数理解析研究所講究録 1451 巻 2005 年 120-130120Special classes of algebraic integers in low-dimensional topology
Eriko Hironaka

April 15, 2005
Abstract
realThis note describes some open problems concerning distributions of special classes of
specialThese
Perronnumbers.
P-V
andSalem,
algebraic integers such as algebraic units, andtopology, We
algebraic integers appear naturally as geometric invariants inrelate properties of Salem, P-V and Perron to minimization problems in various geometricsettings.
$\mathrm{l}\mathrm{o}\mathrm{w}rightarrow \mathrm{d}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l}$Introduction

1is an algebraic integer if it is a root of a monic integer polynomial. Two
A complex number, it a and satisfy the same
algebraic integers a and are algebraically conjugat\^e vxritterr.
unit if
algebraicanirreducible monic integer polynomial. An algebraic integer is:a such that
Let be a real algebraic integer with a $>1$ . Consider all$\alpha\in \mathbb{C}$$\alpha\sim\beta$$\beta$$\beta$$\alpha\sim\alpha^{-1}$$\alpha$$\beta\neq\alpha$$\beta\sim$$\alpha$(i) it$|\beta|<|\alpha|$, then a is a Perron number’(ii) if $|\beta|<1$ , then(iii) if$|\beta|\leq 1$$\alpha$is aP-$V$

number, andwith at least one $|\beta|=1$ , then$\alpha$is a Salem number.

In this short note, we review definitions and known results concerning distributions of P-V,
Salem and Perron numbers (Section 2), and relate them to geometric invariants in low-dimensional
topology, including lengths of geodesies, growth rates of automatic groups, and homological andgeometric dilatations of surface homeomorphisms (Section 3).

2

Distributions of algebraic integers and Lehmer’s problem

be the set of complex roots of
Let be the set of monic integer polynomials. Given $f\in P$ , letoutside the unit circle .
pointsofsubsetthebecounted with multiplicity, and let
For $f\in P$ define$s_{f}$$P$$C$$S^{+}(f)\subset s_{f}$$f$;$N(f)$$=$$|\mathrm{S}^{+}(f)|$$\lambda(f)$$=$$\max\{|\alpha| :$M(f)$$=$$\prod$$|\alpha|$\alpha\in S(f)\}$

; and

.

$\alpha\in \mathrm{S}^{+}(J)$Here an empty product is defined to equal 1. The number $M(f)$ , also an algebraic integer, is called
the Mahler measure of .
The minimal polynomial for a root of unity is called a cyclotomic polynomial. The following are
equivalent tor $f\in P$ :
$f$

121

(i) $f$ is a product of cyclotomic polynomials;(ii) $N(f)=0$ ;(iii) A(f) $=1$ ; and(iv) $M(f)=1$ .

Thus,
and $M(f)$ can be considered as measures of how far $f$ is from being a productof cyclotomic polynomials. Let
he the subset of products of cyclotomic polynomials.
While $N(f)$ takes discrete values andcan get arbitrarily close to one from above it is notknown whether there is a lower bound for Mahler measures greater than one. In 1933 Lehmer [Leh]posed the following problem.
$N(f)\mathrm{X}(\mathrm{f})$$\mathcal{T}\subset P$$\lambda(f)$Question 1 (Lehmer’s problem) Given $\delta>0$ , does there exist a f $\in P$ such that $1<M(f)<$$1+\delta^{\varphi}$, if we fix the degree of
It is not hard to see that for $f\in P$
thenand $M(f)$ arebounded from below by a number greater than one depending on .
Up to degree 40 there is no non-cyclotomic polynomial with Mahler measure less than that of
Lehmer’s candidate polynomial$d$$\backslash \mathcal{T}$$f$$\lambda(f)$}$d$$f_{L}(x)=x^{10}+x^{9}$–$x^{7}-x^{6}-x^{5}-x^{4}-x^{3}+x+1$(see, for example, [Boydl] [Mos]), The Mahler measure $M(f_{L})$ is approximately 1.7628.
By a result of Smyth in 1970, Lehmer’s problem reduces to the case of reciprocal polynomials,

which we describe in Section 2.1. Section 2.2 gives some known results concerning distributions ofPerron, Salem and P-V numbers.

Reciprocal polynomials

2.1

Given $f\in P$ of degree , the reciprocal$d$$f_{*}(x)$of $f(x)$ is defined to be$f_{*}(x)=x^{d}f(1/x)$.

