Annales de la Faculté des Sciences de Toulouse

Viêt Anh Nguyên

El Hassan Youssfi

Optimal Lipschitz estimates for the $\overline{\partial }$ equation on a class of convex domains

Samira Benmouloud-Sbai

Mohamed Guedda

Critical exponent and minimization problem in $\mathbb {R}^N$

Aurore Cabet

Local existence of a solution of a semi-linear wave equation in a neighborhood of initial characteristic hypersurfaces

Anne Duval

Séries de $q$-factorielles, opérateurs aux $q$-différences et confluence

Albert Fathi

Regularity of $C^1$ solutions of the Hamilton-Jacobi equation

Grégoire Allaire

Dispersive limits in the homogenization of the wave equation

Robert O. Bauer

Discrete Löwner evolution

Nafaa Chbili

ANNALES DE LA FACULTÉ DES SCIENCES DE TOULOUSE
NAFAA CHBILI
A new criterion for knots with free periods
Annales de la faculté des sciences de Toulouse 6e série, tome 12,
no 4 (2003), p. 465-477
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p. 465
A new criterion for knots with free periods
NAFAA CHBILI (1)
ABSTRACT. - Let p  2 and q ~ 0 an integer. A knot K in the three-
sphere is said to be a (p, q)-lens knot if and only if it covers a link in
the lens space L(p, q). In this paper, we use the second coefficient of the
HOMFLY polynomial to provide a necessary condition for a knot to be
a (p, q)-lens knot. As an application, it is shown that this criterion rules
out the possibility of being (5,1)-lens for 80 among the 84 knots with less
than 9 crossings.
RÉSUMÉ. - Soient p  2 et q ~ 0 deux entiers. Un noeud K de la sphère
S3 est dit (p, q)-lenticulaire s’il est invariant par l’action lenticulaire pp,q .
Dans ce travail, nous étudions le comportement du polynôme de HOMFLY
des noeuds lenticulaires. Nous démontrons que la symétrie lenticulaire est
reflétée d’une façon très nette par le second coefficient du polynôme de
HOMFLY. Comme application, nous démontrons que 80 parmi les 84
noeuds ayant un nombre de croisements inférieur ou égal à 9, ne sont pas
(5, 1) -lenticulaires.
Annales de la Faculté des Sciences de Toulous Vol. XII, n° 4, 2003
1. Introduction
This paper is concerned with the question of whether the symmetry of
knots and links in the three-sphere is reflected on the quantum invariants.
The symmetry we consider in the present paper is the free periodicity. A
link L in S3 is said to be p-freely periodic (p  2 an integer ) if and only
if L is fixed by an orientation preserving action of the finite cyclic group
G = Z/pZ on the three-sphere without fixed points. It has been conjectured
(*) Reçu le 14 février 2003, accepté le 6 novembre 2003
(1) Département de Mathématiques, Faculté des Sciences de Monastir. Boulevard de
l’environnement, Monastir 5000, Tunisia.
E-mail: nafaa.chbili@esstt.rnu.tn
- 466 -
since many decades that such an action is topologically conjugate to an
orthogonal action. Consequently, we are going to limit our interest to links
which arise as covers of links in the lens space L(p, q). Such a link will be
called here a (p, q)-lens link.
The two variable HOMFLY (called also, skein and HOMFLYPT) polyno-
mial is an invariant of ambiant isotopy of oriented links, which generalizes
both the Alexander and the Jones polynomials, and can be defined by the
following :
where 0 is the trivial knot, L+, L- and Lo are three oriented links which
are identical except near one crossing where they look like in the following
figure:
It is well known that the HOMFLY polynomial [9] takes its values
in the ring Z[v±,z±]. However, if L is a knot then we have PL ( v, z) =
Li?:O P2i,L (v)z22 where P2i,L are elements of Z[v±2].
Knots with free periods were first studied by Hartly [5] who, motivated by a
question of R. Fox, used the Alexander polynomial to provide a criterion for
a knot to be freely periodic. The first criteria for periodicity of links using
