P. -V Koseleff

D Pecker

F Rouillier

The first rational Chebyshev knots *

Keywords: Polynomial curves, two-bridge knots, Chebyshev curves, real roots isolation, computer algebra, algorithms Mathematics Subject Classification 2000: 14H50, 57M25, 14P99

A Chebyshev knot C(a, b, c, ϕ) is a knot which has a parametrization of the form x(t) = T a (t); y(t) = T b (t); z(t) = T c (t + ϕ), where a, b, c are integers, T n (t) is the Chebyshev polynomial of degree n and ϕ ∈ R. We show that any two-bridge knot is a Chebyshev knot with a = 3 and also with a = 4. For every a, b, c integers (a = 3, 4 and a, b coprime), we describe an algorithm that gives all Chebyshev knots C(a, b, c, ϕ). We deduce a list of minimal Chebyshev representations of two-bridge knots with small crossing number.

Introduction

A Chebyshev knot C(a, b, c, ϕ) is a knot which has a parametrization of the form x(t) = T a (t); y(t) = T b (t); z(t) = T c (t + ϕ), where a, b are coprime integers, c is an integer, T n (t) is the Chebyshev polynomial of degree n and ϕ ∈ R. Chebyshev knots are polynomial analogues of Lissajous knots, which admit parametrizations of the form x = cos(at); y = cos(bt + ϕ); z = cos(ct + ψ), where a, b, c are pairwise coprime integers. These knots, first defined in [BHJS], have been studied by many authors: V. F. R. Jones, J. Przytycki, C. Lamm, J. Hoste and L. Zirbel [JP, La, HZ].

It is known that every knot may be obtained from a polynomial embedding R → R 3 ⊂ S 3 , where S 3 is the one-point compactification of R 3 ( [Va, DOS]). In [START_REF] Koseleff | Chebyshev knots[END_REF] we have shown that every knot is a Chebyshev knot.

A two-bridge knot (or a rational knot) is a knot which is isotopic to a compact space curve such that the x-coordinate has only two maxima and two minima. When a = 3 or a = 4 the Chebyshev knot C(a, b, c, ϕ) is a two-bridge knot K. Its projection onto the (x, y)-plane is the Chebyshev curve C(a, b) : T b (x) = T a (y) and we have a regular diagram of K that is in Conway normal form when a = 3 (see [Con, Mu]). This gives us an easy way to identify these knots using their classical Schubert invariant.

Figure 1: Two Chebyshev knots: 7 7 and 7 5 (Conway-Rolfsen numbering) In [START_REF] Koseleff | Chebyshev diagrams for rational knots[END_REF] we gave an explicit parametrization of the torus knots T (2, n) and other infinite families of knots. In [START_REF] Koseleff | Chebyshev diagrams for rational knots[END_REF], we gave a complete classification of harmonic knots C(a, b, c, 0) where a = 3 and a = 4 (see also [FF]). We have shown in [START_REF] Koseleff | Chebyshev diagrams for rational knots[END_REF] that every rational knot of crossing number N admits a polynomial parametrization x = T 3 (t), y = T b (t), z = C(t)

with b + deg C = 3N .
It is proved in [START_REF] Koseleff | Chebyshev knots[END_REF] that every two-bridge knot is a Chebyshev knot with a = 3. We first showed that every two-bridge knot has a Chebyshev curve C(3, b) as projection. Then we used a density argument based on Kronecker's theorem to show that given a knot K whose C(a, b) is a plane projection, there exists c and ϕ such that C(a, b, c, ϕ) is isotopic to K.

Our aim is to give an algorithm that determines a Chebyshev parametrization C(a, b, c, ϕ), (with a = 3 and also with a = 4), for any rational knot.

1. Rational knots are classified by their Schubert fraction. For a = 3 and a = 4 we determine the minimal integer b such that the Chebyshev curve C(a, b) : x = T a (t), y = T b (t) is a plane projection of K. This algorithm is based on continued fraction expansions.

2. Let Z a,b,c be the set of ϕ such that C(a, b, c, ϕ) is singular. Z a,b,c is finite. The knot type of K(ϕ) = C(a, b, c, ϕ) is constant over any interval of R -Z a,b,c . Then we determine a rational number in each component of R -Z a,b,c .

3. We determine the Schubert fraction of the knot C(a, b, c, r) by evaluating the (under/over) nature of the crossings.This amounts to evaluating the signs of polynomials at the real solutions of a zero-dimensional system.

These algorithms are based on three basic black boxes:

• PhiProjection(P a (S, T ), P b (S, T ), As a conclusion we give a list of Chebyshev parametrizations for the first 95 rational knots of crossing number not greater than 10. They admit polynomial parametrizations whose plane projections have few crossing points. This is not the case with Lissajous knots (see [BDHZ]).

Q c (S, T, ϕ) ∈ Q[ϕ]) → R ∈ Q[ϕ], such that Z(R) = π ϕ {P a = 0, P b = 0, Q c = 0} • PhiSampling(R ∈ Q[ϕ]) → r 0 , . . . , r N ∈ Q such that r 0 < ϕ 1 < r 1 < • • • < ϕ N < r N ,

Geometry of Chebyshev knots

Chebyshev curves were defined in [Fi]. Their double points are easier to study than those of Lissajous curves. The classical Chebyshev polynomials T n are defined by T n (t) = cos nθ, where cos θ = t. These polynomials satisfy the linear recurrence

T 0 = 1, T 1 = t, T n+1 = 2t T n -T n-1 , n ≥ 1,
C(a, b, c, ϕ) is at most 1 2 (a -1) × (b -1) × (c -1) + 1. Proof. C(a, b, c, ϕ) is a singular space curve iff there exists s = t ∈ R such that T a (t) -T a (s) t -s = 0, T b (t) -T b (s) t -s = 0, T c (t + ϕ) -T c (s + ϕ) t -s = 0. From Prop. 2.1, the set {s, t}, T a (t) -T a (s) t -s = 0, T b (t) -T b (s) t -s = 0 has 1 2 (a -1)(b -1)
elements. For each of these elements, the set {ϕ ∈ R,