.

A polynomial is reciprocal if $=f_{*}$ . Visually, a reciprocal polynom ial is one for which the coefficients are palindromic, that is, they are the same written from right to left or left to right. Lehmer’spolynomial
is a reciprocal polynomial.
.
It satisfies $f=-f_{*}$ , it is called anti-reciprocal A polynomial is anti-reciprocal if and only
if $f(x)=(x-1)g(x)$ where $g(x)$ is reciprocal. All cyclotomic polynomials are reciprocal except(x–1). A polynom ial is reciprocal or anti-reciprocal if and only if it is a product of irreducible
reciprocal polynomials and (x–1). A separable polynomial is reciprocal or anti-reciprocal if andonly if $S(f)$ is closed under inverses. Thus, an algebraic integer is an algebraic unit if and only
if its minimal polynomiai is reciprocal. An irreducible polynomial with a root on the unit circle isautomatically reciprocal. Thus, minimal polynomials of Salem numbers are always reciprocal, andthe minimal polynomial of a P-V number is reciprocal only if it is quadratic.
Smyth showed [Smy] that if $f\neq\pm f_{*}$ , then the smallest Mahler measure is realized by$f$$f_{L}$$f$$f$$\alpha$$f_{S}(x)=x^{3}-x-1$ ,which has Mahler measure $M(fs)\approx$ 1.32472. Since Lehmer’s polynomial$f_{L}$

satisfies$M(f_{L})<M(fs)$ ,it follows that to solve Lehmer’s problem it is enough to look at reciprocal polynomials.

122

P-V and Salem polynomials

2.2

An interesting special case of Lehmer’s problem is when $N(f)$ $=1$ . The following are equivalent:(i) $N(f)=1$ ;

(ii)

$f$

has a single root outside $C$ , which is (up to sign) a Salem number or a P-V number; and(iii) $f=gh$ where$g\in \mathcal{T}$

and

$h$

is the minimal polynomial of a Salem number or a P-V number.

For quadratic polynomials $N(f)=1$ implies that both roots are real. The reciprocal case isdiscussed in Section 3.1. For irreducible polynomials of degree $>2$ , $N(f)=1$ im lies has a Salemroot (up to sign) if and only if is reciprocal or anti-reciprocal.
The set of P-V numbers is closed [Sal] , and the smallest accumulation point is the golden mean(cf. Section 3.1). A complete set of P-V numbers less than 1.6 was catalogued by Dufresnoy
and Pisot [DP].
The polynomial $fs(z)=x^{3}-$ –1 is the minimal polynomial for the smallest P-V number. It is an open
[Sie], andis the minimal polynomial for the smallest known Salem number
problem whether there is a lower bound larger than one for the set of Salem numbers, or whether
there is a Salem number less than .
In their study of distributions of Salem numbers, Salem [Sal] and Boyd [Boyd2] investigatedsequences polynomials of the form
$f$

$f$

$\alpha_{G}$$\theta 0$$x$$\alpha_{L}$$f_{L}$$f_{L}$$Q_{n}(t)=t^{n}P(t)\pm P_{*}(t)$(1),

for $P\in$ V. The sequence of polynom ials of the form given in (1) is called a Salem-Boyd sequencefor . Salem [Sal] proved that the set of P-V numbers lies in the set of upper and lower limits ofSalem numbers by proving the following result.
$P$

be a Salem-Boyd sequence for P. Then for somepolynomial , let$=1$$N\{Qn$)$N>0$ ,
andfor $n\geq N$ . Fur thermore, for $n>N$ , the Salem$M(Q_{n})=\lambda(Q_{n})$converge monotonically to $M(P)=\lambda(P)$ from above or below depending
numbers
on the sign.

Theorem 1 Given any P-$V$$\mathrm{P}$$Q_{n}$$N\{Qn$ ) $=0$ jar $n<N_{\}}$In the more general setting where P $\in P$ is any element, Boyd showed the following [Boyd2],

Theorem 2 Let
the following.

$Q_{n}$

be a Salem-Boyd sequence for a monic integer polynomial $P(t)$ . Then

(1) $N(Q_{n})\leq N(P)$ for all(2)$\lim_{narrow\infty}\mathrm{N}\{\mathrm{Q}\mathrm{n}$$n$ $\geq 1$$\prime n$

) $e$

have

;

) $=\lambda(P1j$ and(3) $\lim_{narrow\infty}M(Q_{n})=M(P)$ .