the HOMFLY polynomial is due to Przytycki [10]. In [4], we used the first
term of the HOMFLY polynomial to find a necessary condition for a knot
to be p-freely periodic, for p prime. This criterion was applied successfully
to rule out the possibility of being freely periodic for certain knots. The
aim of this paper is to extend this criterion to the second coefficient of the
HOMFLY polynomial. Thus we shall prove that similar conditions hold for
the polynomial P2,x (v). The proof of our main result is based on the three
crucial facts:
. The combinatorial description of lens knots we provided in [1].
. The techniques developed by Traczyk [14] and Yokota [16] in the case
of periodic knots and adapted to freely periodic knots in [4].
- 467 -
2022 The formula for the second term of the HOMFLY polynomial intro-
duced recently by Kanenobu and Miyazawa [8].
An outline of the present paper is as follows. In section 2 we introduce our
main results. In section 2, basic properties of freely periodic knots will be
summarized. Some properties of the HOMFLY polynomial, needed in the
rest of the paper, are given in section 4. In section 5, we shall prove Theorem
2.1. In the last paragraph, our criterion is applied, in the case p = 5, to knots
with less than 9 crossings.
2. Results and applications
Let p be a prime and fp be the cyclic finite field of p elements. Through-
out the rest of our paper we denote by P2,K(v)p the second term of the
HOMFLY polynomial considered with coefficients reduced modulo p. If m
and n are two integers then T(n, m) denotes the torus link of type (n, m).
Recall here that the number of components of T (n, m) is equal to gcd(n, m).
In particular T(n, m) is a knot if and only if n and mare. coprime.
THEOREM 2.1. 2013 Let p &#x3E; 3 be a prime, q = ±1 and K a (p, q)-lens
knot. Then P2,K(V)p E rp,q, where rp,q is the Fp[v±2p]-module generated
by P2,T(a,aq:t:p) (v)p for all 1  03B1  p - 1.
This result is more significant for small values of p. Indeed, for such
values the generators of the module rp,q are easily computed using the
formula given by V. Jones [7] for the HOMFLY polynomial of torus knots.
This fact is illustrated by the following corollary:
COROLLARY 2.2. - Let q = ±1 and K be a (5, q)-lens knot. Then
P2,K(v)5 E F5[v±10]-module generated by vq8.
Proof of Corollary 2.2. - According to Theorem 2.1, the generators of
r5,1 are given by P2,T(03B1,03B1±t)(v)5 for 1  03B1  4. We use the formula given
in section 4 to compute the HOMFLY polynomial of torus knots. These
generators are given by the list below :
A similar computation can be easily made in the case q = -
- 468 -
Remark 2.3. - In the case p = 7, the module 03937,1 is generated by the
two elements: 2v6 + 3V8 and 6V8 + 4v10. Consequently the module F7,-1 is
generated by 2v-6 + 3v-8 and 6v-8 + 4v-1o.
Application. - Corollary 2.2 provides a criterion for a knot of the three-
. sphere to be fixed by the lens transformation ~5,±1. Hence, given a knot K,
if the polynomial P2,K(v)5 does not satisfy the condition given by Corollary
2.2 then K is not a (5, ±1)-lens knot. Let us illustrate this by considering
the knot K = 813. According to the table in [9] we have
As P2,813 (v)5 is not in thelf5 [v±10]-module generated by v8. Then K = 81o is
not a (5,1)-lens knot. It is worth mentioning that the criterion we introduced
in [4] using the first coefficient of the HOMFLY polynomial does not decide
in the case of the knot K = 813. Thus, Corollary 2.2 is not a consequence
of the results we introduced in [4]. More applications are given in the last
section of this paper.
3. Freely periodic links
Symmetry of knots and links is a vast subject that has fascinated
researchers since the early age of knot theory. Problems as chirality and
invertibility have motivated classical knot theory for a long time. Roughly,