T c (t + ϕ) -T c (s + ϕ) t -s = 0} has at most c -1 elements because the leading monomial of T c (t + ϕ) -T c (s + ϕ) t -s is c2 c-1 ϕ c-1 .
Consequently the set Z a,b,c of critical values ϕ has at most 1 2 (a -1) × (b -1) × (c -1) elements. For any ϕ ∈ R -Z a,b,c , the curve C(a, b, c, ϕ) defines a knot. We claim that in any interval included in R -Z a,b,c , the knots are the same because the nature of the crossings is constant. 2

Remark 2.3. We see that C(a, b, c, -ϕ) is the reverse of C(a, b, c, ϕ) (see [Mu]). If a = 3 or a = 4, they define the same knot. Z a,b,c is symmetrical about the origine.

Knot diagrams

We shall study the diagram of the curve C(a, b, c, ϕ), that is to say the plane projection C(a, b) onto the (x, y)-plane and the nature (under/over) of the crossings over the double points of C(a, b). There are two cases of crossing: the right twist and the left twist (see [Mu], P. 178). In the following lemma, we see that the nature of the crossing is given by

D(s, t, ϕ) = T c (t + ϕ) -T c (t + ϕ) t -s T b-a (t) -T b-a (s) t -s . (1)
Then D(s, t, ϕ) > 0 if and only if the crossing is a right twist.

Proof. Let (s, t) be the parameters of a double point of C(a, b). The crossing is a right twist if and only if

z(t) -z(s) y ′ (t) x ′ (t) - y ′ (s) x ′ (s) > 0.
Using Prop. 2.1, we get s = cos σ and t = cos τ . By simple computation we get sin τ x ′ (t) =

sin σ x ′ (s), sin τ y ′ (t) = sin σ y ′ (s) and

y ′ (t) x ′ (t) = - y ′ (s) x ′ (s)
. The slopes of the corresponding tangents are opposite. We therefore deduce that (using x ∼ y for sign x = sign y )

x ′ (t)y ′ (t) ∼ -x ′ (s)y ′ (s) ∼ y ′ (t) x ′ (t) - y ′ (s) x ′ (s) ∼ x ′ (t)y ′ (t) -x ′ (s)y ′ (s). But x ′ (t)y ′ (t) = ab sin aτ sin bτ sin 2 τ ∼ 2 sin aτ sin bτ = T a-b (t) -T a+b (t). Consequently x ′ (t)y ′ (t) -x ′ (s)y ′ (s) ∼ (T a-b (t) -T a+b (t)) -(T b-a (s) -T b-a (s)
). On the other hand, using the identities

T b+a + T b-a = 2T a T b , T a (t) = T a (s) and T b (t) = T b (s), we conclude that x ′ (t)y ′ (t) -x ′ (s)y ′ (s) ∼ T b-a (t) -T b-a (s)
and the announced result. 2

A two-bridge knot (or link) admits a diagram in Conway's normal form. This form, denoted by C(a 1 , a 2 , . . . , a n ) where a i are integers, is explained by the following picture (see [Con], [Mu] p. 187). The number of twists is denoted by the integer |a i |, and the sign of a i is

a 1 a 2 a n-1 a n a 1 a 2 a n-1 a n
Figure 3: Conway's normal forms, n odd, n even defined as follows: if i is odd, then the right twist is positive, if i is even, then the right twist is negative. On Figure 3 the a i are positive (the a 1 first twists are right twists).

The two-bridge links are classified by their Schubert fractions

α β = [a 1 , . . . , a n ], α > 0,
where [a 1 , . . . , a n ] is the continued fraction expansion a 1 + 1

a 2 + 1 a 3 + 1 • • • + 1 a n .
We shall denote by S α β the two-bridge link with Schubert fraction α β . The two-bridge links S( α β ) and S( α ′ β ′ ) are equivalent if and only if α = α ′ and β ′ ≡ β ±1 (mod α). The integer α is odd for a knot, and even for a two-component link. If K = S( α β ), its mirror image is [Mu]).

K = S( α -β ) (see
We shall study knots with a Chebyshev diagram C(3, b) : x = T 3 (t), y = T b (t). It is remarkable that such a diagram is already in Conway normal form (see Figure 4). Consequently, the Schubert fraction of such a knot is given by a continued fraction of the form [±1, ±1, . . . , ±1].

For example we obtain the torus knot 7 1 = C(- 1, -1, -1, 1, 1, 1, -1, -1, -1), andthe knots 6 3 = C(1, 1, 1, 1, 1, 1), 6 1 = C(1, 1, 1, 1, -1, -1, -1) We get for the knot 7 1 (resp. 6 3 , 6 1 ) the fractions 7 -6 ∼ 7, (resp. 13 8 , 9 -5

).

The crossing number of a knot is the smallest number of double points in any plane projection of any isotopic knot. The crossing number of a two-bridge knot S( α β ) is the sum of the integers in the regular continued fraction expansion of α β > 1.

In the case of Chebyshev knots C(3, b, c, ϕ) with a = 3, the Conway notation is given by C(ε 1 , . . . , ε b-1 ) where ε i = -(-1) i sign D i , the D i being ordered by their abscissae.

In the case of Chebyshev knots C(4, b, c, ϕ) with a = 4, we obtain diagrams like Figure 5. Let a i (resp. b i , c i ) be the signs of D i (resp. -D i ) corresponding to the crossing points with y = 0 (resp. y < 0, y > 0). Following Murasugi ([Mu]), the Conway normal form for such a knot is

C(a 1 , b 1 + c 1 , a 2 , b 2 + c 2 , . . . , a n , b n + c n ).
Figure 5 shows the examples 5 2 : C(-1, -2, 1, 2) and 9 20 : C(1, 2, 1, 2, 1, 2). We thus deduce that the Conway notation for a knot

C(4, b, c, ϕ) is C(a 1 , b 1 , . . . , a n , b n ) where a i = ±1, b i = 0, ±2.
In conclusion, we see that, in the particular case when a = 3 or a = 4, the knot C(a, b, c, r) is determined by its Schubert fraction α β corresponding to the nature of the crossings over the double points of the projection C(a, b).