Any reciprocal polynomial can be written in the form offor some and . Thus, although, it does parin terms of $M(P)$ andTheorem 2 doesn’t give a lower bound on $M(Qn)$ or
tition the set of Mahler measures and radii of reciprocal polynomials into (non-disjoint) convergent
families.
A polynomial is a Perron polynomial if there is a simple real root $\alpha>1$ , such that for any
other root of ,. Lehmer’s problem is unsolved for this special subclass of monic integer
matrix is Perron
polynomials. By definition, the characteristic polynomial of a[Gan].
Theorem 1 generalizes to Perron polynomials as follows [Hir4]$Q_{n}$$\mathrm{X}(\mathrm{Q}\mathrm{n})$

$P$

$n$

$\lambda(P)$$f$$\beta$$f$$|\beta|<\alpha$$\mathrm{P}\mathrm{e}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{n}rightarrow \mathrm{F}\mathrm{r}\mathrm{o}\mathrm{b}\mathrm{e}\mathrm{n}\mathrm{i}\mathrm{u}\mathrm{s}$123

Theorem 3 If P is a Perron polynomial andan eventually monotone sequence convergeng to

is a Salem-Boyd sequence for P. Then

$Q_{n}$$\lambda(Q_{n})$is$\lambda(P)$ ,In general, $M(Q_{n})$ is not monotone, eventually monotone, or monotone for an arithmetic subsequence

3

Examples from low-dimensional topology and geometry

In this section, wc list some occurrences of reciprocal, Salem, P-V and Perron polynomials in low
dimensional topology. These exam ples indicate a common underlying structure behind many of
the invariants of low dimensional topology, which is yet to be fully explored.

3.1

Quadratic polynomials

There is a bijective correspondence between
This is defined by$\Gamma=\mathrm{S}\mathrm{L}(2, \mathbb{Z})$$A\mapsto f_{A}$and reciprocal quadratic polynomials.

,

is the characteristic polynomial for $A\in\Gamma$ . The characteristic polynomial
whereis reciprocal, since the two eigenvalues of must multiply to 1.
element
The inverse map is defined as follows. Let A be a quadratic such thatdefineA $=($ $-1a$$f_{A}$$f_{A}$of any$A$$A\in\Gamma$$a=\lambda+1/\lambda\in \mathbb{Z}$$01$

are the roots ofThen A and
The correspondence$\lambda^{-1}$$f_{A}$

, and

).

.
$\mathrm{T}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}(A)\mapsto$(2)

A $(fA)$. Thus, non-cyclotomic reciprocal
is order preserving, and $\lambda(f_{A})=1$ if and only if
quadratics correspond to hyperbolic elements of .
. Then hyperbolic elements
Consider the action of as isometries on the hyperbolic disk
is given, whose length
correspond to closed geodesies
on the quotient spaceby.
$|’\mathrm{b}\mathrm{a}\mathrm{c}\mathrm{e}(A)|\leq 2$$\Gamma$$\mathbb{H}^{2}$$\Gamma$$A\in\Gamma$$\Gamma_{\backslash }^{\backslash }\mathbb{H}^{2}$$\ell(\gamma A)$$\gamma_{A}$$\ell(\gamma_{A})=\log(’\mathrm{b}\mathrm{a}\mathrm{c}\mathrm{e}(A))$

The correspondence given in (2) also gives rise to an ordering preserving correspondence between.
and lengths of closed geodesies on, the smallest Mahler measure greater thanpolynomials
reciprocal
If follows that for quadratic
one that occurs is realized by$\Gamma\backslash \mathbb{H}^{2}$$\lambda(f_{A})$$f$$f_{0}(x)=x^{2}-3x+1$

and $M(fo)$$=\lambda(J_{0}^{\cdot})=(3 +\sqrt{5})/2$.

Remark 4 There a similar correspondence between Salem numbers and lengths of closed geodesies(see, for example, [G-H]). Thus, the minimization
on more general arithmetic quotients ofproblemtotheisrelatedproblem for Salem numbersof finding a minimum length geodesic on anplanearithmetic quotient of the hyperbolic$\mathrm{o}s$$\mathbb{H}^{2}$

124

The smallest Mahler measure greater than one among all quadratics is realized by the nonreciprocal polynom ial$f_{g}(x)=x^{2}-$ $x-1$.

