On polynomial Torus Knots Pierre-Vincent Koseleff, Daniel Pecker
To cite this version:
Pierre-Vincent Koseleff, Daniel Pecker. On polynomial Torus Knots. Journal ofKnot Theory and Its Ramifications, World Scientific Publishing, 2008, 17 (12), pp.1525-1537. 10.1142/S0218216508006713. hal-00108510v2
HAL Id: hal-00108510 https://hal.archives-ouvertes.fr/hal-00108510v2 Submitted on 18 Oct 2007
HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. hal-00108510, version 2 - 18 Oct 2007 ftesmls nt,tetrskoso ye(2 type of knots torus the knots, simplest the of aaerztoso h rfi n ftefiueegtknot. eight figure the of and trefoil the kno of polynomial parametrizations the of projections plane the t interval, and compact large, quite a on theorem approximation Weierstrass nt a eatie rmplnmaso ere (3 degrees of polynomials from attained be can knots u ehdi ae ntefc htalpaepoetosof projections plane all that fact the on based is method Our (3 yecnb bandfo oyoilembedding polynomial p He a [Va]. from Vassiliev obtained with be began can knots compact type non of study The Introduction 1 2000: Classification Subject Mathematics Theorem keywords: ye(hr sa nlgu usini asle’ ae [V paper Vassiliev’s in question repres analogous polynomials the an of is degrees (there minimal type the find to is which fcosnshv setal h aedarm hsi con a is This diagram. same the essentially have crossings of oyoil fdges(3 degrees of polynomials aaerzto ftetrefoil the of parametrization ti o icl ose sn ´zu hoe,ta hspla this that (3 B´ezout theorem, than using smaller degrees see, of to polynomials difficult not is It n , hsi htmtvtdA ajnadRm hka[S ofidsm find to [RS] Shukla Rama and Ranjan A. motivated what is This neednl,Sati[h aeadtie ro fti th this of proof detailed a gave [Sh] Shastri Independently, nti ae,w hl rv h olwn result following the prove shall we paper, this In h ubro rsig fapaepoeto of projection plane a of crossings of number The aual,Rm iha(M] se hte twspossible was it whether asked ([Mi]) Mishra Rama Naturally, 1 + embedding xmlswt iia eiorpi degree. lexicographic minimal with examples m , eso htn ou nto ye(2 type of knot torus no that show We . ) . ntter,plnma uvs ou nt,parametrized knots, torus curves, polynomial theory, Knot If n t 7→ 3 6= ( f ( sod h ou knot torus the odd, is t e-mail: ) g , n , ( ,paeJsiu -55 ai ee 05 Cedex Paris F-75252 Jussieu, place 4, t ) 1 + nplnma ou Knots Torus polynomial On h , K ( m , nvri´ iree ai Curie Marie et Universit´e Pierre 3 t { )wee(deg( where )) koseleff,pecker nlgu oSatisoe hyas se h aua ques natural the asked also They one. Shastri’s to analogous when ) .-.Kslff .Pecker D. Koseleff, -V. P. n , coe 9 2007 19, October 1) + n ≡ Abstract n , K f . ,o o 3 mod 0 or 1, ) ), n , n , 45,1D0 61,57M25 26C10, 12D10, 14H50, 1 deg( > n ) antb ersne yplnmaso degrees of polynomials by represented be cannot n , K } g , d,cnb bandfo polynomial a from obtained be can odd, 3 d,dntdby denoted odd, )) @math.jussieu.fr n t 2 n ≤ 7→ sa least at is − (3 ( a]). 2 f n , sqiecomplicated. quite ts edgeso h oyoil a be may polynomials the of degrees he oe,h logv ipepolynomial simple gave also he eorem, , . nigagnrltrsko fagiven a of knot torus general a enting ( ecrecno eprmtie by parametrized be cannot curve ne 2 euneo h o ovdclassical solved now the of sequence t ) vnuly egv explicit give we Eventually, 1). + n ) g , oe htaynncmatknot non-compact any that roved − K ( n oprmtieteknot the parametrize to t n 1) ) Bnwt hoe,se[Re]). see theorem, (Bankwitz h , . ihtemnmlnumber minimal the with K npriua,te bana obtain they particular, In ( l ereparametrizations degree all t pc curve space n )) hypoe htthese that proved They . t , ∈ IR . h ro uses proof The K n tion by n 2 P. -V. Koseleff, D. Pecker, On polynomial Torus Knots Tait’s conjectures [Mu, Ka, Pr, MT]. This allows us to transform our problem into a problem of real polynomial algebra. As a conclusion, we give explicit parametrizations of K3, K5 and K7. By our result, they are of minimal degrees. We also give an explicit parametrization of K9 with a plane projection possessing the minimal number of crossing points. This embedding is of smaller degree than those already known.
