50 research outputs found
Chord Diagrams and Coxeter Links
This paper presents a construction of fibered links out of chord
diagrams \sL. Let be the incidence graph of \sL. Under certain
conditions on \sL the symmetrized Seifert matrix of equals the
bilinear form of the simply-laced Coxeter system associated to
; and the monodromy of equals minus the Coxeter element of
. Lehmer's problem is solved for the monodromy of these Coxeter links.Comment: 18 figure
Salem-Boyd sequences and Hopf plumbing
Given a fibered link, consider the characteristic polynomial of the monodromy
restricted to first homology. This generalizes the notion of the Alexander
polynomial of a knot. We define a construction, called iterated plumbing, to
create a sequence of fibered links from a given one. The resulting sequence of
characteristic polynomials has the same form as those arising in work of Salem
and Boyd in their study of distributions of Salem and P-V numbers. From this we
deduce information about the asymptotic behavior of the large roots of the
generalized Alexander polynomials, and define a new poset structure for Salem
fibered links.Comment: 18 pages, 6 figures, to appear in Osaka J. Mat
Lehmer's Problem, McKay's Correspondence, and
This paper addresses a long standing open problem due to Lehmer in which the
triple 2,3,7 plays a notable role. Lehmer's problem asks whether there is a gap
between 1 and the next smallest algebraic integer with respect to Mahler
measure. The question has been studied in a wide range of contexts including
number theory, ergodic theory, hyperbolic geometry, and knot theory; and
relates to basic questions such as describing the distribution of heights of
algebraic integers, and of lengths of geodesics on arithmetic surfaces. This
paper focuses on the role of Coxeter systems in Lehmer's problem. The analysis
also leads to a topological version of McKay's correspondence
Boundary Manifolds of Line Arrangements
In this paper we describe the complement of real line arrangements in the
complex plane in terms of the boundary three-manifold of the line arrangement.
We show that the boundary manifold of any line arrangement is a graph manifold
with Seifert fibered vertex manifolds, and depends only on the incidence graph
of the arrangement. When the line arrangement is defined over the real numbers,
we show that the homotopy type of the complement is determined by the incidence
graph together with orderings on the edges emanating from each vertex.Comment: Latex, 22 pages, 15 figure
Digraphs and cycle polynomials for free-by-cyclic groups
Let \phi \in \mbox{Out}(F_n) be a free group outer automorphism that can be
represented by an expanding, irreducible train-track map. The automorphism
determines a free-by-cyclic group
and a homomorphism . By work of Neumann,
Bieri-Neumann-Strebel and Dowdall-Kapovich-Leininger, has an open cone
neighborhood in whose integral points
correspond to other fibrations of whose associated outer automorphisms
are themselves representable by expanding irreducible train-track maps. In this
paper, we define an analog of McMullen's Teichm\"uller polynomial that computes
the dilatations of all outer automorphism in .Comment: 41 pages, 20 figure
Braid Group Actions on Rational Maps
Rational maps are maps from the Riemann sphere to itself that are defined by ratios of polynomials. A special type of rational map is the ones where the forward orbit of the critical points is finite. That is, under iteration, the critical points all eventually cycle in some periodic orbit. In the 1980s Thurston proved the surprising result that (except for a well-understood set of exceptions) when the post-critical set is finite the rational map is determined by the “combinatorics” of how the map behaves on the post-critical set. Recently, there has been interest in the question: what happens if we just fix the degree and impose the condition that only one critical orbit is finite. In that case, the family of rational maps defined by the combinatorics is a complex manifold naturally acted on by subgroups of the pure spherical braid group on n-strands where n depends on the order of the orbit and the degree, In this talk, we discuss the question: what is the global topology of this manifold
Small dilatation pseudo-Anosov mapping classes coming from the simplest hyperbolic braid
In this paper we study the minimum dilatation pseudo-Anosov mapping classes
coming from fibrations over the circle of a single 3-manifold, the mapping
torus for the "simplest pseudo-Anosov braid". The dilatations that arise
include the minimum dilatations for orientable mapping classes for genus
g=2,3,4,5,8 as well as Lanneau and Thiffeault's conjectural minima for
orientable mapping classes, when g = 2,4 (mod 6). Our examples also show that
the minimum dilatation for orientable mapping classes is strictly greater than
the minimum dilatation for non-orientable ones when g = 4,6,8.Comment: 16 pages. 5 figures. Contains minor corrections to previous
submission