) $=(1+\sqrt{5})/2$ is the golden mean. The smallest Mahler measures greater than one
Here,for reciprocal polynomials of degrees 4 and 6 are approximately 1.72208 and 1.40127 respectively,
and hence are also larger than the smallest non-reciprocal Mahler measure $M(fs)$ . For degrees 8
and higher there always exists a reciprocal polynomial (not necessarily irreducible) with Mahler
measure greater than one and less than $M(fs)$ .
$M\langle f_{\mathit{9}}$Transformations preserving lattices

3.2

preserves, and suppose
Let $B$ be a non-degenerate symmetric bilinear form on$B$. Thento$B$ . Equivalently, $M$ preserves the lattice inproduct
associateddefined by the inner$M$
repeated
$M$
no
plc
has
is closed under inverses. Thus, if for exam
the set of eigenvalues ofis reciprocal oris separable, theneigenvalues, or equivalently the characteristic polynomial
anti-reciprocal (see, for example, [G-Mc], Theorem 2.1).
Consider the Coxeter element of a Coxeter system (see [Hum] for definitions). The Coxeter
element preserves an associated symmetric bilinear form defined by the Coxeter system. If $(W, S)$
is the characteristic polyonomial of the Coxeter element
is an irreducible Coxeter system, and. If $(W, S)$ is irreducibleof $(W, S)$ , then $\lambda(f_{(W,S)})=1$ if and only if $(W, S)$ is spherical or affine, and
Coxeter system
is minimized by theand not spherical or affine, then$f_{(W_{0},S_{0})}=f_{L}$ is Lehm cr’s polynom ial ([Me] Theorem 6.1). It follows that for any Coxeter system$(W, S)$ , either
or $M(f_{(W,S)})\geq M(f_{(W_{0},S_{0})}$ , which solves Lehmer’s problem for this classexamples.
of$M\in \mathrm{S}\mathrm{L}(n, \mathbb{Z})$$\mathbb{R}^{n}$$\mathbb{R}^{n}$$f_{M}$$f_{M}$$\mathrm{f}(\mathrm{w},\mathrm{s})$$[\mathrm{A}’ \mathrm{C}]$$B_{10}$$\lambda(f_{(W,S)})$$(W0, S_{\mathit{0}})$$f_{(W,S)}\in \mathcal{T}$

3.3
Let

M-matrices$T\in \mathrm{S}\mathrm{L}(n, \mathbb{Z})$be a matrix satisfying$T=\pm M^{\mathrm{t}\mathrm{r}}M^{-1}$(3),and henceis conjugate to, andis the transpose of $M$ . Then
where
has reciprocal or anti-reciprocal characteristic polynomial. Such matrices are called M-matrices.
Howlett [How] showed that if is the jCoxeter element of a simply-laced Coxeter system associated to a graph , then the Coxeter element can be written in terms of the adjacency matrix.
for . Letbe its upper triangular part. Th en setting $M=I-A^{+}$ , we haveThis gives another proof that the characteristic polynomial of a Coxeter element is reciprocal in
the simply-laced case.
Another well-known case is the Alexander matrix of a knot $(S^{3}, K)$ (see [Rolf] for definitions).
Let
be the Seifert matrix for $(S^{3}, K)$ . Then the Alexander polynomial for $(S^{3}, K)$ is given by, up to multiples of. If $V$ is invertible, it follows that the characterisis reciprocal and has a monic representative. Furtherm ore , $\Delta(S^{3},K)(1)=\pm 1$ .
tic polynomial
such that) $=\pm 1$ , then there is a knot
Conversely, if is a reciprocal polynomial with$\Delta_{(S^{3},K)}=f$ [SeiQ.
$M\in \mathrm{S}\mathrm{L}(\mathrm{n}, \mathbb{Z})$$T^{-1}$$M^{\mathrm{t}\mathrm{r}}$$\pm T^{\mathrm{t}\mathrm{r}}$

$T$

$T$

$A$

$\Gamma$$\Gamma$$T=-M^{\mathrm{t}\mathrm{r}}M^{-1}$$A^{+}$$V$$\Delta_{(S^{3},K)}(t)=|\det(tV-V^{\mathrm{t}\mathrm{r}})|$$t^{\pm 1}$$\Delta(S^{3},K)$$J^{\cdot}$$f\langle 1$$(S^{3}, K)_{\mathrm{t}}$125Growth rates of automatic groups

3.4

Let be a finitely presented group, with generating set
The growth series of is the formal sum
$G$

$S$

, such that$S$

is closed under inverses.

