be deformed continuously, in the set of tangles with non-fixed endpoints, to a tangle consisting of two unlinked and unknotted segments. Examples. The tangles in Figure are not rational, since the white strand of the left tangle is knotted, and the two strands of the tangle at right cannot be unlinked if we allow the endpoints of the strands to move on the bounding sphere. See Figure for an example of rational tangle. The main result on rational tangles is a theorem by Conway (announced in []) stating that it is possible to associate with every rational tangle one and only one rational number, so that two rational tangles are isotopy equivalent if and only if they are represented by the same rational number. The idea of the elementary and self-consistent proof of this theorem given in [] is that any rational tangle is isotopy equivalent to exactly one canonical rational tangle (alternating). To such a tangle one associates exactly one continued fraction, and hence a rational number; conversely, from a rational number, via its continued fraction, the canonical representative of exactly one class of rational tangles is constructed. There exist other proofs of this theorem in literature, see for instance another combinatorial proof in [] and topological proofs in [] and [5]. See [] also for a wide list of references. We show here that the Conway theorem can be obtained in a simple way by considering the group of transformations of isotopy classes of rational tangles, called T, (see Section ), which is isomorphic to the modular group. In fact, this isomorphism is obtained by associating to two basic moves (and their inverses) changing the isotopy classes of rational tangles, two elements (and their inverses) which generate the modular group (see Section ). Every rational tangle, in its standard representation, can be transformed into another rational tangle by a (not unique) series of basic moves. This allows to associate constructively to every transformation between rational tangles in standard representation an element of the modular group, namely the element obtained as product of the generators corresponding to the basic moves in exactly the same order they were applied. Similarly, to every word in the two generators and their inverses there correspond exactly one sequence of basic moves. Observe that there are infinitely many ways to transform, via basic moves, a given rational tangle into another given rational tangle. Because of the isomorphism, each series of moves realizing the transformation gives the same element of the modular group (the same element of the modular group, indeed, can be written in infinitely many ways as product of the two considered generators and of their inverses). For instance, this element will be the identity if and only if the rational tangles are isotopic. The set of isotopy classes of rational tangles is thus obtained as the orbit of T on a basic rational tangle, namely a trivial tangle (with untwisted strands). In Section we prove that every isotopy class of rational tangle is represented by an

3 RATIONAL TANGLES AND THE MODULAR GROUP element of T, defined up to multiplication on the right by a power of one generator, and vice versa. From this fact the Conway theorem follows, since there is a way to associate to an element of T, modulo such a multiplication on the right, exactly one rational number and vice-versa. The proposition that every rational tangle is isotopy equivalent to an alternating rational tangle is here obtained as a corollary of the isomorphism theorem, using the fact that every element of the modular group can be essentially written as product of only positive powers (or only negative powers) of the two basic elements we have considered (Section ). In this section, again using the isomorphism theorem and the properties of continued fractions, we obtain how to associate with a rational number a unique alternating rational tangle and vice versa. The isomorphism we have introduced (via the basic elements generating the modular group) is helpful to prove the Conway theorem and other properties of rational tangles. We show however (Section 5) that two non-constructive proofs of this isomorphisms can be obtained in two other simple ways, both involving the spherical braid group with four strands. The second of such proofs was indicated by V.I. Arnold. Wewouldliketounderlinethefactthatthispaperisbasedentirelyonelementary facts. We have provided our statements with simple proofs, without claiming that the same statements are new (except for the main theorem on the isomorphism, that is perhaps and in that case surprisingly new).. Representations of Rational Tangles Let our sphere be centered at the origin of the three space with Cartesian coordinates, Y and Z, and the endpoints of the strands lie in the plane Z = 0. We depict a tangle projected to the Y-plane, which contains the endpoints of the strands. Without loss of generality, we put these endpoints at the intersection of the sphere with the main diagonals of the plane (Fig. ). Y Y Y Figure. Isotopic r-tangles For short we write r-tangle for rational tangle, and we denote by R the space of rational tangles. We denote by and = the simplest r-tangles (see Fig. ) and by,,, the endpoints of the strands, starting with point ( > 0, Y > 0) and going to in the clockwise direction.