a knot in S3 is said to be symmetric if and only if K is fixed by an action of
a finite cyclic group on S3. According to the set of fixed points of the action,
we can distinguish many kinds of symmetry. In this section we focus on the
case where the action has no fixed points. We define freely periodic knots
then we review some basic properties of this family of knots and links.
DEFINITION 3.1. - Let p  2 be an integer. A link L in 53 is said
to be p-freely periodic if and only if there exists an orientation preserving
diffeomorphism h : S3 ~ S3 such that:
1) hi has no fixed points for all 1  i  p - 1,
Example 3.2. - Let L be the torus knot T(2,5). It is well known that
L can be seen as the intersection between an appropriate three-sphere and
the complex surface defined by :
- 469 -
Let us consider the diffeomorphism :
Obviously h satisfies conditions 1 and 2 of Definition 3.1. Moreover, one may
easily check that h(L) = L. Thus L is a freely periodic knot with period 3.
Remark 3.3. - Let p  2 and q an integer such that gcd (p, q) = 1.
Consider pp,q the diffeomorphism given by:
It is easy to see that cpp,q is an orientation preserving diffeomorphism of
order p and that ~p,q has no fixed point. Moreover, we have a p-fold cyclic
covering (7rp, q, S3, L (p, q)).
DEFINITION 3.4. - Let p  2 and q an integer such that gcd(p, q) = 1.
A link L ouf 83 is said to be a (p, q)-lens link if and only if L is mapped onto
itself by ~p,q.
It is worth mentioning that lens links are the only examples we know of
freely periodic links. More precisely we have the following conjecture proved
for p = 2 and 3.
CONJECTURE 3.5 [13]. 2013 Let p be a prime and h : S3 ~ S3 an orien-
tation preserving diffeomorphim of order p such that for all
1  i  p - 1, hi has no fixed points. Then there exists an integer q such
that h is topologically conjugate to ~p,q.
Let n  1 be an integer. An n-tangle T is a submanifold of dimension one
in R2 I such that the boundary of T is made up of 2n points {A1, .... Anlx
{0, 1}. If T and T’ are two n-tangles we define the product T T’ by putting
T over T’ as follows:
As in the case of braids we define the closure of T and we denote by
T the link obtained from T by joining Ai x 1 to Ai x 0 by a simple arc
- 470 -
without adding any crossing. Throughout the rest of this paper Bn denotes
the n-string braid group. It is well known that this group has the following
presentation:
For n &#x3E; 2, the group Bn is not abelian. Its center is known to be gen-
erated by the element Ç2n = (03C3103C32...03C3n-1)n. The following theorem gives a
combinatorial description of lens links.
THEOREM 3.6. [1]. - A link K of S3 is a (p, q)-lens link if and only il
there exists an integer n ~ 0 and an n-tangle T such that:
Remark 3.7. - Let n and m be two integers. The torus link T(n, m)
is the closure of the braid (03C3103C32...03C3n-1)m. Using elementary techniques we
can prove that T(n, m) is a (p, q)-lens link if and only if p divides m - nq.
4. The HOMFLY polynomial
The discovery of the Jones polynomial [7] led to a significant progress in
knot theory. The Jones polynomial was followed by a family of invariants of
knots and three-manifolds called the quantum invariants. Among this fam-
ily of invariants the HOMFLY polynomial which is an invariant of ambiant
isotopy of oriented links. This invariant is a two-variable Laurent polyno-
mial which can be seen as a generalization of the Jones and the Alexander
polynomial. This section is to review some properties of this polynomial
needed in the sequel. At the beginning let us fix some notations.
- 471 -
Let L = 11 U l2 U ... U ln be an n-component link of the three-sphere.
Throughout the rest of this paper Ai,j denotes the linking number of the
two components li and h and A denotes the total linking number of the link
L. It is well known that the HOMFLY polynomial takes values in the ring
