Math. Proc. Camb. Phil. Soc. (2000), 129, 477

Printed in the United Kingdom c© 2000 Cambridge Philosophical Society

477

On the character variety of periodic knots and links

By HUGH M. HILDEN

Department of Mathematics, University of Hawaii, Honolulu, HI 96822, U.S.A.

MARIA TERESA LOZANO†Departamento de Matematicas, Universidad de Zaragoza, 50009 Zaragoza, Spain

and JOSE MARIA MONTESINOS-AMILIBIA†Departamento de Geometrıa y Topologıa, Facultad de Matematicas,

Universidad Complutense, 28040 Madrid, Spain

(Received 22 March 1999; revised 16 September 1999)

Abstract

For a hyperbolic knot, the excellent component of the character curve is the one

containing the complete hyperbolic structure on the complement of the knot. In this

paper we explain a method to compute the excellent component of the character var-

iety of periodic knots. We apply the method to those knots obtained as the preimage

of one component of a 2-bridge link by a cyclic covering of S3 branched on the othercomponent. We call these knots periodic knots with rational quotient. Among this

class of knots are the ‘Turk’s head knots’. Finally we give some invariants deduced

from the excellent component of the character curve, such as the h-polynomial andthe limit of hyperbolicity for all the periodic knots with rational quotient, up to 10

crossings, which are not 2-bridge or toroidal.

Introduction

The set of representations of a finitely generated group G into SL (2,C) (orPSL (2,C)), up to conjugation, has the structure of a closed algebraic variety. Thesubset of the abelian representations is a closed algebraic subvariety ([CS]; cf. [GM]).

If G is a knot or link group π1(S3\K), we introduce the following definitions and

notations. The union of components of the algebraic variety of representations other

than the subvariety of abelian representations will be called the character variety of

K into SL (2,C) (or PSL (2,C)) and denoted respectively, by C(K) and C(K). Thereis a natural map from C(K) to C(K).If K is a 2-bridge knot, then all components of C(K) have dimension 1. However,

there are many hyperbolic knots for which C(K) contains components of dimensiongreater than 1 ([CL]). Nevertheless it is known that if L is a hyperbolic link of ncomponents and ρ0 ∈ C(L) is the holonomy of the complete hyperbolic structure of

† This research was supported by grant no. PB95-0413.

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478 H. M. Hilden, M. T. Lozano and J. M. Montesinos-Amilibia

finite volume of S3\L, the algebraic component, CE(L), of C(L) containing ρ0 hascomplex dimension n. Therefore, for a hyperbolic knot K this excellent component

CE(K) is an algebraic curve, called the excellent curve of K.The excellent curve CE(K) contains also the holonomies of the hyperbolic cone

manifold structures (S3,K, α) of angle α around K ([HLM3]). A fair amount of

information, like volume, Chern–Simons invariant and arithmetic properties can be

drawn off from CE(K) (or CE(K)) ([HLM3], [HLM6], [HLM4], [HLM5]). Thismakes this excellent curve especially important (see also [CS], [O]), and we are

seeking for practical computational methods.

If m is a meridian of K, there is a natural map

p: CE(K) −→ C

defined by p(ρ) = tr (ρ(m)) = tr (ρ(m−1)) where ρ is an element of CE(K) and trdenotes the trace function. By desingularization of the projective curve associated

to CE(K) we obtain a Riemann surface Σ(K) together with a meromorphic map

p: Σ(K) −→ CP 1.

The pair (Σ(K), p) is a knot invariant. We are interested in its calculation.There exist general methods to compute the character variety of a knot or link

K (see [GM], [BH]), but the number of polynomials obtained can be cumbersome.

Therefore we seek better methods of computation.