4 F. AICARDI x xxxxxx x x x xxxxx Figure. The basic tangles and = and the scheme of the tangle By the constructive definition of r-tangles in [], an r-tangle is obtained from or = by a series of moves, consisting in twisting pairs of adjacent endpoints. We denote by + i and i, i =,,,, the positive and the negative twists, as shown in Fig Figure. The moves transforming into ± i Given an r-tangle, the r-tangle obtained by applying the move σ i (σ = + or ) to is denoted by σ i. Thus any expression σ i σ i σ n i n 0, () where 0 = or =, represents the r-tangle obtained from 0 by applying the moves ± i in the order from σ n i n to σ i. Remark.. Observe that and = satisfy ± = ± =, ± = = ± = = =.

5 RATIONAL TANGLES AND THE MODULAR GROUP Any r-tangle, written in terms of the ± i, can be therefore represented such that each crossing corresponds to one of the moves ± i and its sign is positive or negative according to the scheme of Figure 5. We call such representations standard representations. positive negative Figure 5. Positive and negative crossings in standard representations of r-tangles Isotopic r-tangles may have different standard representations. Example. The representations of isotopic r-tangles in the middle and on the right of Figure are standard, and they are respectively: =, =. We recall that an r-tangle in a standard representation, where the crossings are either all positive or all negative, is said alternating.. The Group of Transformations of Isotopy Classes of r-tangles The following lemma is known (see for instance [], page 05). It is essential for simplifying the set of moves, and we give here a proof in order that this paper be self-consistent. Lemma.. Every r-tangle is isotopic to the tangle obtained from by a rotation by π about the Y-axis and to the tangle obtained from by a rotation by π about the -axis. Proof. The basic tangles and = are invariant under these rotations. The tangles with only one crossing, obtained as ± =, ± = or as ±, ± (see Remark.), are also invariant under such rotations. Given a tangle, we write its expression () in terms of n moves. Therefore we have = σ i Φ, where the r-tangle Φ is expressed in terms of n moves. Let us suppose that the isotopy class of Φ is invariant under the considered rotations, that is, Φ Φ Φ. The following figure shows that is isotopy equivalent, by consequence, to and to, when = + Φ. Indeed: = + Φ + Φ =, and = + Φ + Φ + Φ =.

6 F. AICARDI Φ Φ = + Φ = + Φ + Φ = Φ Φ + Φ + Φ = and Similarly, if = + Φ: = + Φ + Φ + Φ =, = + Φ + Φ =. The cases with negative twists, as well as the cases with = ± Φ and = ± Φ are analogous. The lemma follows. From the lemma above we deduce evidently also the following Proposition.. For every r-tangle R: ± ±, ± ±. Therefore, in order to classify the r-tangles up to isotopies, it is possible to consider only two basic moves that change the isotopy class of an r-tangle. We call them A and B, and precisely: A := +, B := +. Their inverses are denoted, respectively, A and B and satisfy A =, B =. Examples. The r-tangles in the middle and on the right of Figure have the following representation: A B B A =, A B A =. Note that the moves A and B keep invariant the tangle endpoint numbered by lying in the first quadrant of the Y-plane. There is another move keeping invariant this endpoint. It is the twist of the endpoints and by a rotation by π about the diagonal = Y, as shown in Fig. 6. We denote this move by R. We now define a group of moves, generated by A, B, A and B, and we denote it by T.

7 RATIONAL TANGLES AND THE MODULAR GROUP 5 Y Y Y Z xxxx xxxxxx xx xx xx xx xx xx xxxxxx xxxx xx xxxx xxxxx xxxxxx xxxxxx xxxxxx x Z xx x xxx xxxx xxxx xxxx xxx xx xxxxxx xxxx Z xx A B R Figure 6. The moves A, B and R obtained by rotating by π one half of the sphere containing the end-points of the r-tangle A B R Figure 7. Action of the moves A, B and R on a tangle The identity E is the absence of any move. AnywordW = T T T n wheret i areeitheraorb, ora, orb, represents an element of the group, and will be interpreted as the composition of the moves T i, which are applied to any r-tangle starting from T n and ending with T : T T T n = T (T ( (T n )))). Two words W and Q represents the same element if and only if, for every R, Q W. For instance, the equation AA = E means that, for any r-tangle, AA E =. This equivalence is indeed obtained by the Reidemeister move which eliminates two crossings. A sequence of k consecutive A (B) moves will be denoted by A k (B k ) and its inverse by A k (B k ). Any word W = T T T n will be thus written as W = A a B a A a B a... or W = B a A a B a A a..., () where a i are nonzero integers. The word W has the inverse W =...B a A a B a A a or W =...A a B a A a B a satisfying evidently Q Q = QQ = E.