2 The principal result
If n is odd, the torus knot Kn of type (2,n) is the boundary of a Moebius band twisted n times (see [Re, Ka, St]). The recently proved Tait’s conjectures allow us to characterize plane projections
Figure 1: Kn, n = 3, 5, 7. of Kn with the minimal number of crossings. Lemma 1 Let be a plane curve with n crossings parametrized by (t) = (x(t),y(t)). If is the C C C projection of a knot Kn then there exist real numbers s1 < < sn < t1 < < tn, such that · · · · · · (si)= (ti). C C Proof. Let be a plane projection of a knot of type Kn with the minimal number n of crossings. C Using the Murasugi’s theorem B ([Mu]) which says that a minimal projection of a prime alter- nating knot is alternating, we see that is alternating. C Then the Tait’s flyping conjecture, proved by Menasco and Thistlethwaite ([MT, Pr]), asserts that is related to the standard diagram of Kn by a sequence of flypes. Let us recall that a flype C is a transformation most clearly described by the following picture.
Figure 2: A flype
The standard diagram 0 of Kn has the property (cf [Re]) that there exist real numbers s1 < < S · · · sn sa s s′ A B C A C ′ t t B Figure 3: Flype on the part B After the flype on B, we have new parameters corresponding to the crossing points satisfying ′ ′ ′ ′ ′ ′ ′ ′ sa ′ The transformed diagram has the same property: there exist real parameters s1 < < sn < ′ S ′ · · · t1 < In this paper we shall consider polynomial knots, that is to say, polynomial embeddings IR 3 3 −→ IR , t (x(t),y(t), z(t)). Polynomial knots are non-compact subsets of IR . The closure of a 7→ 3 3 polynomial knot in the one point compactification S of the space IR is an ordinary knot (see [Va, Sh, RS] and figures at the end). Lemma 2 Let be a plane polynomial curve with n crossings parametrized by (t) = (x(t),y(t)). C C Suppose that is the projection of Kn and deg x(t) deg y(t). Then we have deg x(t) 3. If C ≤ ≥ deg x(t) = 3, then deg y(t) n + 1. ≥ Proof. x(t) must be non-monotonic, so deg x(t) 2. Suppose that x(t) is of degree 2. Then ≥ x(ti) = x(si) implies that ti + si is constant, and so the parameter values corresponding to the crossing points are ordered as s1 < 3 Proof of the main result Our proof makes use of Chebyshev (monic) polynomials. 4 P. -V. Koseleff, D. Pecker, On polynomial Torus Knots 3.1 Chebyshev Polynomials sin((n + 1)θ) Definition 1 If t = 2 cos θ, let Tn(t) = 2 cos(nθ) and Vn(t)= . sin θ Remark 1 Tn and Vn are both monic and have degree n. We have V0 = 1, V1 = t, Vn+1 = t Vn Vn−1. (1) − We have also T0 = 2, T1 = t, Tn+1 = t Tn Tn−1. − n n−2 n−4 For n 2, let Vn = t + ant + bnt + . Using recurrence formula 1, we get ≥ · · · an+1 = an 1, bn+1 = bn an−1 − − so by induction, n n−2 1 n−4 Vn = t (n 1)t + (n 2)(n 3)t + . (2) − − 2 − − · · · We shall also need the following lemmas which will be proved in the next paragraph. Lemma A. Let s = t be real numbers such that T3(s)= T3(t). For any integer k, we have 6 Tk(t) Tk(s) 2 kπ − = sin Vk−1(s + t). t s √3 3 − Lemma B. Let n 3 be an integer. ≥ Let s1 3.2 Proof of the theorem Proof. We shall prove this result