$G$$\Psi_{(G,S)}(x)=\sum_{\mathrm{i}=0}^{\infty}a_{n}x^{n}$where
rate of$a_{n}$$G$is the number of words in
is given by$G$of minimal word length$n$in the generating set $S$ . The growth.

$\lambda(G, S)=\lim\sup|a_{n}|^{\frac{1}{n}}$isfor more details.) The growth series $\Phi(G,S)$ is rational, for example, whenIn
details).
, [Can], for more
hyperbolic, automatic, or a Coxeter group (see for example
isthe characteristic polynomial$\lambda(f(G,S))$ whereas
realize
can
case,
we
the automaticof an associated matrix.
beLehmer’s polynomial appears among these examples as follows. Letangleswithplanehyperbolic
theinpolygonaofthroughsides
the Coxeter group of reflections, , where
,
(See

$G$

$[\mathrm{E}\mathrm{C}\mathrm{H}^{+}]$$[\mathrm{E}\mathrm{C}\mathrm{H}^{+}]$$\lambda(G, S)$$\mathrm{f}(\mathrm{a},\mathrm{s})$$(G , Sp1,\ldots,Pk\mathcal{P}1,.

$\ldots$$\frac{\pi}{p_{1}}$,\mathrm{P}k)$$\frac{\pi}{p_{k}}$$\frac{1}{p_{1}}+\cdots+\frac{1}{p_{k}}<k-2$,

series, andThen Cannon and others [F-P], [C-W], [Floy] calculate the denominators of the growthoccurs as
polynomal
Lehmer’s
is a Salem number. In particular,
show thatarea
thesmallestrise
togivingset
and corresponds to the angle
the denominator for
hyperbolic polygon.
andThere is a close relation between the automatic group structure of
Coxeter
simply-lacedis thewhere
the Coxeter element offrom a central vertex ofemanating-branches
with
graph
“star-like”
system associated to the. . ’ . (See, for example, [Hir2]. Although the star-like graphs do not directly delengths, calculations in [Hirl] show that the sequencefine the auotomatic structures ofcomputed directly from the Coxeter elementbecanfor the growth series.
$\lambda(G_{p1},.

$f_{L}$,\mathrm{P}k , S_{p1}, 1p\lambda )$$(G_{2,3,7}, \mathrm{S}_{2,3,7})$$(G_{\mathcal{P}1}, .,Pk ,\mathrm{P}1,.

$(\mathcal{G}_{\rho_{1}},.

.,pk S)$$(\mathcal{G}_{\mathrm{p}1{}_{\ddagger}Pk}..,, \mathrm{S}_{\mathcal{P}1,\ldots,\mathrm{P}\mathrm{L}}.)$,p_{k}’ S_{p_{1},..,pk}))$$\mathrm{f}\mathrm{c}$$p_{11}$$Pk$$(G_{p1},.

,\rho_{k}’ S_{\rho 1,.,T^{J}k})$$\Psi(G_{p_{1}}, ,\mathrm{p}_{k)}^{S}\mathrm{p}_{1}.

,p_{k})$$a_{n}$$(\mathcal{G}_{p_{1,\}}\mathrm{p}k}., \mathrm{S}_{\mathcal{P}1\}}.,p_{k})$Dilatations of pseudo-Anosov maps

3.5

: $Farrow F$
characteristic, and let
Let $F$ be a compact orientable surface with negative Euler
anysurfaceforthatbe a homeomorphism. The Thurston-Nielsen theory [Thu] [FLP] [CB] states
following:homeomorphism : $Farrow F$ , is isotopic to some 0 satisfying one of the$\phi$$\phi$$\phi$(i)$\Phi$

is periodic, i.e.,$\Phi^{n}$is the identity;(ii) (X) is irreducible, i.e., there is a closed curve on invariant undercomponents have negative Euler characteristic; or$F$(iii) (X) is pseudo-Anosov, i.e., there is a number
foliations such that$\lambda>1$

and a pair$\Phi(\mathcal{F}^{\pm}\}=\lambda^{\pm 1}F^{\pm}$

4 such that complementary$F^{\pm}$of transverse measured.

class of with smallest topologicalIn the pseudo-Anosov case, is the unique element in the isotopy
is called the dilatation of .
entropy [FLP]. The number$\Phi$$\phi$$\lambda(\phi)=\lambda(\Phi)$$\phi$