8 6 F. AICARDI We define now a homomorphism from T to PSL(, Z). Let A and B be the following elements of PSL(, Z), which are defined up to multiplications by : A = ( ), B = 0 ( ) 0. () Definition. The map µ: T PSL(, Z) is defined by µ(a) = A, µ(b) = B, µ(a ) = A, µ(b ) = B, and µ(e) = E, E being the identity ( )-matrix, up to multiplication by. Moreover, µ sends any sequence of moves A and B and their inverses to the operator of PSL(, Z) obtained as the corresponding product of A, B and their inverses. ( ) 0 Recall that the standard basis for PSL(, Z) is {A, S}, where S =, 0 up to multiplication by, and satisfies S = E () Observe that S is written in terms of the basis {A, B, A, B } as S = AB A = B AB. (5) Lemma.. Every operator Q PSL(, Z) can be written as VT, where T is either a word in sole A, B or a word in sole A, B, and V = E or V = S. Proof of the lemma. It follows from the following relations holding in PSL(, Z), which are derived from (5): SA = B S, SB = A S, SA = BS, SB = AS. (6) We are now ready to prove that the homomorphism µ is an isomorphism. Theorem.. The group T is isomorphic to the modular group. Proof. To prove that µ is an isomorphism, we have to prove that, for any two different words W and W in T µ(w) = µ(w ) R W W. (7) Proof of (7). To every element of PSL(, Z) there correspond different words in the generators A, B, A and B, since they are not independent. We have therefore to prove that whenever µ(w) = µ(w ), W W for every r-tangle. This is true if the relations 5 and are the image by µ of relations holding in T. We define therefore S := AB A T, and we prove: (i) B AB = S; (ii) S = SS. In this paper the term modular group means the projective special linear group PSL(, Z), because of the isomorphisms between these groups.

9 RATIONAL TANGLES AND THE MODULAR GROUP 7 Proof of (i). The next figure shows that for any tangle, Hence item (i) is satisfied by S = R. AB A R, B AB R. AB - A B - A xxxx xxxxx xxxxx xxx A xxx xxxxx xxxxxx - xxxx B - - AB - B AB xxxxx xxxxxx xxxxx xxx xxxxx R Proof of (ii). Y Y Y R R R - - R The tangles R and R are isotopic to the tangle obtained by rotations by ±π of the whole ball containing about the diagonal Y =. Observe that such rotations, unlike the vertical and horizontal rotations by π considered in Lemma., do not preserve the isotopy class of. However, from R R we conclude R R, that is, R = E (a rotation by π is the identity). Proof of (7). To conclude the proof of the theorem, we have to exclude that there exist some words W and W in T such that, for every r-tangle, the isotopy classes of W and W coincide, but µ(w) µ(w ). In this case there should be a move Q, namely Q = W W, which is isotopy equivalent to the identity, and satisfies µ(q) E. Using Lemma., we write the operator Q := µ(q) in the form Q = VT. T is a word either in A, B or in A, B. Suppose that the word T ends with A a,

10 8 F. AICARDI for some nonzero a Z. Consider the r-tangle Q =, where Q = VT, T being the word in the moves A and B obtained translating A to A and B to B from the word T, and V is equal to R or to E, according to V. The crossings of T are either all positive or all negative, and, since the move R is equivalent to a rotation changing the sign of all crossings, the r-tangle Q is alternating. It is therefore evident that Q = =. But we have Q = Q =, since µ(q ) = µ(q). Therefore Q = =. ThiscontradictsthehypothesisthatthemoveQisisotopyequivalentto the identity. If T ends with B a for some nonzero a Z, then we consider the analog move Q and the tangle Q, getting the relations Q and Q Q, whence Q, contradicting the hypothesis. This concludes the proof of the theorem. Let us denote 0 the tangle =. Observe that We obtain the following corollary: B A 0, or A B 0 (8) Corollary.5. The space of the isotopy classes of rational tangles coincides with the orbit of 0 under the action of the group T. Proof. Every r-tangle is isotopy equivalent to an r-tangle reached from 0 or by a series of twists i. By Corollary., to generate all the isotopy classes in R the moves A, B and their inverses are sufficient. Also can be obtained from 0 by A, B and their inverses, by (8). Since every word in A, B and their inverses is an element of T, and all classes are representable starting from 0, the corollary follows.. Rational Tangles and Rational Numbers In this section we prove that every class of r-tangles different from that of is uniquely represented by a rational number, and, vice versa, that every rational number represents uniquely a class of rational tangles, i.e., we give a proof of the Conway theorem. We denote by the isotopy class of the r-tangle, and by R the space of the isotopy classes of r-tangles. Let Q := {(p, q) Z \(0, 0) modulo (p, q) (rp, rq), r Z\0}. Let v 0 = (0, ) and v = (, 0). Observe that (0, r), for all nonzero r Z, represent the same element as v 0 in Q, and (r, 0), for all nonzero r Z, represent the same element as v. Definition. We define the map ρ: R Q by the following equations: if = Q 0, then ρ( 0 ) = v 0 ; ρ( ) = µ(q)v 0.