On the other hand, we show that any rational number α β may be expressed as continued fractions corresponding to Chebyshev diagrams C(a, b) with a = 3 and a = 4.

Algorithm 3.2. Let α β be a rational number.

1. There exists a sequence ε 1 , . . . , ε n ,

ε i = ±1, such that α β = [ε 1 , . . . , ε n ].
2. If β is even, there exists a sequence ε 1 , . . . , ε 2n ,

ε i = ±1, such that α β = [ε 1 , 2ε 2 , . . . , ε 2n-1 , 2ε 2n ].
Proof. Let us prove the existence by induction on the height h(

α β ) = max(|α| , |β|). 1. If h = 2 then α β = 1 = [1] and the result is true. If α > β, we have α β = [1, β α -β ]. Since h( β α -β ) < h( α β
), we get our continued fraction by induction.

If β > α we have α β = [1, -1, - β -α α ].
And we also get the continued fraction.

This completes the construction of our continued fraction expansion [±1, . . . , ±1].

2.

If h( α β ) = 2, then α = 1 and β = 2 and we have r

= [1, -2]. If α > 2β > 0 then we write α β = [1, 2, -1, 2, α -2β β ]. We have h( α -2β β ) < h( α β
) and we conclude by induction.

If β < α < 2β then we write α β = [1, 2, α -β 3β -2α ].
We have |3β -2α| ≤ α and |α -β| < α and we conclude by induction.

If β > α > 0 we write α β = [1, -2, α -β 2α -β ]. From |2α -β| ≤ β we have h( α -β 2α -β ) < h( α β
) and we conclude by induction.

The existence of a continued fraction [1, ±2, . . . , ±1, ±2] is proved. 2

Note that we have proved in [START_REF] Koseleff | Chebyshev diagrams for rational knots[END_REF] that the continued fraction expansion

α β = [ε 1 , . . . , ε n ],
ε i = ±1, is unique if there is no two consecutive sign changes and ε n-1 ε n > 0. We also proved that the continued fraction expansion

α β = [ε 1 , 2ε 2 , . . . , ε 2n-1 , 2ε 2n ], ε i = ±1, is
unique if there is no three consecutive sign changes. Proof. Let us consider a knot K = S( α β ). Using Algorithm 3.2, we can write

α β = [ε 1 , . . . , ε n ], ε i = ±1. One can see that (see [KP2]) n ≡ 2 (mod 3) iff α is even and, since K is a knot, this is not the case. K is isotopic to C(ε 1 , . . . , ε n ) which corresponds to a Chebyshev diagram C(3, n + 1) : x = T 3 (t), y = T n+1 (t). Using Algorithm 3.2, we can write α β = [ε 1 , 2ε 2 , . . . , ε 2n-1 , 2ε 2n ], ε i = ±1. The knot K is isotopic to C(ε 1 , 2ε 2 , . . . , ε 2n-1 , 2ε 2n ) which corresponds to a Chebyshev diagram C(4, 2n + 1) : x = T 3 (t), y = T 2n+1 (t). 2 Corollary 3.4. Every two-bridge knot is a Chebyshev knot C(3, b, c, ϕ). Every two-bridge knot is a Chebyshev knot C(4, b, c, ϕ).
Proof. Using a density argument (Kronecker theorem), we proved in [START_REF] Koseleff | Chebyshev knots[END_REF] that if ϕ is small enough, then there exists c such that

C(3, b, c, ϕ) = C(ε 1 , . . . , ε n ). The case a = 4 is similar. 2
Unfortunately, this last corollary will not provide c and ϕ and not even any bound for c. We want to give the minimal Chebyshev parametrization for every rational knot with a small crossing number. We shall describe all rational knots C(a, b, c, ϕ) with given a, b, c.

Description of Chebyshev knots

Let us consider the curve C(a, b, c, ϕ) with a = 3 or a = 4. From section 3., we know that 1. The curve is singular iff it has double points.

2. If the curve is not singular, the knot C(a, b, c, ϕ) is determined by the sequence of crossings of the projection C(a, b).

We will use the symmetric variables S = s + t and T = st. Let us define

P n (S, T ) = T n (t) -T n (s) (t -s) , Q n (S, T, ϕ) = T n (t + ϕ) -T n (s + ϕ) (t -s) . ( 2
) Lemma 4.1. There exists R a,b,c ∈ Q[ϕ] with degree deg R a,b,c ≤ 1 2 (a -1)(b -1)(c -1) such that Z a,b,c = Z(R a,b,c ). Proof. From Prop. 2.1, {P a (S, T ) = 0, P b (S, T ) = 0} is 0-dimensional and has degree 1 2 (a -1)(b -1). V a,b,c = {P a (S, T ) = 0, P b (S, T ) = 0, Q c (S, T, ϕ) = 0} is 0-dimensional and has degree 1 2 (a -1)(b -1)(c -1) because Q c has c(2ϕ) c-1 as leading term. From lemma 2.2, Z a,b,c = π ϕ (V a,b,c ) is finite. It is for example Z(R) where R = V a,b,c Q[ϕ].
2

We obtain R a,b,c by using the black-box PhiProjection(P a , P b , Q c ).

The set

Z a,b,c = {ϕ 1 , . . . , ϕ N }, for which C(a, b, c, ϕ) is singular, is exactly π ϕ (V a,b,c ), from Prop. 2.2. From section 3., the knot K(r) = C(a, b, c, r) is constant over any interval ]ϕ i , ϕ i+1 [. We obtain r i in ]ϕ i , ϕ i+1 [ by using PhiSampling(R a,b,c ).
In the case when a = 3 or a = 4, and r ∈ R-Z a,b,c , the knot K(r) is uniquely determined by its Schubert fraction. Let (S i , T i ) i=1,...,n be the points of {P a (S, T ) = 0, P b (S, T ) = 0}. The Conway notation of the knot K(r) is deduced from the sequence [sign D(S i , T i , r) ] i=1,...,n = SignSolve(r, P a , P b , Q c • P b-a ).