128$F$
is pseudo-Anosov. Then there is an embedded graph in representing the spine
Supposefor restricted to is a Perron-Frobenius matrix. Thus,
of $F$ , such that the transition matrix
is a Perron polynomial. Furthermore, the Perron numberofthe characteristic polynomial$\mathcal{G}$$\Phi$$\Phi$$M_{\Phi}$$\mathcal{G}$$M_{\Phi}$$f\phi$has the property that
. The transition matrix
equals the dilatation
associated tois reciprocal.
and hence its characteristic polynomial
1s conjugate to$M_{\Phi}$$\lambda(\phi)$$f_{\phi}$$M_{\Phi}^{\mathrm{t}\mathrm{r}}$$f\emptyset$$M_{\Phi}^{-1}$(b)$\langle \mathrm{a})$Figure 1: Braids with small dilatation.

In [HK] we study two families of pseudo-Anosov maps on a marked disk associated to the braids
drawn in Figure 1, and show that their dilatations satisfy the characteristicandequations$\beta_{m,n}(\mathrm{a})$$\sigma_{m,n}(\mathrm{b})$$x^{n+1}R_{m}(x)+(R_{m})_{*}(x)$for$\beta_{m,n}$and, when $n\geq m+2$ ,$x^{n+1}R_{m}(x)-(R_{m})_{*}(x)$for$\sigma_{m,n}$, when$\mathrm{e}$$R_{m}(x)=x^{m}(x-1)-2$ .

roots outside the unit circle. Thus, $R_{1}(x)$ is a P-V polynomial, andhave
The polynomials
and $\sigma_{1,n}(n\geq 3)$ are Salem numbers and are monotone
hence by Theorem 1 the dilatations of
are Perron polynomials,). For all $m$ , the polynomials
and increasing for(decreasing for
and
for 5
decreasing
and hence by Theorem 3 the dilatations are eventually monotone (again,,
anygoes to infinity, it follows that for
approaches 1 as). Since
increasing for
it is possible to make $m$ and large enough so that the dilatations are within of 1.
and , lift to homeomorphisms of genus compact
The pseudo-Anosov maps defined byandcovering, where, if $m+n$ is even,doublesurfaces with boundary components via
Then
lift
of
$b=2$
the
.
be
Let
and
$b=1$ , and if$+n$ is odd,
$R_{m}$$\tau r\iota$$\beta_{1,n}$$R_{m}$$\beta_{1,n}$$\sigma_{1,n}$$n\iota,r\iota$$\epsilon>0$$\lambda(R_{m})$$\sigma_{m,n}$$77\mathrm{L}$$\epsilon$$n$$\beta_{m,n}$$\sigma_{m}$

$g$

$n$

$g= \frac{m+n}{2}$$b$$g= \frac{r\mathrm{n}+n-1}{2}$$\prime r\iota$$\sigma_{g-1,g+1\sim}$$\phi_{g}$$\log(\lambda(\phi_{g}))=\log(\lambda(\sigma_{g-1,g+1}))-\cdot\frac{1}{g}$,and we recover Penner’s result on least dilatations of pseudo-Anosov maps on orientabie genus
are the monodromy of fibered two-bridge links (see also [Bri] and [Hir4]).
surfaces. The lifts ofacross the axis containing the marked points, we see that the two braids are
Reflecting, predicts that underin the form given in Figure 2. A conjecture of de Carvalho and Halland for large enough, braids of the form given in Figure 2 (left) willcertain conditions on
determine pseudo-Anosov maps whose invariant fibrations will have nice limiting behavior. Thus,
it is natural to ask the following.
$g$$\beta_{m,n}$$\beta_{m,n}$$[\mathrm{d}\mathrm{C}\mathrm{H}]$

$B$

$n$

Question 2 Under what conditions on B will the braids in the form given in Figure 2 (left) havedilatations satisfying a Salem-Boyd sequence$\rho$

127($|\begin{array}{l}-|\prime\prime,r_{\prime}.\prime\end{array}|$$\zeta \mathrm{a})$1(b)$\frac{\mathrm{m}}{\mapsto^{1}}||_{\tau}|\ldots..|$$‘\ovalbox{\tt\small REJECT}_{1}|$Figure 2: Braid form.