Z[v±1,z±1]. Moreover, we can write PL(v,z) = 03A3i0P1-n+2i,L(v)z1-n+2i
where Pl-n+2i,L C Z[v±2] if n is odd and P1-n+2i,L ~ Z[v±1] if n is even.
PROPOSITION 4.1 [9]. - Let L = 11 U 12 U ... U ln be an n-component
link then: 
Motivated by this proposition, Kanenobu and Miyazawa [8] introduced
a similar formula for the polynomial P3-n,L.
THEOREM 4.2 [8]. 2013 Let n  3 be an integer and L = li U l2 U ... ~ ln
an n-component link then:
where Li,j denotes the 2-component link li U lj. 
The HOMFLY polynomial of torus knots was computed by V. Jones.
To introduce the Jones formula, we find it more convenient to use the poly-
nomial XL (q, t). This is a version of the HOMFLY polynomial related to
PL (v, z) by the variable changes: z = q1/2-q-1/2 and v = (tq)1/2. Let k j 1
be an integer, we define: [k]! = (1 - g)(1 - q2)...(1 - qk) and [k] = 1-qk 1-q.
THEOREM 4.3 [7]. 2013 For the torus knot T(n, m) we have:
- 472 -
5. Proof of Theorem 2.1
Most of the techniques used in this section were first developed by
Traczyk [14] to study the HOMFLY polynomial of periodic knots (in some
sens this class of knots corresponds to the (p, 0)-lens knots). In this section
we aim to adapt these techniques to the case of freely periodic knots. This
will be done in two steps. In the first one, we prove that PD belongs to 0393’p,q,
where Fp is the Fp[v±2p]-module generated by the polynomials of torus
knots T(n, nq + p). The second step explains how to extract a finite set of
generators for 0393’p,q. Let us fix some notations. By T+, T- and To, we denote
three tangles which are identical except near one crossing where they look
like in figure 1. By D+ (respectively D-, Do ) we dénote a diagram of the
(p, q)-lens link D+ = Tp+03A9qn (respectively D- = T and Do = T)). It
is worth mentioning that if D+ is a knot then D- is also a knot. However,
Do is a link with 2 or p+1 components. In the case Do has p + 1 components
D,UD2U ... U Dp+1, then one component (say D1) is invariant by ’Pp,q, the
others are cyclically permuted by ’Pp,q’ We shall prove by induction on the
number of crossings of D, that P2,D E Fp g. Let D be a (p, q)-lens diagram.
Assume that for all (p, q)-lens diagram D’ with less crossings than D we
have P2,DI E hp,q. In [4], the following lemma was proved:
LEMMA 5.1. Let p be a prime. The following congruence holds
modulo p:
PROPOSITION 5.2. - Let p  5 be a prime.
. i) If Do has two components then:
ii) If Do has p + 1 components then
Proof. - According to Lemma 5.1 we have the following congruence
modulo p:
Obviously, if Do has two components then P3-(p+1),D0 is zero. Assume nom
that Do has p + 1 components Di,D2,... ,Dp+i. Let we denote by Di,j thE
- 473 -
two-component link Di U Dj and define G and H as follows:
It can be easily seen from Theorem 4.2 that
By the fact that components D2,D3,... ,Dp+1 are identical and cyclically
permuted by the action of Z/pZ we can write:
Using the fact that Ai,2 = Aij and that 1/1,2 = 1JI,j tor all 2  j  p + .
we get:
On the other hand :
One may check easily that 03A3v-203BBi,jP1,Di,j(v) ~ 0 modulo p. Thus
G(v) is zero modulo p. A similar computation shows that:
- 474 -
Therefore: H(v) == v203BB(v-1 - v)pP2,D1(v)(P0,D2(v))p modulo p. This
ends the proof of Proposition 5.2.
Proof. - It is easy to see that the result is true in the case Do has two
components. If Do has p + 1 components Dl, D2, ..., Dp+1, then Dl is a
(p, q)-lens diagram with less crossings than Do. Thus P2,Dl E I",q by the
induction assumption. Moreover, an easy computation shows that p divides
the total linking number À. According to Proposition 5.2 we have:
Obviously, V2À(V-1 - v)p(P0,D2(v))p belongs to 0393’p,q. Therefore, the second
term in the previous identity belongs to the module rp,q . Consequently,
P2,D+ E 0393’p,q if and only if P2,D- E rp,q. 
Notation. - Throughout the rest of this paper, we denote by D+ ~