We will now show how to determine these quantities in the case a = 3 and a = 4.

Case a = 3

We get

P 3 (S, T ) = 4T -4S 2 +3 that is T = S 2 -3 4 . The set {(S, T ), P 3 (S, T ) = 0, P b (S, T ) = 0} has cardinal b -1. Its elements satisfy P b (S, S 2 -3 4 ) = 0, T = S 2 -3 4 . We deduce that deg(P b (S, S 2 -3 4 )) = b -1. The set of critical values ϕ is Z 3,b,c = π ϕ {P b (S, S 2 - 3 4 ) = 0, Q c (S, S 2 - 3 4 , ϕ) = 0} .
It is exactly the roots of the polynomial of degree (b -1)(c -1):

R 3,b,c = Res S P b (S, S 2 - 3 4 ), Q c (S, S 2 - 3 4 , ϕ) .
Let A(S) be a crossing point corresponding to parameter S (and

T = S 2 -3 4 ). Its abscissa is T 3 (t) = T 3 (s) = -T 3 (S) = S(3 -4S 2 ). A b-2 A b-1 A 1 A 2 A 3
(S) < 3 A(S ′ ) if T 3 (S) > T 3 (S ′ ). The Conway notation of C(3, b, c, r) is C(D(A 1 ), -D(A 2 ), . . . , (-1) b-1 D(A b-1 )) where A 1 < 3 A 2 < 3 • • • < 3 A b-1 and D(A) is D(s, t, r) defined in Formula 1.

Case a = 4

We get P 4 (S, T ) = 8 S S 2 -2 T -1 . We thus obtain two families of double points

A : {S = 0, P b (0, T ) = 0}, B : {T = 1 2 S 2 -1 , P b (S, 1 2 (S 2 -1)) = 0}.
Let n = 1 2 (b -1). We have |A| = n from which we deduce that deg T P b (0, T ) = n. From |B| = 2n, we deduce also that deg

S P b (S, 1 2 (S 2 -1)) = 2n. As the leading coefficient of Q c (S, T, ϕ) is c(2ϕ) c-1 we deduce that R 1 (ϕ) = Res T (P b (0, T ), Q c (0, T, ϕ)) has degree n(c -1) and R 2 (ϕ) = Res S P b (S, 1 2 S 2 -1 ), Q c (S, 1 2 S 2 -1 , ϕ) has degree 2n(c -1). Z 4,b,c is the set of real roots of R 4,b,c = R 1 × R 2 . B n B ′ n B n-1 B ′ n-1 A n-1 A n B 1 A 1 B ′ 1 A 2 B 2 B ′ 2 Figure 7: C(4, 2n + 1)
The abscissa of

A(T ) ∈ A is given by T 4 (t) = T 4 (s) = 1 + 8 T + 8 T 2 . The abscissa of B(S) ∈ B is given by T 4 (t) = T 4 (s) = -1 + 4 S 2 -2 S 4
. We have to sort separately the crossing points of A and B by increasing abscissae: A 1 , . . . , A n and B 1 , B ′ 1 , . . . , B n , B ′ n . Note that B i and B ′ i have the same abscissa. The Conway notation for the knot we obtain is then

C D(A 1 ), -(D(B 1 ) + D(B ′ 1 )), . . . , D(A n ), -(D(B n ) + D(B ′ n )) ,
where D(A) (resp. D(B)) is D(s, t, r) defined in Formula 1.

Remark 4.2. We have here deg R a,b,c = 1 2 (a -1)(b -1)(c -1). We could have computed also R a,b,c by eliminating S and T using Gröbner Basis (see [CLOS]). It may happen that deg R a,b,c < 1 2 (a -1)(b -1)(c -1).

Computation of the polynomials

As T n satisfies the linear recurrence of order 2:

T n+1 + T n-1 = 2tT n we deduce that Q n (Form.
2) satisfies the linear recurrence of order 4:

Q 0 = 0, Q 1 = 1, Q 2 = 2S + 4ϕ, Q 3 = -4 T + 12 ϕ S + 4 S 2 + 12 ϕ 2 -3. Q n+4 = 2 (S + 2 ϕ) (Q n+3 + Q n+1 ) -2 2 ϕ 2 + 2 T + 2 ϕ S + 1 Q n+2 -Q c . (3)
For P n (S, T ) = Q n (S, T, 0) we find P 0 = 0, P 1 = 1, P 2 = 2S, P 3 = -4 T + 4 S 2 -3.

P n+4 = 2S (P n+3 + P n+1 ) -(4 T + 2) P n+2 -P n . (4)
or T = 1 2 (S 2 -1) or S = 0. These polynomials satisfy also linear recurrences. As T m (T n ) = T mn we deduce that Q n |Q nm . We can therefore obtain factors of the polynomials Q n (S, T, ϕ).

In the particular case where a = 3 or a = 4, we obtain our resultants R a,b,c = PhiProjection(P a , P b , Q c ) by computing the resultants between factors of P b (S, T ) and factors of Q c (S, T, ϕ) that depend only on S and ϕ or on T and ϕ.

We have to determine the Schubert fraction of any knot of the type C(a, b, c, r) where r is a given rational number in Q -Z a,b,c . Such rational number is given by PhiSampling(R a,b,c ).