11 RATIONAL TANGLES AND THE MODULAR GROUP 9 Theorem.. The map ρ associates with every isotopy class of r-tangles one and only one element of Q. Before proving this theorem, we make some observations. Remark.. The r-tangles 0 and possess the following invariances: for every m Z B m 0 0, A m. No other element of T keeps invariant 0 nor. Observe that, from Eq. 8 and Remark. it follows also that R 0, R 0. (9) Lemma.. There is no element of T keeping invariant a class of R different from 0 or. Proof. Suppose the equation W be fulfilled by some r-tangle and some word W E. Let = Q 0 for some Q T. By hypothesis we have WQ 0 Q 0, and, by consequence, Q WQ 0 0. By Remark. we obtain that Q WQ = B k, for some integer k. The last equation is fulfilled either if both W and Q are powers of B, and hence 0, and W = B k, or if Q = A j R, and hence, by Eq. 9,, and W = A k, being RA j A k A j R = B k. Lemma.. Every class in R is uniquely represented by an element of T, modulo multiplication from the right by B m, m Z. Proof. Every class of r-tangles can be represented as Q 0, for some Q T, by Corollary.5. With the r-tangle = Q 0 we associate the element Q T. By Remark., = QB m 0, therefore if we associate Q to, we may as well associate QB m, forall m Z. On the otherhand, if Q QB m for some m, thenq 0 Q 0, by the same remark. Proof of Theorem.. We observe first that the map ρ respects the invariances of Remark.. ( ) ( Indeed, ρ( B m ) = B m v 0 = = = mv m )( m) 0 v 0 = ρ( 0 ). Moreover, using Eq. 8, ( ) ( ρ( ) = B 0 Av 0 = = = v 0)( 0). (0) ( ) ( m m Thereforeρ(Ãm ) = A m B Av 0 = A m v = = = mv 0 )( 0 0) v, that is, ρ(ãm ) = ρ( ). We have to prove that for every pair of classes and in R = ρ( ) = ρ( ). ()

12 0 F. AICARDI Proof of (). ( ByLemma., ) if = then ( = Q )( 0 and ) = Q ( 0 withq ) = QB m a b. If µ(q) =, we obtain µ(q a b 0 a+mb b ) = =. c d c d m c+md d Therefore ρ( ) = µ(q)v 0 = (b, d) and ρ( ) = µ(q )v 0 ( = (b,) d). ( ) Proof of (). Let Q 0, Q a b 0, µ(q) =, µ(q a b ) = c d c d. We suppose that ρ( ) = ρ( ), that is, (b, d) (b, d ) Q. This equality implies b = b and d = d, because b, d as well as b, d are relatively prime, constituting a column of a discriminant one matrix. The matrices Q := µ(q) and Q = µ(q ) may differ for the first column, and they have the same unit discriminant. We obtain therefore a = a+bk and c = c+dk for some integer k, and this implies Q = QB k. We thus obtain Q 0 QB k 0 Q 0, and hence. Consider the map t sending every element of Q with q 0 to a rational number: t: (p, q) p/q. Every rational number is the image by t of one and only one element of Q. We will write p/q = t(v), where v = (p, q) Q. Evidently mv = (mp, mq) v in Q. Theorem. has the following corollary: Corollary.5. The map ρ t: R Q associates with every isotopy class of r- tangles different from one and only one rational number. Proof. By Theorem., the map ρ associates with every isotopy class of r-tangles an element of Q. The map t associates with every element of Q, q 0, a rational number. The element (r, 0) Q is the image by ρ of the class of, by Eq. 0. The corollary follows. Corollary.6. The map (ρ t) : Q R associates with every rational number one and only one isotopy class of r-tangles. Proof. With the rational number p/q, t associates v = (p, q) Q. Since q 0, and p and q are coprime, the pair (p, q) defines a matrix Q PSL(, Z) such that Qv 0 = v, up to a right factor equal to B m, for some m Z as we have seen in the proof of Eq.. Therefore also the element Q T such that µ(q) = Q is defined up to a right factor equal to B m, for some m Z. But all elements QB m define the same class Q 0, therefore the image of v by ρ is well defined and unique by Remark... Alternating Rational Tangles and Continued Fractions In this section we obtain first that every r-tangle is isotopy equivalent to an alternating r-tangle, as a consequence of the results of the preceding section. Corollary.. A rational tangle different from is equivalent to a tangle obtained by 0 either by sole positive twists or sole negative twists. Proof. We write Q 0. Hence we consider Q = µ(q). Q is different from A m S since, according to Eq. 8, Remark. and Eq. 9. By Lemma. and Theorem., Q can be written as VT, where T is a word either in A and