In particular we define that the curve Σ(K) iswell-computedwhen it is given by onlyone polynomial. Thus the pair (Σ(K), p) will be conveniently given by a polynomialrE[K](y, z) in two variables with integer coefficients defining the curve CE(K). Oneof the variables should be y = tr (ρ(m)). Thus the projection to the y-variable definesp. Among the singular points of p there exists one, yh = 2 cos (αh/2) defining anangle 0 < αh < 2/π, to which we conjecture that the hyperbolic cone-manifoldstructures aroundK degenerate. We therefore call this knot invariant αh the limit of

hyperbolicity. On the other hand, yh is an algebraic number and we want to computeits minimal polynomial hK(y). This polynomial (an important knot invariant also)will be called (superseding similar names used in [HLM3]) the h-polynomial of K.So far, only the character variety of knots or links with a 2-generator group has

been computed. The character curve of a 2-bridge knot has been well-computed ([B],

[H] and [K] for representations in SU (2); see also [HLM3]). The character surface of

a 2-bridge link has been computed in [HLM3] ([HLM2]). This information was used

in [HLM6] and [HLM5] to obtain volume and Chern–Simons invariant of hyperbolic

manifolds and orbifolds and in [S] to obtain Dirichlet polyhedra for cone-manifolds.

Also, the character curve of tunnel number 1 knots has been well-computed in the

forthcoming paper [HLM7].

In this paper we shall compute rE[K](y, z) and hK(y) for some bridge number > 2knots. We shall obtain several knots which, in contrast with 2-bridge knots, have the

limit of hyperbolicity αh > π.Here is a brief sketch of our method. We shall start with a 2-bridge link L = KxU ,

where the trivial knot U is the branching set of a g-fold branch covering pg:S3 → S3.

If g and the linking number L(K,U ) are relatively prime then p−g (K)÷ Kg will be

a g-periodic knot with rational quotient. Since we know the character surface of L([HLM3], [HLM2]) we shall be able to compute the subset of C(Kg), of g-periodic

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On the character variety of periodic knots and links 479

representations of S3\Kg. This subset, Cs(Kg), is an algebraic variety containing

CE(Kg) because, by a result of Thurston, the cyclic action of order g fixing Kg will

be an isometry of the complete hyperbolic structure of finite volume of S3\Kg (if it

exists).

In this paper we develop a method to compute Cs(Kg) and hK(y). We apply it to,

among others examples, the seven hyperbolic g-periodic knots with rational quotientin Rolfsen’s table and with bridge number > 2.

1. The symmetric curve Cs(K) of a g-periodic knot or link

Let (S3, U, g) be the orbifold with singular set the unknot U and isotropy cyclic

of order g. Let pg:S3 → (S3, U, g) be its universal covering space and let Cg be the

group of covering transformations. Let

(pg)♯:π1(S3\Kg) −→ π01(S

3\K ,U, g)

be the homomorphism, induced by pg, between the group of the g-periodic knot or

link Kg = p−1g (K) (see, e.g. [BZ]) and the orbifold group of (S

3\K ,U, g), where K is

a knot in S3\U . The number of components of Kg is equal to the g.c.d. of the linking

number L(K,U ) and g.If M is a meridian of U , the orbifold group π01(S

3\K, U, g) is equal toπ1(S

3\K x U )/〈Mg〉(cf. [MM], for instance).

We are interested in the subset of C(Kg) of representations in PSL (2,C) which

factor through (pg)♯. This subset will be denoted by Cs(Kg) and will be called the

symmetric curve of Kg (with respect to pg). (Note that a link might be periodic indifferent ways.)

Remark 1·1. We want to use the homomorphismπ1(S

3\Kg x U ) −→ π1(S3\K x U )

induced by the covering map, to obtain representations π1(S3\Kg)→ PSL (2,C) (or

SL (2,C)) and, in particular, to obtain the holonomy ρ0 of the complete hyperbolicstructure of finite volume of S3\Kg (if Kg is a hyperbolic knot).

We do this by identifying π1(S3\Kg) with π1(S

3\Kg x U )/〈M〉, where M is a

meridian of U . We, then, are led to consider the homomorphism

(pg)♯:π1(S3\Kg) =

π1(S3\Kg x U )

〈M〉−→ π1(S

3\K x U )

〈Mg〉 .

But the holonomy ρ0 of Kg cannot, in general, be obtained as the composition of

(pg)♯ with the lifted holonomy ω of the orbifold (S3,K∞ x Ug), because if g is even,

then ω:π1(S3\K x U ) → SL (2,C) sends M to an element of order 2g, hence ω(Mg)

is an element of order 2, so that ω does not induce an homomorphism

π1(S3\K x U )

〈Mg〉 −→ SL (2,C).