Examples

5.1 The family of knots C(3, 5, 7, ϕ)

We get P 3 (S, T ) = T 3 (t) -T 3 (s) t -s = 4(T -S 2 + 3 4 ), so T = S 2 -3 4 and P 5 (S, S 2 -3 4 ) = -16S 4 + 12S 2 -1, Q 7 (S, S 2 -3 4 , ϕ) = 64 S 6 -16 84 ϕ 2 + 5 S 4 -112 ϕ 1 + 20 ϕ 2 S 3 + 24 42 ϕ 2 + 1 S 2 +28 ϕ 48 ϕ 4 + 3 + 80 ϕ 2 S -1 + 1120 ϕ 4 + 448 ϕ 6 + 84 ϕ 2 .
C(3, 5, 7, ϕ) is singular iff ϕ is a root of R 3,5,7 = Res S P 5 (S, S 2 -3 4 ), Q 7 (S, S 2 -3 4 , ϕ) .

R 3,5,7 has degree 24 = 1 2 (3 -1)(5 -1)(7 -1) and 12 real roots ϕ 1 , . . . , ϕ 12 . We choose 13 rational values

r 0 < ϕ 1 < r 1 < • • • < ϕ 12 < r 12 .
Let us determine now the nature of C(3, 5, 7, r). We have to evaluate D(s, t, ϕ) = Q 7 (S, S 2 -3 4 , r)•P 2 (S, S 2 -3 4 ) when P 5 (S, S 2 -3 4 ) = 0. Let S 1 < S 2 < S 3 < S 4 be the 4 real roots of P 5 . They correspond to parameters (s 1 , t 1 ), . . . , (s 4 , t 4 ) such that s i +t i = S i , s i t i = T i = S 2 i -3 4 . We have T 3 (-S 2 ) < T 3 (-S 1 ) < T 3 (-S 4 ) < T 3 (-S 3 ) and the knot C(3, 5, 7, r) is given by the continued fraction expansion α

/β = [D 2 , -D 1 , D 4 , -D 3 ] where D i = sign Q 7 (S i , S 2 i - 3 4 , r) • P 2 (S i , S 2 i -3 4
) . We obtain

r 7 = 0 r 8 = 1 15 r 9 = 1 5 r 10 = 1 4 r 11 = 1 2 r 12 = 2 3 r 13 = 1 α β = 5 3 α β = -1 3 α β = -1 α β = -1 α β = 1 α β = 1 α β = 1
The only non trivial knot is obtained for r = 0. It is the figure-eight knot 4 1 = S( 5 2 ).

S 1 [144,72,72,2,1]. We compute R 3,14,292 as the product of 20 resultants between factors of P 14 and Q 292 . R 3,14,292 (ϕ) has degree 3783 and exactly 2185 distinct real roots. We compute the 1093 Schubert fractions C(3, 14, 292, r i ) where r i > 0. We obtain 275 non trivial knots and eventually 34 distinct knots. One of these has crossing number greater that 10, it is the knot 12 518 = S( 157 34 ).

S 2 S 3 S 4 ϕ 1 ϕ 2 ϕ 3 ϕ 4 ϕ 5 ϕ 6 ϕ 7 ϕ 8 ϕ 9 ϕ 10 ϕ 11 ϕ 12
5.3 A much more complicated example, the family of knots C(4, 13, 267, ϕ)

P 13 (S, T ) is irreducible so as P 13 (0, T ) (that has degree 6) and P 13 (S, 1 2 (S 2 -1)) (that has degree 12). We have 267 = 3 • 89. We know that Q 89 |Q 267 . Q 267 is the product of Q 3 , Q 89 and a polynomial of degree 176 in ϕ. Q 267 (S, T, ϕ) is a product of polynomials of degrees [176,88,2].

We thus obtain R 1 = Res T P 13 (0, T ), Q 267 (0, T ) as a product of polynomials of de-

Results

In this section we present some results we have obtained using certified implementations of the three black-boxes on which our algorithms are based. They are easily implementable in any high level language.

Implementations

There are numerous choices for the implementations, but our requirements are strict: we must certify all the results since our goal is to obtain a classification; bearing in mind that the systems of polynomial equations have thousands of roots.

• PhiProjection(P a (S, T ),

P b (S, T ), Q c (S, T, ϕ) ∈ Q[ϕ]) → R ∈ Q[ϕ], such that Z(R) = π ϕ {P a = 0, P b = 0, Q c = 0}
A straightforward way is to compute a Gröbner basis of P a (S, T ), P b (S, T ), Q c (S, T, ϕ) for a so called elimination order (see [CLOS]). Triangular decompositions provide a suitable alternative, or, more basically, iterative resultants in generic situations. Resultants can be used efficiently for our problem, the system being sufficiently generic.

Our choice is then an ad-hoc method based on resultants computations in the same spirit as in section 4.

• PhiSampling(R ∈ Q[ϕ]) → r 0 , . . . , r N ∈ Q such that r 0 < ϕ 1 < r 1 < • • • < ϕ N < r N , where ϕ 1 < • • • < ϕ N are the real roots of R
Such a function can easily be implemented using any solver that is able to isolate real roots of univariate polynomials (say providing non overlapping intervals with rational bounds around all the real roots). It must be able to discriminate multiple roots from clusters of roots, real roots from complex roots with a small imaginary part (which excludes many numerical methods and most of implementations using hardware floats). One can use methods based on Sturm sequences or the Descartes rule of signs (see [BPR] for an overview), but also many strategies using interval analysis. Due to the high degree of the polynomials, our choice is to use algorithms based on the Descartes rule of signs using multi-precision interval arithmetic as in [RZ].

• SignSolve(r ∈ Q, P a (S, T ), P b (S, T ), D a,b,c (S, T, ϕ) ∈ Q[S, T, ϕ]) → [sign D a,b,c (S, T, r) ], (S, T ) ∈ {P a (S, T ) = P b (S, T ) = 0}
The determination of the sign of a polynomial over a zero-dimensional system is difficult to certify when using numerical method. There are few exact/certified existing methods/implementations for this problem. The strategy is naturally linked to the implementation of the function PhiProjection since the zero-dimensional system to be considered by SignSolve is a subsystem of the one which is to be considered by PhiProjection. One can use the generalized Hermite method for zero-dimensional systems as in [PRS], which makes use of Gröbner bases. One can use also any method that first rewrites the system as a rational parametrization (as in [Rou] or [GLS]) and then apply any algorithm that computes the sign of an univariate polynomial at a real algebraic number. This last step can be done by extending methods based on Sturm theorem or based on the Descartes rule of signs. Due to our implementation of PhiProjection and to the degrees of the polynomials, we base our implementation on the Descartes rule of signs.