by reducing it to a contradiction. Suppose the plane curve C parametrized by x = P (t), y = Q(t) where deg P = 3, deg Q = n + 1 is a plane projection of Kn. By translation on t, one can suppose that P (t)= t3 αt+β. If the polynomial P was monotonic, − would have no crossings, which is absurd. Therefore α > 0. Dividing t by ρ = √3/√α, one C 3 3 has P (t) = ρ (t 3t)+ µ. By translating the origin and scaling x, one can now suppose that 3 − P (t)= t 3t = T3(t). − By translating the origin and scaling y, we can also suppose that Q(t) is monic and write P (t)= T3(t), Q(t)= Tn+1(t)+ anTn(t)+ + a1T1(t). · · · By B´ezout theorem, the curve has at most (3 1)(n + 1 1)/2= n double points. As it has at C − − least n crossings, we see that it has exactly n crossings and therefore is a minimal crossing diagram of Kn. According to the lemma 1, there exist real numbers s1 < < sn, t1 < < tn, si < ti · · · · · · such that P (si)= P (ti), Q(si)= Q(ti). Let ui = ti + si, 1 i n. We have ≤ ≤ n Q(ti) Q(si) Tn+1(ti) Tn+1(si) Tk(ti) Tk(si) − = − + ak − . ti si ti si =1 ti si − − Xk − Accepted for publication in J. of Knot Th. and Ram. 5 so by lemma A, u1,...,un are the distinct roots of the polynomial n R(u)= εn+1 Vn(u)+ akεk Vk−1(u), (3) =1 Xk where 2 kπ εk = sin . √3 3 Remark 2 Note that εk = Vk−1(1) is the 6-period sequence ε0 = 0, ε1 = 1, ε2 = 1, 0, 1, 1,.... − − We have to consider several cases. £ Case n 2 mod 3. εn+1 = 0 and R(u) has degree at most n 1. This is a contradiction. ≡ − £ Case n 1 mod 3. In this case, n 1 mod6 and εn+1 = εn = 1, εn−1 = 0. Thus R(u) can ≡ ≡ be written as R(u) = Vn(u)+ anVn−1(u) an−2Vn−3(u) + a2V1(u)+ a1 n n−1 − n−2 −··· = u + anu (n 1)u + . − − · · · using equation 2. Therefore we get ui = an, uiuj = (n 1), − − − 1≤i≤n 1≤i R(u) = Vn(u) an−1Vn−2(u) an−2Vn−3(u)+ a2V1(u) a1. − − − ···− − Let σi be the coefficients of n n k n−k R(u)= u + ( 1) σku . − =1 − Xk From the equation 2, we see that (n 2)(n 3) σ1 = 0, σ2 = (an−1 + n 1), σ4 = (n 3) an−1 + − − . − − − 2 n k Let Sk be the Newton sums i=1 ui of the roots of the polynomial R. Using the classical Newton formulas ([FS]), we obtain P 2 2 S1 = σ1 = 0, S2 = σ 2σ2 = 2σ2, S4 = 2σ 4σ4, 1 − − 2 − and then 2 S4 = 2(an−1 + 2) + 6n 18 6n 18. − ≥ − By the lemma B, we deduce that 22 + n 6n 18, i.e. n 8 so n = 3. 2 ≥ − ≤ 6 P. -V. Koseleff, D. Pecker, On polynomial Torus Knots 3.3 Proof of lemmas A and B We shall use the following lemma Lemma 3 (Lissajous ellipse) Let s = t be complex numbers such that 6 T3(t)= T3(s). There exists a complex number α such that s = 2 cos(α + π/3), t = 2 cos(α π/3). − Furthermore, α is real if and only if s and t are both real, and then t>s if and only if sin α> 0. Proof. We have T3(t) T3(s) − = t2 + s2 + st 3. (4) t s − − Then, if T3(t)= T3(s), t = s, we get 6 3 1 (t + s)2 + (t s)2 = 2(t2 + s2 + st) = 6. 2 2 − That means (s,t) 2 2 2 t + s t s + − = 1. 2 √ 2 3 1 Then there exists a complex number α such that t + s t s cos α = , sin α = − , (s, 2 cos α) 2 2√3 ±2 ±1 1 2 that is t = 2 cos(α π/3), s = 2 cos(α + π/3). ±1 − α is real if and only if cos α and sin α are both real that is to say, iff s and t are real. In this case: t>s sin α> 0. 