However if we consider the holonomy ω of the orbifold (S3,K∞ xUg), which is an

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480 H. M. Hilden, M. T. Lozano and J. M. Montesinos-Amilibia

homomorphism

ω:π1(S3\K x U ) −→ PSL (2,C)

we have that ω(Mg) is the identity. Thus ω does induce an homomorphism

π1(S3\K x U )

〈Mg〉 −→ PSL (2,C)

whose composition with (pg)♯ is the holonomy ρ0 of Kg. (Note that this ρ0 lifts toSL (2,C)! [HLM6].)

Therefore, using this method, we are obliged to use representations in PSL (2,C)to be sure of not missing the excellent component.

Proposition 1·2. If Kg is a hyperbolic knot, the symmetric curve Cs(Kg) contains

the excellent component CE(Kg). Therefore the covering transformation group Cg acts on

every hyperbolic cone-manifold structure (S3, Kg, α) by isometries.

Proof. Since Kg is a hyperbolic knot, there is a neighbourhood N of α = 0 such

that (S3, Kg, α) is hyperbolic for α ∈ N . By [T] Cg acts on (S3, Kg, α) by isometries.

Therefore the holonomy of (S3, Kg, α) belongs to Cs(Kg) w CE(Kg). Then, since

CE(Kg) is irreducible, CE(Kg) ⊂ Cs(Kg).

The elements of Cs(Kg) are equivalence classes of elements of the form ω ◦ (pg)♯,

where ω:π1(S3\KxU )/〈Mg〉 → PSL (2,C) is a representation. ThusC

s(Kg) coincides

with the representation variety of the group π1(S3\KxU )/〈Mg〉 into PSL (2,C). The

image ω(M ) of the meridian M of U is conjugated to an element of the form

±[eiα/2 0

0 e−iα/2

], α =

2πk

g, 0 < k < g.

(Note that if k is equal to zero and K is trivial, then ω is an abelian represen-

tation.) Let b = tr (ω(M 2)) = 2 cos (2πk/g), 0 6 k < g. These are the roots ofψg(b) =

∏h|g φh(b) where φh(b) is the minimal polynomial of 2 cos (2π/h). The rep-

resentation variety of the link K x U is the algebraic variety V 〈f1, . . . , fn〉 of someideal 〈f1, . . . , fn〉 of a polynomial ring in a set of variables including b.

Theorem 1·3. The symmetric curve Cs(Kg) is given by the ideal 〈f1, . . . , fn, ψg(b)〉.

Therefore

(1) Cs(Kg) is the union of the varieties defined by the ideals 〈f1, . . . , fn, φh(b)〉, h|g.

(2) Cs(Kg) contains as an algebraic subvariety the symmetric curves C

s(Kh) where

h|g.Note that C

s(K1) = C(K).

Proof. The inclusion of ψg(b) in the set of polynomials generating the ideal guaran-tees that ω(Mg) is the identity in PSL (2,C).

Remark 1·4 (Heusener and Klassen, unpublished). If K is not a hyperbolic knot

(a convenient double, for instance) C(K) can have complex dimension > 1. ThereforeC(Kg), which contains C

s(K1) = C(K), might have complex dimension > 1, even if

Kg is a hyperbolic knot. (Cf. [CL].)

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On the character variety of periodic knots and links 481

2. A normal form for 2-bridge links

Proposition 2·1. Let p, q be a pair of relatively prime integers p > q > 0, p even.Then p/q has a unique continued fraction expansion of the form

[2, n1,−2, n2, 2, n3, . . . , nr, (−1)r2],where n1, n2,. . . ,nr are nonzero integers.

Corollary 2·2. The 2-bridge link p/q has the normal forms depicted in Figs. 2·1and 2·2.Proof. See [BZ] or [Ro] for the form depicted in Fig. 2·1. The forms in Fig. 2·2

are obtained by isotopy.