For the experiments, we used the Maple environment. PhiProjection is based on resultants computation (see 4.1 and 4.2). We use the Maple function Isolate for PhiSampling (without constraints) and for SignSolve (with constraints). In the univariate case, this function is based on the algorithm described in [RZ]. Other computations have been straightforwardly implemented according to the descriptions proposed in section 4.

Experiments

Let us remind in the next table the number K N of two-bridge knots with crossing number N , up to mirror symmetry (see [ES] for a formula).

N 3 4 5 6 7 8 9 10 K N 1 1 2 3 7 12 24 45

The minimal b for a Chebyshev

diagram C(3, b) of K = S( α β ), α β > 1, is obtained with b = n + 1
where n is the length of the continued fraction of α β or α αβ (see [START_REF] Koseleff | Chebyshev diagrams for rational knots[END_REF]). This allows us, using Algorithm 3.2, to know the minimal b for which C(3, b) is a projection of a given rational knot K.

In a similar manner, let

K = S( α β ), α β > 1, β even. The minimal integer b = 2n + 1 for which there exists a continued fraction expansion r = [ε 1 , 2ε 2 , . . . , ε 2n-1 , 2ε 2n ], ε i = ±1, such that S(r) is equivalent to K, is the smallest length of the continued fraction expansion [±1, ±2, . . . , ±1, ±2] of either α β , α 2α -β , α β ′ or α 2α -β ′ where 0 < β ′ < α, β ′ even and ββ ′ = ±1 (mod α).
It happens that for some knots, there is a continued fraction expansion with smaller length including 0 instead of ±2. The list of these knots (up to crossing number 10) is 8 12 , 9 13 , 9 15 , 9 26 , 10 8 , 10 12 , 10 13 , 10 25 , 10 29 , 10 38 , 10 42 . For example for the knot 8 12 = S( For example the knot 9 5 admits the Chebyshev parametrization C(3, 17, 45, 1/364). It is not minimal and we know that there is some other parametrization C(3, 13, c, r) where c > 300 and r is some rational number. This knot admits also the parametrization C(4, 11, 152, 1/44) which is minimal with respect to b and C(4, 19, 22, 1/20) that has minimal degree.

We get both 10 20 and 10 29 with minimal b = 14 for the same value c = 292. We had to compute the polynomial R 3,14,292 of degree 3783 and 2185 real roots (see 5.2). We obtain the knot 10 20 = S( 3511 ) with C(3, 14, 292, 1/94) and the knot 10 29 = S( 63 17 ) with C(3, 14, 292, 1/93).

Conclusion

We have shown that any two-bridge knot is a Chebyshev knot with a = 3 and also with a = 4. For every a, b, c integers (a = 3, 4 and a, b coprime), we have described an algorithm that gives all Chebyshev knots C(a, b, c, ϕ).

Our experiments fully justify the use of certified algorithms and exact computations since numerical methods would have certainly failed in finding for example the knot 10 20 = S( 35 11 ) with C(3, 14, 292, 1/94) and the knot 10 29 = S( 63 17 ) with C(3, 14, 292, 1/93). Also, an objective is now to consolidate and speed up our algorithms in order to increase its capabilities.

As the zero-dimensional systems we study have a triangular structure, we could try to get directly an exhaustive list of Chebyshev knots C(a, b, c, ϕ) without computing additional resultants. In case when a, b and c are relatively coprime, we can expect that the real variety V a,b,c has only single points and try to get directly all possible signs. In that case, our three black-boxes could be implemented using exclusively univariate functions that compute recursively the roots of the systems to be solved without any additional rewriting.

Chebyshev parametrizations of the first rational knots

K minimal b α/β DP deg min. (b -1)(c -1) α/β DP deg 3 1 C(3, 4, 5, 0) 3/2 3 12 C(3, 4, 5, 0) 3/2 3 12 C(4, 3, 5, 0) 3/

  where ϕ 1 < • • • < ϕ N are the real roots of R • SignSolve(r ∈ Q, P a (S, T ), P b (S, T ), D a,b,c (S, T, ϕ) ∈ Q[S, T, ϕ])→ [sign D a,b,c (S, T, r) ], (S, T ) ∈ {P a (S, T ) = P b (S, T ) = 0}

  from which we deduce that T n is a polynomial of degree n and leading coefficient 2 n-1 . Proposition 2.1 ([KP1]). Let a and b be relatively prime integers. The affine Chebyshev curve C(a, b) defined by C(a, b) : T b (x) -T a (y) = 0 admits the parametrization x = T a (t), y = T b (t). C has 1 2 (a -1)(b -1) singular points which are crossing points. The pairs (t, s) giving a crossing point are a and b are coprime integers, the projection of C(a, b, c, ϕ) onto the (x, y)-plane is the plane Chebyshev curve C(a, b). The curve C(a, b, c, ϕ) is a knot if and only if C(a, b, c, ϕ) has no double point. We thus deduce Proposition 2.2. Let a, b and c be integers (a and b being relatively prime). The number of Chebyshev knots

Figure 2 :

 2 Figure 2: The right twist and the left twist

  Figure 4: Chebyshev diagrams

  Figure 5: a = 4

Corollary 3. 3 .

 3 Every two-bridge knot has a Chebyshev diagram C(3, b), b ≡ 0 (mod 3). Every two-bridge knot has a Chebyshev diagram C(4, b), b ≡ 1 (mod 2).