Forof$R_{m}$

and, the limiting behavior is the same, and this occurance explains the appearance
for the characteristic equations of both braid families.
$\beta_{m,n}$$\sigma_{m}$$n$Problem 3 Characterize pairs of distinct braids which if plugged into B will give rise to SalemBoyd sequences associated to the same polynomial P and differing y the sign in front of P.
$\dot{\mathit{0}}$Homological dilatation

3.6

be the restriction of to the first homology
Let : $Farrow F$ be a surface homeomorphlism, and letmeasuresisotopy class of , then since
theof. If is a pseudo-Anosov representative
Hx[BH]) we
[FLP],
example,
for
(see,
of
under iterations
the growth rate of word lengths ofhave in general$\phi$$\phi_{*}$$\phi$$M_{\Phi}$$\phi$$\Phi$$(\mathrm{F};\mathbb{R})$$\Phi$$\pi_{1}(F)$$\lambda(\phi_{*})\leq\lambda(\phi)$.

of ,are orientable, then the largest eigenvalue
If, in addition, the invariant foliations.
called the homological dilatation of , equals
smoothly embedded$(S^{3}, K)$ is a pair where $K$ is the disjoint union of a finite number ofA link
$K$ inis$U(K)$,
ofneighborhood$(S^{3},K)$
is fibered if for a regularcircles in . A link
uniquenecessarily
not
is
over
a locally trivial fiber bundle over . This fiber bundle structureisbe a fiber. Then
when $K$ has more than one compoent. Given a fibration let
is$(x,1)=(\phi(x),
0)$
,
where
$[0,1]$ modulo an identification
homeomorphic to the product of
$K$
then
knot,
a
is
a surface homeomorphism from to itself, i.e., it is the mapping torus for . If
$f$ is a monic reciprocal integer
equals). In general, if
the characteristic polynomial ofofuP
polynomial, then there is a fibered link such that equals the characteristic polynomialbecanintegers
algebraic
about
question
any
that
$(x$
–1 [Kan]. It thus follows
to a multiple oflinks.
fiberedoftranslated to a question about homological dilatations$\lambda(\phi_{*})$$F^{\pm}$$\phi$$\phi_{*}$$\lambda(\phi)[\mathrm{R}\mathrm{y}\mathrm{k}]$$S^{3}$$S^{3}\backslash U(K)$$S^{3}$$S^{1}$$S^{1}$$s^{3}\backslash U(K)$$\Sigma$$\phi$$\Sigma \mathrm{x}$$\phi$$\Sigma$$\phi_{*}$$\Delta\langle S^{3},K$$\phi_{*}$

$J$

$)$

Problem 4 Characterize fibered links with small Mahler measure greater thanone.

Using the similar forms of Alexander matrices and Coxeter elements describedlinks) such thatis possible to construct many examples of fibered knots (and$f_{\phi}.(x)=f_{(W,S)}(-x)$

where $(W, S)$ is a simply-laced Coxeter system [Hir3],

in Section 3.3, it

128, which is a fibered knot associated to the
One such example is the (-2, 3, 7)-pretzel knot
component,
only
one
has
Coxeter system. Since$K_{2,3,7}$$K_{2,3,7}$$E_{10}$$f_{\phi}.(x)=\Delta_{K_{2,3,7}}(x)$.

$(-x)$ $=f_{L}(x)$ is the Lehmer polynomial (cf. Section 3.2).
Thus, Aby plumbing a $(2-n)$ -torus link (see, for example,
Ifis obtained from a fibered linkby iterated Hopf plumbing.
is obtained from[Mur] and [Har] for definitions), we say thatmonodromy of such ahomological
In [Hir4] , we show that the characteristic polynomial of the
2 define via double
Figuregiven
inis a Salem-Boyd sequence. As an example, braids of the form
plumbing.
Hopf
iteratedcovering homeomorphisms of surfaces, which are obtained by$K_{2,3.7}$$K_{0}$$K_{n}$$K_{0}$$K_{n}$$K_{n}$:
Problem 5 Let : $Farrow F$ be a homeomorphism of a surface with boundary, and let
pseudo-Anosov,
be obtained by iterated Hopf plumbing. Under what conditions on and is
and in this case do the dilatations satisfy a Salem-Boyd sequence?
$\phi_{n}$$F_{n}arrow F_{n}$$\phi$$\phi$$n$$\phi_{n}$References[A’C]

N. A’Campo. Sur les valeurs propres de la transformation de Coxeter. Invent Math. 33(1976), 61-67,[BH]

M. Bestivina and M. Handel. Train tracks and automorphism s of free groups. Ann. Math.
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