2 ⇔ ±2 In order to prove lemma B, we shall use the following lemma which describes the geometrical configuration. Let us denote s(α) = 2 cos(α + π/3) and t(α) = 2 cos(α π/3). − Lemma 4 Let α, α′ [0,π] be such that s(α) < s(α′), and t(α) < t(α′). Then α > α′ and 2π α + α′ π ∈ > > . 3 2 3 Proof. We have 2 cos α = s(α)+ t(α) Proof of lemma B. Let s1 < At least two of the αi’s lie in the intervals ]0, π/2] or [π/2,π[. We have only two cases to consider: π π π > α1 > α2 , or αn−1 > αn > 0. ≥ 2 2 ≥ On the other hand, we get the equality cos2 x + cos2 y = 1 cos2(x + y) + 2 cos x cos y cos(x + y). (5) − π £ Case 1. αn−1 > αn > 0. 2 ≥ 1 We get cos αn 0, cos αn−1 0 and cos(αn−1 + αn) < so eq. 5 becomes ≥ ≥ −2 2 2 2 3 cos αn−1 + cos αn 1 cos (αn−1 + αn) ≤ − ≤ 4 and n n−2 2 2 2 2 2 cos αi = cos α1 + cos αi + (cos αn−1 + cos αn) i=1 i=2 X X 1 3 1 1 + (n 3) + = (n + 4), ≤ − · 4 4 4 that is n n 2 2 S2 = u = (2 cos αi) n + 4. i ≤ i=1 i=1 X X π £ Case 2. π > α1 > α2 . ≥ 2 1 We get cos α1 0, cos α2 0 and cos(α1 + α2) < so eq. 5 becomes ≤ ≤ −2 2 2 2 3 cos α1 + cos α2 1 cos (α1 + α2) ≤ − ≤ 4 and similarly, we get n 2 S2 = (2 cos αi) n + 4. ≤ i=1 X Analogously, we get cos4 x + cos4 y (cos2 x + cos2 y)2, and we deduce: ≤ n 4 S4 = (2 cos αi) n + 22. ≤ i=1 X 8 P. -V. Koseleff, D. Pecker, On polynomial Torus Knots 2 Proof of lemma A. Let s t = 2 cos(α π/3), s = 2 cos(α + π/3). − π We have s + t = 2 cos α and t s = 4sin sin α, so − 3 kπ sin kα sin Tk(t) Tk(s) 2 cos k(α π/3) cos k(α + π/3) · 3 − = − π− = π t s 4sin sin α sin α sin − 3 · 3 2 kπ = sin Vk−1(2 cos α). √3 3 2 4 Parametrized models of K3,K5,K7 and K9 We get parametrizations of Kn: = (x(t),y(t), z(t)), with n crossings obtained for parameter C values satisfying the hypothesis of lemma 1. According to lemma A, we choose n distinct points 1 u1 < < un 1. We look for Q1 and Q2 of minimal degrees, such that − ≤ · · · ≤ Q1(t) Q1(s) Q2(t) Q2(s) R1(t + s)= − , R2(t + s)= − t s t s − − satisfy, for i = 1,...,n, i R1(ui) = 0, R2(ui) = ( 1) . − We then choose y(t) = Q1(t) and z(t) = Q2(t). We also add some linear combinations of T6i efficiently. We then obtain a knot whose projection is alternating, when R1 has no more roots in [ 2, 2]. As we have chosen symmetric ui’s, all of our curves are symmetric with respect to the − y-axis. 4.1 Parametrization of K3 We can parametrize K3 by x = T3(t),y = T4(t), z = T5(t). It is a Lissajous space curve (compare [Sh]). The plane curve (T3(t), T4(t)) has 3 crossing points. The plane curve (T3(t), T5(t)) has 4 crossing points corresponding to parameters (si,ti) with s1 bottom view of K3 face view of K3 Figure 4: The trefoil knot K3 4.2 Parametrization of K5 Let us consider the curve of degree (3, 7, 8): x = T3(t), y = T8(t) 2 T6(t) + 2.189 T4(t) 2.170 T2(t), − − z = T7(t) 0.56 T5(t) 0.01348 T1(t). − − The curve (x(t),y(t)) has exactly 5 double points when the projection (x(t), z(t)) has exactly 6. Note here that deg z(t) < deg y(t). In conclusion we have found a curve of degree (3, 7, 8). Using bottom view face view Figure 5: Knot K5 our theorem, we see that this curve has minimal degree. A. Ranjan and R. Mishra showed the existence of such an example ([RS, Mi]). 