Figure 6 :

 6 Figure 6: C(3, b), b even

Figure 8 :

 8 Figure 8: The curve Q 7 (S, S 2 -3 4 , ϕ) = 0 and the lines P 5 (S, S 2 -3 4 ) = 0

Figure 9 :

 9 Figure 9: Number of real roots, number of non trivial knots, proportion of distinct knots

we have 29 12

= [1, 0, 1, 2, 1, 0, 1, 2] while it is not possible to get a shorter continued fraction corresponding to 8 12 . We have enumerated all possible continued fraction expansions corresponding to diagrams C(4, b) to determine the minimal b corresponding to a rational knot K.

In the next table we give the number of two-bridge knots of crossing number N that have a projection C(3, n) with n ≤ b and a projection C(4, n ′ ) with n ′ ≤ b.

Minimal b a = 3 a = 4 b 4 5 7 8 10 11 13 14 3 5 7 9 11 13 We can also observe that for b = 10 we have all knots with crossing number not greater than 6, 7 knots with crossing number 7, 3 knots with crossing number 8 and 1 knot with crossing number 9. This last one is the Fibonacci knot C(1, 1, 1, 1, 1, 1, 1, 1, 1) ∼ C(3, 10, 17, 0) (see [START_REF] Koseleff | Chebyshev knots[END_REF]).

Looking at the 45 knots with crossing number 10 we see that: 13 of them have a projection C(4, 9) (12 crossing points), 37 of them have a projection C(4, 11) (15 crossing points), 45 of them have a projection C(4, 13).

Let K be a two-bridge knot. Once we know what we can expect as a diagram C(a, b) for K, we look for it as a Chebyshev knot C(a, b, c, ϕ). We see from the previous table that every rational knot with crossing number N ≤ 10 is a Chebyshev knot C(3, b, c, r) with b ≤ 14 and a Chebyshev knot C(4, b, c, r) with b ≤ 13. We proved in [START_REF] Koseleff | Chebyshev diagrams for rational knots[END_REF] that a two-bridge knot with crossing number N admits a plane Chebyshev projection C(3, b) with b < 3 2 N . In comparison, the number of crossing points for Lissajous diagrams is far greater (see [BDHZ]).

We have limited ourselves to the bounds b ≤ 21, c ≤ 300 and (b-1)(c-1) ≤ 13•299 = 3887. The degrees of the polynomials R(ϕ) giving the critical values are bounded by 3887 when a = 3 and 5382 when a = 4. A remarkable fact is that these polynomials have a large number of real roots (in average 58% when a = 3 and 57% when a = 4, see Figure 9, first column). The proportion of non trivial knots C(a, b, c, r) is approx. 25% when a = 3 and


A Zghiche Lapp

Search for ν τ appearance via neutrino oscillations in the ν µ CNGS beam with the OPERA experiment

The OPERA neutrino detector in the underground Gran Sasso Laboratory (LNGS) is designed to realize the first detection of neutrino oscillations in the appearance mode through the study of νµ → ντ oscillations. The apparatus consists of an emulsion/lead target associated to electronic detectors and is placed in the high energy long-baseline CERN to LNGS beam (CNGS) 730 km away from the neutrino source. Runs with CNGS neutrinos were successfully carried out in 2007 and 2008 with the detector fully operational with its related facilities for the emulsion handling and analysis. After a brief description of the beam and of the experimental setup we report on the collection, reconstruction and analysis procedures of first samples of neutrino interaction events.

Introduction

Recent 'Solar' and 'atmospheric' neutrino experiments discovered that lepton flavour is not conserved, thus identifying the new physics responsible of this anomaly has become the aim of active calculations. Although observed flavour conversions could be produced by different mechanisms, simplicity suggests oscillations of massive neutrinos to explain the data, provided that mixing angles among the Standard Model neutrinos are sufficiently large. Present data strongly disfavor alternative exotic possibilities, such as neutrino decay or oscillations into extra 'sterile' neutrinos and show some hints for the characteristic features of oscillations.

Oscillations can be directly seen by precise reactor and long-baseline beam experiments, that are respectively testing the solar and atmospheric anomalies. In the case of the atmospheric neutrino sector, accelerator experiments can probe the same oscillation parameter region as atmospheric neutrino experiments. This is the case of the OPERA experiment dedicated to the first direct detection of ν µ → ν τ appearance, not yet observed, while being the most probable explanation of the atmospheric data.

The OPERA 1 experiment is located in the Gran Sasso underground laboratory (LNGS) in Italy. The detector developed by the OPERA international collaboration is designed to search for ν τ appearance in the high energy ν µ CERN to Gran Sasso (CNGS) beam, 730 km downstream of the neutrino source. The ν τ direct appearance search is based on the observation of leptonic and hadronic τ decays in the ν τ charge current interaction (CC) events. Because of the weak neutrino cross section and the τ short lifetime, the OPERA detector 2 must combine a huge mass with a high spatial resolution. Both requirements are fulfilled using nuclear emulsions combined with lead plates and assembled into detector units called "bricks".

The CNGS ν µ beam is designed to provide 4.5 × 10 19 proton-on-target/year (p.o.t./y) with a running time of 200 days per year. The beam parameters have been tuned to optimize the number of ν τ charged current interactions in the OPERA detector. Thus, the average neutrino energy is < E >= 17 GeV for an almost pure ν µ beam only with a small contamination of νµ (4%) and of ν e and νe (less than 1%). The average L/E ratio is 43 km/GeV, far from the oscillation maximum, due to the high energy required for ν τ appearance. In 5 years of data taking, OPERA foresees to observe 10 to 15 ν τ events from oscillation at full mixing in the range 2.5 × 10 -3 < ∆m 2 < 3.0 × 10 -3 eV 2 , with a total background of 0.75 events.