2

-2

±2

n3

1n

2n

2

n 0 , n 0n

Fig. 2·1. The link {n1, n2, . . . , nk}.

...

n

n

nn

1

2

34

nk

...

...

n

n

n

1

2

3

...

n 4

nkK

US S

m m

U

K

M M

Fig. 2·2. The link {n1, n2, . . . , nk}.

Notation 2·3. The link p/q will be denoted by {n1, n2, . . . , nk}. The g-periodic knotor link Kg = p

−1g (K), for the cyclic branched covering pg:S

3 → S3 branched over U

will be denoted by g{n1, n2, . . . , nk}. The number of components of Kg is equal to

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482 H. M. Hilden, M. T. Lozano and J. M. Montesinos-Amilibia

the g.c.d. of the linking number L(K,U ) and g. The linking number L(K,U ) canbe read off directly from {n1, n2, . . . , nk}. The knot Kg is obtained from Fig. 2·2 byrepeating the segment S g times before closing the ‘braid’.

Remark 2·4. The link g{±1,±1, . . . ,±1} is truly braided around U = p−1g (U ). Thusg{1,−1, . . . , (−1)k−1} is the traditionally called Turk’s head knot or link of k + 1strands and g bights, T (k + 1, g) (cf. [CF] and [A]). Also g{1, 1, . . . , 1} is the torusknot (k + 1, g).

Remark 2·5. The meridian m of Fig. 2·2 is then repeated cyclically g times in Kg.

These g meridians m1, m2, . . . ,mg generate the group of Kg, π1(S3\Kg). In fact, the

bridge number of Kg 6 g (see Fig. 2·3).

m31m 2m

Fig. 2·3. The knot 941 = 3{1,−2}.

Remark 2·6. If p/q = K x U = {n1, n2, . . . , nk} then the 2-periodic knot or linkK2 = 2{n1, n2, . . . , nk} is the 2-bridge knot or link (p/2)/q. Thus the methods in thispaper allow the study of the curve C

s((p/2)/q) from C(p/q).

Example 2·7. We list some examples. The first column contains the seven non-toroidal g-periodic knots with rational quotient with bridge number > 2 and 6 10

crossings. These are the knots ([KS, table 3·1]) with period > 2 which are hyperbolic(namely non-toroidal). The second column contains its rational quotient.

818 = 4{1,−1}, 10/3 = {1,−1}935 = 3{3}, 12/5 = {3}940 = 3{1,−1, 1}, 24/7 = {1,−1, 1}941 = 3{1,−2}, 18/5 = {1,−2}947 = 3{1, 1,−1}, 16/3 = {1, 1,−1}949 = 3{1, 2}, 14/3 = {1, 2}

10123 = 5{1,−1}, 10/3 = {1,−1}.

Example 2·8. The two toroidal knots with bridge number > 2 and 6 10 crossings

which are g-periodic knots with rational quotient are the following:

819 = 3{−1,−1,−1} = 4{1, 1}10124 = 3{−1,−1,−1,−1} = 5{1, 1}.

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On the character variety of periodic knots and links 483

3. Computing the symmetric curve Cs(Kg) of Kg = g{n1, n2, . . . , nk}

Let p/q = K x U be a 2-bridge link. In [HLM3] we have computed the repre-

sentation surface C(p/q). Between C(p/q) and C(p/q) there exists a C2 × C2 regularbranched covering

f : C(p/q) −→ C(p/q).

In fact, C(p/q) is a surface given by a polynomial r[p/q](y1, y2, v) in the variablesy1 = tr (M ), y2 = tr (m) and v = tr (mM ), where M is a meridian of U and m is a

meridian ofK (see Fig. 2·2). This polynomial has been obtained in [HLM3] (see also[HLM2]). The action of C2 × C2 in C(p/q) is generated by the maps (y1, y2, v) →(−y1, y2,−v) and (y1, y2, v)→ (y1,−y2,−v). In fact, π1(S3\p/q) has the presentation|m,M :mw = wm|, wherew is a word in {m±1,M±1}. This shows that the polynomialr[p/q](y1, y2, v) is composed of monomials y

i1y

j2v

k where i ≡ j ≡ k mod2. Therefore, to

obtain the equation for C(p/q) from the equation r[p/q](y1, y2, v) of C(p/q) we simplymake the following change of variables

w = y1v,

b = y21 − 2 (= tr (M 2)).