4.3 Parametrization of K7 We choose x = T3 (t) , y = T10 (t) 2.360 T8 (t) + 4.108 T6 (t) 6.037 T4 (t) + 7.397 T2 (t) , − − z = T11 (t) + 3.580 T7 (t) 3.739 T5 (t) T1 (t) . − − 10 P. -V. Koseleff, D. Pecker, On polynomial Torus Knots Bottom view Zoom on the bottom view Figure 6: Knot K7 The values of the parameters corresponding to the double 2 points are obtained as intersection points between the ellipse t2 + s2 + st 3 = 0 1.5 − t and the curve of degree 9: 1 (y(t) y(s))/(t s) = 0 − − 0.5 The curve (x(t),y(t)) has exactly 7 double points correspond- ing to cos(α)= 1/2, 3/10, 2/10, 0 . {± ± ± } In conclusion we have found a curve of degree (3, 10, 11). ±2 ±1.5 ±1 ±0.5 0 s Using our theorem, this curve has minimal degree. 4.4 Parametrization of K9 We choose polynomials of degree (3, 13, 14). x = T3 (t) , y = T14 (t) 4.516 T12 (t) + 12.16 T10 (t) 24.46 T8 (t) + 39.92 T6 (t) − − 55.30 T4 (t) + 66.60 T2 (t) , − z = T13 (t) 2.389 T11 (t) 5.161 T7 (t) + 5.161 T5 (t) + 1.397 T1 (t) . − − The curve (x(t),y(t)) has exactly 9 double points corresponding to cos(α)= 1/2, 3/10, 2/10, 1/10, 0 . {± ± ± ± } One can prove that it is minimal under the assumption that the projection (x(t),y(t)) has exactly 9 double points. Conclusion We have found minimal degree polynomial curves for torus knots Kn, n = 3, 5, 7. For degree 9, one can prove that it is minimal under the assumption that the projection (x(t),y(t)) has exactly 9 double points. We have similar constructions for higher degrees. Acknowledgments We would like to thank Julien March´efor fruitful discussions on knot theory. Accepted for publication in J. of Knot Th. and Ram. 11 Bottom view Zoom on the bottom view Figure 7: Knot K9 References [FS] Faddeev, D., Sominskij, I. Problems in higher algebra, W. H. Freeman, 1965. [Fi] G. Fischer, Plane Algebraic Curves, A.M.S. Student Mathematical Library Vol 15, 2001. [Ka] L. Kaufmann, On knots, Princeton University Press, 1987. [MT] W.M. Menasco, M.B. Thistlethwaite, The Tait Flyping conjecture, Bull. Amer. Math Soc. 25 (2) (1991), 403-412. [Mi] R. Mishra, Minimal Degree Sequence for Torus knots, Journal of Knot Theory and its Ramifications, Vol. 9, No 6 (2000), 759-769. [Mu] K. Murasugi, Jones polynomials and classical conjectures in knot theory, Topology, 26 (1987), 187-194. [Pr] J.H. Przytycki, History of the knot theory from Vandermonde to Jones, Aportaciones Matem´aticas Comunicaciones, 11 (1992), 173-195. [RS] A. Ranjan and R. Shukla, On polynomial Representation of Torus Knots, Journal of knot theory and its ramifications, Vol. 5 (2) (1996) 279-294. [Re] K. Reidemeister, Knot Theory, Springer-Verlag, Berlin, Heidelberg New-York, 1974. [Sh] A.R. Shastri, Polynomial Representation of knots, Tˆohoku Math. J. 44 (1992), 11-17. [St] H. Steinhaus, Mathematical Snapshots, 3rd edition, New York: Dover 1999. [Va] V. A. Vassiliev, Cohomology of Knot Spaces, Theory of singularities and its Applications, Advances Soviet Maths Vol 1, (1990)