The OPERA Detector

OPERA is a hybrid detector (Figure 1) made of two identical Super Modules (SM) each consisting of a target section of about 625 tons made of emulsion/lead bricks, of a scintillator tracker detector (TT) needed to trigger the read-out and localize neutrino interactions within the target, and of a muon spectrometer. The latter, made of two parts located downstream of each SM, provides the charge and momentum informations of the penetrating tracks. Each spectrometer is equipped with RPC bakelite chambers and High Precision Tracker (HPT) composed of drift-tubes. The spectrometer reduces the charge confusion to less than 0.3%, gives a muon momentum measurement better than 20% for a momentum less than 50 GeV/c and reaches a muon identification efficiency of 95%. The rejection of the charged particles originating from outside the target fiducial region coming from neutrino interactions in the surrounding rock material is provided by a large size anti-coincidence detector (VETO), made of two glass RPC planes mounted in front of the first part of the target.

The target section is composed of 29 walls. Each one is made of an X-Y double layered wall of scintillator strips (the TT) and a vertical supporting steel structure that contains the basic target detector units called Emulsion Cloud Chambers (ECC) bricks. A total of 150036 ECC bricks with a total mass of 1.25 ktons have been produced to make the OPERA target. Each ECC brick is a sequence of 57 emulsion films interleaved with 56 (1 mm thick) lead plates. An emulsion film is composed of a pair of 44 µm thick emulsion layers deposited on a 205 µm plastic base. Downstream of each brick (Figure 2), an emulsion film doublet called Changeable Sheet (CS) is attached in a separate enveloppe. The CS doublet can be detached from the brick for analysis to confirm and locate the tracks produced in the electronic detectors by neutrino interactions. The CS doublet is the interface between the ECC brick and the electronic detector. The ECC bricks have been assembled underground at an average rate of 700 per day by a dedicated fully automated Brick Assembly Machine (BAM).

The detector is equipped with an automatic machine, the Brick Manipulator System (BMS) that achieved the entire brick insertion and will perform the extraction of bricks from the detector during the foreseen data taking period. In order to cope with the analysis of the large number of emulsion sheets related to neutrino interactions, large automated facilities are used for the handling, the development and the scanning of the emulsion films. For the latter task, two new generation computer driven fast automatic optical microscopes have been developed: the European Scanning System ESS 3 and the Japanese S-UTS 4 . While the implementation differs in both the hardware and software architectures, the two systems have comparable performances ensuring a scanning speed up to 20 cm 2 /h, and a spatial and angular resolution of the order of 1µm and 1 mrad, respectively.

The OPERA Event Selection

For a given event, the electronic detectors are used to compute a probability map of the location of the interaction brick. The brick with the highest probability is extracted by the BMS. Then the CS doublet is detached from the brick and developed in the underground facility. The two emulsion films are scanned by using fast automated microscopes. If the track candidates found on the CS doublet match with the electronic data then the brick is exposed for 12 hours to cosmic rays in order to provide the film-to-film alignment in the brick. Subsequently the brick is developed in an automated facility and sent to the scanning laboratories either in Europe (ESS) or in Japan (S-UTS). The vertex finding strategy consists in following back, film by film, the tracks found on the CS doublet until the tracks stop inside the brick. To confirm the stopping track, an area scan of several mm 2 around the stopping point of the tracks is performed for 5 films upstream and downstream. Then an interaction vertex can be reconstructed and a topology compatible with the decay of a τ lepton is sought.

Run Status and Real Data Events

After a short commissioning run in 2006, the CNGS operation started on September 2007 at rather low intensity with 40% of the total OPERA target mass. Due to the operational problems of the CNGS, the physics run lasted only a few days. During this run 0.082 × 10 19 protons on target (p.o.t.) were accumulated and 465 events were recorded, but only 35 in the target region. The other events originated in the spectrometers, the supporting structures, the rock surrounding the cavern and the hall structure. From June to November 2008, 1.782 × 10 19 p.o.t. were delivered by the CNGS. OPERA collected 10100 events with 1663 interactions in the target region. This number is compatible with the MC (Monte Carlo) simulation prediction of 1723 interactions. All electronic detectors were operational and the live time of the data acquisition system exceeded 99%. For the events classified as CC interactions in the target the muon momentum and the muon angle in the vertical (y-x) plane with respect to the horizontal (z) axis distributions were found in agreement with the MC simulation expectation [START_REF] Agafonova | [END_REF] . By midjune 2009, 1019 bricks have been developed and around 746 events have been located. The brick finding efficiency value is around 80%. The vertex finding efficiency in the selected bricks for CC events is between 84-95 % while a value of 93% is predicted by the MC simulation. For NC events, the efficiency ranges between 70 and 91 % while the MC simulation prediction is 81%. In a sample of 550 ν µ CC interactions and among the located events, 8 events show a charm-like decay topology in agreement with the MC simulation predicted value of 10. Because charm decays exhibit the same topology as ν τ decays and because they are a potential source of background if the muon at the primary vertex is not identified, Charm production and decay events constitute the best calibration channel for the OPERA detector. A charm-like topology is shown in Figure 3 where a track exibits a decay-kink.

Conclusion

During the 2008 CNGS run all the electronic detectors performed well. The OPERA strategy has been validated and the vertex location was successfully accomplished for CC and NC events.

In the analyzed data sample of 550 ν µ CC interactions, 8 events with a charm-like topology were found. This is in agreement with the expectations and shows the success of combining the topological and kinematical analyses. The 2008 run constitutes an important milestone for the OPERA experiment. For the 2009 run, around 3.5 × 10 19 p.o.t. are foreseen to be acquired. The integrated statistics would be sufficient to expect the observation of two events and give a precise estimation of detector efficiency, background and sensitivity.

Figure 1 :

 1 Figure 1: View of the OPERA detector

Figure 2 :

 2 Figure 2: Schematic view of two bricks with their Changeable Sheets and target tracker planes.

Figure 3 :

 3 Figure 3: Display of the OPERA electronic detector of a νµ charged current interaction with a charm like topology (top panel). The emulsion reconstruction is shown in the bottom panels where the charm-like topology is seen with a kink: top view (bottom left), side view (bottom center), frontal view (bottom right). The dots in the lower panel are due to an electromagnetic shower.