D- the operation that consists of modifying p crossings to transform the
diagram D+ into the diagram D_ or vice-versa.
The following two lemmas, explain how to use the operation D+ ~ D-
to transform a lens diagram D into a torus knot. Details about these tech-
niques can be found in [4]. 
LEMMA 5.4. 2013 Every (p, q)-lens diagram may be transformed into a
(p, q)-lens closed braid by a series of operations D+ ~ D- without increas-
ing the number of crossings.
LEMMA 5.5. 2013 Let B be an n-braid. The (p, q)-lens braid BPç2q may
be transformed into the torus knot T(n, nq + p) by a series of operations
D+ ~ D_.
It remains now to explain how to extract a finite set of generators for
the module 0393’p,q. Our approach here will be based on some combinatorial
elementary properties of torus knots. Namely, we shall adapt the D+ ~ D-
operation to diagrams of torus knots. Therefore, an easy induction will end
the proof of Theorem 2.1. We refer the reader to [4] for more details.
- 475 -
6. More applications
This section is devoted to some applications of Theorem 2.1. As ex-
plained earlier in this paper, we can use the criterion Theorem 2.1 provides
to decide if a knot K is not a (p, q)-lens knot. Let us first recall that in [4],
we introduced a criterion for free periodicity using the first coefficient of the
HOMFLY polynomial P0,K (this criterion will be called the Po-criterion).
In the case p = 5, this criterion writes as follows:
The Po-criterion
If K 1s a (5,1)-lens knot, then PK = 03A3 a2iv2i with al0k+4 = 2al0k+2
and aiok+6 = 2al0k+8 for all k E Z.
Our aim here is to understand how powerful is the criterion introduced
in section 2 in detecting free periodicity. Namely, we shall compare the
condition obtained in the present paper to the Po-criterion. To do, let us
apply both of them to the 84 knots with less than 9 crossings. This is
explained in the following table where:
The first column gives the prime knot according to the notations used
in [12].
The second column (resp. the third) provides informations about the
Po-criterion (resp. the P2-criterion introduced by Corollary 2.2) as follows:
D means that the criterion decides that the knot is not a (5,1 )-lens knot.
ND means that the criterion does not decide that the knot is not a (5,1 )-
lens knot.
- 476 -
Bibliography
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[2] CHBILI (N.), Le polynôme de Homfly des n0153uds librement périodiques, C. R. Acad.
Sci. Paris t. 325, Série I, p. 411-414 (1997).
[3] CHBILI (N.), The Jones polynomials of freely periodic knots, Journal of Knot Theory
and its Ramifications 9(7), p. 885-891 (2000).
[4] CHBILI (N.), The skein polynomial of freely periodic knots, Topology Appl. 121/3,
p. 535-542 (2002).
[5] HARTELY (R.), Knots with free period, Can. J. Math. XXXIII(1), p. 91-102 (1981).
[6] FREYD (P.), YETTER (D.) ; HOSTE (J.) ; LICKORISH (W.B.R.), MILLETT (K.) ;
OCNEANU (A.), A new polynomial invariant of knots and links, Bulletin of the
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[7] JONES (V.F.R.), Hecke algebra representations of braid groups and link polynomials,
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polynomial of a link, Kobe J. Math. 16, p. 147-159 (1999).
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[10] PRZYTYCKI (J.H.), On Murasugi’s and Traczyk’s criteria for periodic links, Math.
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[11] PRZYTYCKI (J.H.), An elementary proof of the Traczyk-Yokota criteria for periodic
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[13] RUBINSTEIN (J.H.), An algorithm to recognize the 3-sphere, Proc. Int. Con. Math.
Zürich, p. 601-611 (1994).
[14] TRACZYK (P.), A criterion for knots of period 3, Topology appl 36, p. 275-281 (1990).
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