Then we obtain a new polynomial in the variables w, b, y2 which we denote byr[p/q](y2, b, w). Observe that using this change of variables we are assuming y1 � 0.

Thus r[p/q](y2, b, w) contains all the elements of C(p/q) except the representationsgiven by y1 = 0. Then

C(p/q) = V 〈r[p/q](y2, b, w)〉 x V 〈r[p/q](y1, y2, v), y1〉.Now,

〈r[p/q](y1, y2, v), y1〉 = 〈r[p/q](y1, y2, v), φ2(b)〉 = Cs(K2)

because φ2(b) = b + 2 = y21.

Thus, to obtain an ideal defining Cs(Kh), h|g, in the natural variables y=tr (ρ(m1)),

z = tr (ρ(m1m2)), where m1, m2 are the meridians of Kg defined in Remark 2·5, wedo the following.

Notice that (pg)♯(m1) = m and (pg)♯(m2) =MmM−1. Therefore

y = tr (ρ(m1)) = tr ((ω ◦ (pg)♯)(m1)) = tr (ω(m)) = y2

and an easy computation, using the formula tr (XY ) = tr (X)tr (Y ) − tr (XY −1) for

traces of matrices X,Y ∈ SL (2,C), implies thatz = tr (ρ1(m1m2)) = tr ((ω ◦ (pg)♯)(m1m2)) = tr (ω(mMmM−1))

= tr (ω(mM )) tr (ω(mM−1))− tr (ω(M 2)) = v(y1y2 − v)− (y21 − 2).Call p(y1, y2, v, z) the polynomial −vy1y2 + v2 + y21 − 2 + z. Changing to variables w,b this polynomial becomes

p(y2, b, w, z) = (b + 2)(wy2 − b− z)− w2.

Then

Cs(K2) = V 〈r[p/q](y1, y2, v), p(y1, y2, v, z), y1〉 = V 〈r[p/q](0, y, v), p(0, y, v, z)〉

= V 〈r[p/q](0, y, v), v2 − 2 + z〉.

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484 H. M. Hilden, M. T. Lozano and J. M. Montesinos-Amilibia

We have proved the following theorem.

Theorem 3·1. Let Kg be a g-periodic knot or link with rational quotient the 2-bridge

link p/q. The symmetric curve Cs(Kg) is the following curve

g odd Cs(Kg) =

⋃

h|g, h�1

V 〈r[p/q](y, b, w), p(y, b, w, z), φh(b)〉,

g even Cs(Kg) =

⋃

h|g, h�1

V 〈r[p/q](y, b, w), p(y, b, w, z), φh(b)〉⋃

Cs(K2),

where Cs(K2) = V 〈r[p/q](0, y, v), v2 − 2 + z〉.

We then eliminate variables to obtain a polynomial in the variables {y, z}. Thiswill be done in the following sections for some Turk’s head knots and links g{1,−1}and the hyperbolic g-periodic knots up to 10 crossings with bridge number biggerthan 2. Of course, the method is general.

4. Computing Cs(g{1,−1}), g = 2, 3, 4, 5

The links of the family g{1,−1} are Turk’s head links. Namely, 10/3 = {1,−1};41 = 2{1,−1} (the figure-eight knot); 632 = 3{1,−1} (the Borromean rings); 818 =4{1,−1}; 10123 = 5{1,−1}.From theorem 2·2 of [HLM2] (see also proposition 5·3 of [HLM3]) we have

r[10/3](y1, y2, v) = v4 − v3y1y2 + v

2y22 + v2y21 − 3v2 − vy1y2 + 1.