13 RATIONAL TANGLES AND THE MODULAR GROUP B or in A and B, and V = R or V = E. If V = E, the proof is finished, since the tangle T is alternating. If V = R, then we apply to every generator in the word T the equations obtained by the relations in T corresponding to the relations (6) in PSL(, Z). Therefore we obtain RT = TR, where T is obtained by T exchanging B with A and A with B. Hence TR 0 T. Since A m, we can write T, where T is a word whose last element is B k, with k 0. If k, and hence T is a word in A and B, we use the relation, from Eq. 8, B A 0, to obtain the relation T = T 0, where T is obtained substituting the last B with A. In this way T 0 is obtained from 0 by sole negative twists. If k, and hence T is a word in A and B, we use the relation, from Eq. 8, B A 0, to obtain the relation T T 0, where T is obtained substituting the last B with A. Therefore T 0 is obtained from 0 by sole positive twists... Alternating rational tangles from continued fractions. We prove now that the continued fraction procedure allows to associate with every rational number one and only one alternating rational tangle. Let p, q and a i (i =,..., n) represent natural numbers. A positive rational number has a unique representation by a continued fraction with positive elements a i : p q = a + a + a + a + + a n As a consequence, a negative rational number has a unique representation by a continued fraction with negative elements a i : p q = a a + a + a + + a n. = ( a )+. ( a )+ ( a )+ ( a )+ + ( a n ) A continued fraction with strictly positive elements is denoted by p/q = [a ;a,..., a n ]. Note that a may be zero, but all a i, for i >, are positive, and a n is bigger or equal to. Therefore, we can always write an equivalent expression for p/q: p q = [a, a, a,..., a n, ].

14 F. AICARDI Example. 7/ = [, ] = [,, ], since + = + +. If the fraction is negative, we will write evidently p q = [ a, a, a,..., a n +, ]. In this way we may always suppose n be odd, because if n is even, then we decrease a n by one and we add the (n+)-th element equal to (or ). We call such a continued fraction odd continued fraction. Proposition.. Let the odd continued fraction of p/q be [a ; a, a,..., a n ], () where the a i are either all strictly positive or all strictly negative but a may be zero. Then p/q = t(ρ( Q 0 )), where Q 0 is the alternating r-tangle obtained by applying to 0 the sequence of moves A a B a A a B a n A a n. Proof. We define the map τ Q : Q Q in the following way: for every z = p q Q τ Q (z) = t(q( p q)). Evidently, t(q( p q)) = t(q( z )). For every α Z we have: τ A α(z) = z +α, τ B α(z) = α+. z If α = 0, we obtain τ E (z) = z. Moreover, if Q = QQ, then τ Q (z) = τ Q (τ Q (z)). Therefore, if Q = A a B a A a B a n A a n then τ Q (0) = a +. a + a + a + + a n We thus associate with the odd continued fraction () of p/q the sequence of moves Q = A a B a A a B a n A a n and the alternating tangle Q 0. By construction, p/q = t(ρ( Q 0 )). 5. Rational Tangles and the Spherical Braid Group with Four Strands We give here a short proof of Theorem., using the spherical braid group with four strands, that is, the group of braids whose strands have the upper and lower endpoints lying on two concentric -dimensional spheres. Let F be the manifold of four nonordered distinct points in S.