Therefore

r[10/3](y, b, w) = 4 + 4b + b2 − 2w2 + bw2 + b2w2 + w4 − 4wy−4bwy − b2wy − 2w3y − bw3y + 2w2y2 + bw2y2.

g = 2: the figure-eight knot 5/3 = 41 = 2{1,−1}

Cs(2{1,−1}) = C

s(41) = V 〈r[10/3](0, y, v), v2 − 2 + z〉= V 〈v4 + v2y2 − 3v2 + 1, v2 − 2 + z〉 = V 〈r[5/3](y, z)〉,

where

r[5/3](y, z) = −1 + 2 y2 − z − y2 z + z2

(cf. [W]).

g = 4: the knot 818 = 4{1,−1}According to Theorem 3·1, C

s(818) is the union of Cs(2{1,−1}) and

V 〈r[10/3](y, b, w), p(y, b, w, z), b〉

since φ4(b) = b.To obtain a polynomial in the variables {y, z} for this component of C

s(818) we

eliminate w between the polynomials

r[10/3](y, 0, w) = 4− 2w2 + w4 − 4wy − 2w3y + 2w2y2

p(y, 0, w, z) = 2(wy − z)− w2

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On the character variety of periodic knots and links 485

by taking the resultant in w. We obtain:

rE[818](y, z) = −1 + 4y2 − 2y4 − 2z + 2y4z − 3z2 − y4z2 − 2z3 + 2y2z3 − z4.

This polynomial defines the excellent component CE(818). Thus the symmetric

curve of the knot 818 is

Cs(818) = V 〈r[5/3](y, z)rE[818](y, z)〉.

To obtain the limit of hyperbolicity αh and the h-polynomial (minimal polynomialof yh = 2 cos (αh/2)) we remark that this polynomial is a factor of the result ofeliminating z between the two polynomials rE[818](y, z) and (d/dz)rE[818](y, z). Weobtain the polynomial

d(y) = y4(y2 − 6)2(y2 − 3)2(y4 − 6y2 + 1).

-2 -1 0 1 2

-0.5

0

0.5

1

1.5

-2 -1 0 1 2

-2

-1

0

1

2

Fig. 4·1. The excellent component CE(818).

Fig. 4·1 shows part of the curve rE[818](y, z) = 0. Because we are interested in

representations corresponding to cone-manifold structures of angle α, we considerreal values of y = 2 cos (α/2) in the interval [−2, 2]. The first graph is the real partof the complex variable z and the second graph is the imaginary part of z. Observethat the complete hyperbolic structure of finite volume of S3\818 corresponds to thepoint P (y0, z0):

y0 = 2, z0 = 1·5± 2·36187iwhich is the only complex root of the polynomial rE[818](2, z) = 0. The point P =

(y0, z0) belongs to the branch of hyperbolicity (see [HLM3]). The limit of hyperbolicityis given by the point Q = (yh, zh)

yh = 1−√2 = −0·414214 = zh

where yh is a root of the h-polynomial h[818](y) = y2 − 2y − 1 which is a factor ofd(y); since yh = 2 cos (αh/2), the limit of hyperbolicity αh is approximately 3·55883radians which is bigger than π. This is the first knot one finds in the table of knots[Ro] with this property. Here is the meaning of αh. It follows from the version of

the Thurston’s Orbifold Theorem proved by Boileau and Porti ([BP]), that the orbi-

fold (S3, 818, π) is hyperbolic. Then by a result of Kojima [Ko] the cone manifolds

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486 H. M. Hilden, M. T. Lozano and J. M. Montesinos-Amilibia

(S3, 818, α), 0 6 α 6 π, are hyperbolic also. As a matter of fact (S3, 818, α) is alsohyperbolic in a small neighbourhood of π, by [K]. Our computation of αh indicates

that the following proposition is probably true:

(S3, 818, α) has hyperbolic cone manifold structure for 0 6 α < αh and a Euclidean

cone manifold structure for α = αh. (If so, it is spherical for some α’s bigger than αh

[P].)

In this example, the fact that (S3, 818, π) is a hyperbolic orbifold can be proveddirectly as follows. The orbifold (S3, 818, π) has a 2-fold manifold covering,M , whichat the same time is the 4-fold cyclic covering of (S3, 41, (2π/4)), since 818 = 4{1,−1},41 = 2{1,−1} and the components of the link 10/3 = {1,−1} are interchangeable.(See Fig. 4·2.) Because (S3, 41, (2π/4)) is a hyperbolic orbifold (cf. [HLM1]), it followsthat M , and therefore (S3, 818, π), is a hyperbolic orbifold. (This is a fairly generalresult that applies to any interchangeable link, like 2-bridge links, for instance.)