15 RATIONAL TANGLES AND THE MODULAR GROUP Every transformation between rational tangles defines a closed path in F, by definition of rational tangles, and hence an element of π (F). Consider any tangle. An expression like = σ i σ i σ n i n, () defines a transformation of the isotopy class of to another, that is, an element of T. Observe that the elements i, i =,,,, generate the spherical braid group with four strands. This defines a natural isomorphism between π (F) and the group T. The isomorphism between the spherical braid group with four strands and PSL(, Z) results by the following (known) lemmas: Lemma 5.. The spherical braid group with four strands is isomorphic to the group B := B / ω, where B is the group of braids with three strands, generated by σ and σ : σ σ and ω is the relation Lemma 5.. B is isomorphic to PSL(, Z) σ σ σ σ σ σ =. () Lt us give the idea of proof of Lemma 5.. Let the upper end-points of the fours strands of the spherical braid be the points,,, and shown in Fig. 8a. Any rotation of the internal sphere permuting cyclically the lower endpoints does not change the isotopy class of the spherical braid (Fig. 8b). Therefore, we may assume that a strand connects the upper endpoint to a fixed lower point, say the lower endpoint (Fig. 8c). Any braid can be then transformed to an element of B, by removing any linking with the first strand, as shown in Fig. 8d,e. a b c d e Figure 8. The element σ σ σ σ σ σ is equivalent to a rotation of the inner sphere by π about the central vertical axis (containing the first strand), rotation that evidently does not change the isotopy class of the spherical braid, and that commutes with

16 F. AICARDI all elements of B. Any other element of B, different from a power of this element, cannot be removed by a rotation of the sphere. σ σ σ σ σ σ 5.. The isomorphism between B and T. There is a natural way to associate with every element of T an element of B and vice-versa, as the following figure shows. Observe that this provides a proof of Lemma 5., in virtue of Theorem.. Q The construction can be described as follows: the endpoints, and of the r-tangle become, respectively, the upper endpoints of the strands of the braid, and the endpoints, and of the r-tangle Q become, respectively, the lower endpoints of the strands of the braid. The topological resolution of each crossing remains unchanged in the transformation, so that we obtain: A σ, B σ. (5) To any word of T in A and B and their inverses there corresponds therefore the word in B where the generators are translated according to (5) and are put in the inverse order, if the order from left to right of the generators in a word of B is (as usually) interpreted as their order from top to bottom in the braid. The relation ω is therefore the translation of Eq. by means of (5). Observe that the construction above is similar to that showed in [], but we haven t any ambiguity problem, since we deal with T, and not with rational tangles. 5.. A remark by Arnold. We conclude this paper with a remark by Arnold, indicating another way to prove Theorem.. The space F of configurations of four nonordered points in S can be considered as the base of a fibration p: E F, whose fiber is the torus, twice covering the

17 RATIONAL TANGLES AND THE MODULAR GROUP 5 Riemann sphere with four ramification points. The fundamental group π (F) acts in a natural way on this fibration: every element of π (F) defines a map: T T over the same base point f F, and hence a map of H (T ) to itself, which is an element of the modular group. In the following figures we illustrate this remark. a b c d e Figure 9. (a b) The sphere with four ramification points, (c d) the cylinder obtained cutting along two segments joining respectively the points and, (e) the torus T two times covering the cylinder. e e a b c Figure 0. (a) The torus T obtained as R /(Z) : the double segments indicates the generators of π (T ). (c d) Action on T of the moves A and B (exchanging respectively the point with the point and the point with the point ), coinciding with the action of the operators A, B PSL(, Z) on the generators e, e of Z. Acknowledgements. I am grateful to the referee for remarks and for signaling other proofs of the Conway theorem on rational tangles.

18 6 F. AICARDI References [] G. Burde, Verschlingungsinvarianten von Knoten und Verkettungen mit zwei Brücken, Math. Z. 5 (975), no., 5. MR [] J. H. Conway, An enumeration of knots and links, and some of their algebraic properties, Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 967), Pergamon, Oxford, 970, pp MR 0580 [] J. R. Goldman and L. H. Kauffman, Rational tangles, Adv. in Appl. Math. 8 (997), no., 00. MR 68 [] L. H. Kauffman and S. Lambropoulou, On the classification of rational tangles, Adv. in Appl. Math. (00), no., MR 0797 [5] J. M. Montesinos, Revêtements ramifiés des noeuds, espaces fibrés de Seifert et scindements de Heegaard, Publicaciones del Seminario Mathematico Garcia de Galdeano (98). ICTP, Strada Costiera,, I 5 Trieste Italy address: faicardi@ictp.it