Mp

q 2

4p2

q4

1818

4

10/3

Fig. 4·2. Relations between branching sets of the above coverings.

Thus (S3, 818, π) and (S3, 41, (2π/4)) are commensurable and therefore (S

3, 818, π) isarithmetic, because (S3, 41, (2π/4)) is arithmetic ([HLM2]). The computation of vol-umes and Chern–Simons invariants can be made using the method given in [HLM4].

From Cs(818) = V 〈r[5/3](y, z)rE[818](y, z)〉 we obtain C

s(818) taking as new variable

x = y2 − 2(= tr (ρ(m21)))

instead of y. Thus

Cs(818) = V 〈(3 + 2x− 3z − xz + z2)(1 + 4x + 2x2 − 6z − 8xz

−2x2z + 7z2 + 4xz2 + x2z2 − 2z3 − 2xz3 + z4)〉.

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On the character variety of periodic knots and links 487

Recall that there exists a relation between the Alexander polynomial of any knot

K and the representation curve C(K) ([B], see also [HLM3]): the roots of the sym-metrized Alexander polynomial ∆S[K]((t+ t

−1)) correspond to reducible metabelian

representations ρ:π1(S3\K)→ SL (2,C) with

ρ(m1) =

(λ 0

0 λ−1

)

and

ρ(m2) =

(λ ∗0 λ−1

)

[DR].For these representations z = tr (ρ(m1m2)) = tr (ρ(m21)) = (λ

2+λ−2) = t+t−1 = x,where t = λ2. Therefore ρ corresponds to a point in the representation curve C(K)such that z = x. Now, put z = x = t + t−1 in the equation of C(K), multiply by theconvenient power of t, in order to obtain a polynomial without negative powers, anddenote it by rK(t). We conclude that ∆S[K]((t + t

−1)) should be a factor of rK(t).In the case of the knot 818 we have

rE[818](t) = −(1− t + t2)2

r[5/3](t) = (1− 3t + t2)∆S[818]((t + t

−1)) = −(1− 3t + t2)(1− t + t2)2

which explains geometrically why the Alexander polynomial ∆S[818]((t + t−1)) con-

tains the factor ∆S[41]((t + t−1)) = (1− 3t + t2).

g = 3: the Borromean rings 632 = 3{1,−1}Here C

s(632) is the curve

V 〈r[10/3](y, b, w), p(y, b, w, z), b + 1〉since φ3(b) = b + 1. Eliminating b and w as before, we obtain:

Cs(632) = V 〈r[632](y, z) = −4y2 + 3y4 − 3y4z + 3y2z2 + y4z2 − 2y2z3 + z4〉.

The h-polynomial is h[632](y) = y.

-2 -1 0 1 2

-0.5

0

0.5

1

1.5

2

-2 -1 0 1 2

-2

-1

0

1

2

Fig. 4·3. The excellent component CE(632).

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488 H. M. Hilden, M. T. Lozano and J. M. Montesinos-Amilibia

Here (see Fig. 4·3) P = (y0, z0) = (2, 2) and Q = (yh, zh) = (0, 0). The limit ofhyperbolicity is αh = π. For this example we know that (S3, 632, α) is a hyperboliccone manifold for 0 6 α < π; for α = π is a Euclidean orbifold; for π < α < 2π is aspherical cone manifold; and for α = 2π is spherical, but the link 632 degenerates inthree mutually intersecting geodesics.

g = 5: the knot 10123 = 5{1,−1}Here

Cs(10123) = V 〈r[10/3](y, b, w), p(y, b, w, z), φ5(b)〉.

where φ5(b) = b2 + b− 1. Eliminating b and w as before, we obtain:

Cs(10123) = V 〈r[10123](y, z) = 1− 68y2 + 242y4 − 298y6 + 165y8 − 42y10 + 4y12

+4z − 195y2z + 406y4z − 277y6z + 72y8z − 6y10z + 4z2 − 260y2z2

+336y4z2 − 159y6z2 + 44y8z2 − 6y10z2 − 2z3 − 207y2z3 + 243y4z3

−118y6z3 + 19y8z3 − 6z4 − 110y2z4 + 109y4z4 − 23y6z4 + y8z4

−4z5 − 34y2z5 + 15y4z5 − 4y6z5 + z6 − 7y2z6 + 6y4z6 + 2z7

−4y2z7 + z8〉

The h-polynomial is given at the end of the paper. The limit of hyperbolicity αh is

approximately 3·76991 > π radians.As in the case of the knot 818, (S

3, 10123, π) has amanifold 2-fold covering which is atthe same time a 5-fold cyclic covering of (S3, 41, 2π/5) and this is a hyperbolic orbifold([T]). Moreover (S3, 10123, π) being commensurable with (S

3, 41, 2π/5) is arithmetic[HLM1].

5. Results for other periodics knots

We limit ourselves to the tabulation of some relevant results of our calculations

for hyperbolic knots with bridge number > 2 up to ten crossings which are periodicknots with rational quotient. These are the knots listed in Example 2·7.Table 1 offers the following information:

p/q is the rational notation of the 2-bridge link quotient;(S3,K, π) denotes the geometry of the orbifold structure in S3 with the knot as

singular set of order 2. It is the geometry of the 2-fold covering of S3 branched overthe knot;

αh is the angle which we conjecture is the limit of the hyperbolic orbifold structures

in S3 with the knot as singular set (limit of hyperbolicity). It is given in radians.

Table 1

Knot p/q (S3,K, π) αh

818 = 4{1,−1} 10/3 Hyperbolic 3·55883935 = 3{3} 12/5 Nil π940 = 3{1,−1, 1} 24/7 Hyperbolic 3·71294941 = 3{1,−2} 18/5 Hyperbolic 3·5259947 = 3{1, 1,−1} 16/3 Hyperbolic 3·5193949 = 3{1, 2} 14/3 Hyperbolic 3·4102810123 = 5{1,−1} 10/3 Hyperbolic 3·76991

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On the character variety of periodic knots and links 489

Table 2 shows two points P andQ in the excellent component of the representationcurve C

s(K). The point P (y0, z0) corresponds to the complete hyperbolic structureof finite volume in the complement of the knot, and the point Q(yh, zh) correspondsto the hypothetic limit of hyperbolic cone-manifold structures. The branch of hyper-

bolicity is the one joining P and Q.

Table 2

Knot P (y0, z0) Q(yh, zh)

818 (2, 1·5± 2·36187i) (−0·414214,−0·414214)935 (2, 4·565198± 1·0434274i) (0, 1)940 (2, 1·67265± 2·48233i) (−0·563512, 0·213438)941 (2, 1·19098± 2·4899i) (−0·381949,−0·126921)947 (2,−1·44666± 1·52141i) (−0·375468,−0·916712)949 (2,−0·447279± 1·86942i) (−0·267881,−0·681665)10123 (2, 1·19098± 2·4899i) (−0·618034,−0·618034)

Here are the h-polynomials.

h[818](y) = −1− 2y + y2

h[935](y) = y

h[940](y) = 512− 2016y2 + 1365y4 − 298y6 + 21y8

h[941](y) = 20 502 784− 272 105 600y2 + 1 224 158 176y4 − 2 608 885 732y6

+3 033 448 957y8 − 2 164 119 142y10 + 1 016 298 597y12 − 325 952 010y14

+72 264 540y16 − 10 949 670y18 + 1 087 101y20 − 63 990y22 + 1701y24

h[947](y) = 1 660 608− 16 789 392y2 + 40 767 080y4 − 39 932 507y6 + 21 095 926y8

−6 807 665y10 + 1 400 862y12 − 180 723y14 + 13 356y16 − 432y18

h[949](y) = 33 856− 543 808y2 + 1 061 724y4 − 835 043y6 + 348 214y8 − 85 005y10

+12 390y12 − 1011y14 + 36y16

h[10123](y) = −1− y + y2.

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