1,994 research outputs found
Analogies between the crossing number and the tangle crossing number
Tanglegrams are special graphs that consist of a pair of rooted binary trees
with the same number of leaves, and a perfect matching between the two
leaf-sets. These objects are of use in phylogenetics and are represented with
straightline drawings where the leaves of the two plane binary trees are on two
parallel lines and only the matching edges can cross. The tangle crossing
number of a tanglegram is the minimum crossing number over all such drawings
and is related to biologically relevant quantities, such as the number of times
a parasite switched hosts.
Our main results for tanglegrams which parallel known theorems for crossing
numbers are as follows. The removal of a single matching edge in a tanglegram
with leaves decreases the tangle crossing number by at most , and this
is sharp. Additionally, if is the maximum tangle crossing number of
a tanglegram with leaves, we prove
. Further,
we provide an algorithm for computing non-trivial lower bounds on the tangle
crossing number in time. This lower bound may be tight, even for
tanglegrams with tangle crossing number .Comment: 13 pages, 6 figure
The Jones polynomial of ribbon links
For every n-component ribbon link L we prove that the Jones polynomial V(L)
is divisible by the polynomial V(O^n) of the trivial link. This integrality
property allows us to define a generalized determinant det V(L) :=
[V(L)/V(O^n)]_(t=-1), for which we derive congruences reminiscent of the Arf
invariant: every ribbon link L = (K_1,...,K_n) satisfies det V(L) = det(K_1)
>... det(K_n) modulo 32, whence in particular det V(L) = 1 modulo 8.
These results motivate to study the power series expansion V(L) =
\sum_{k=0}^\infty d_k(L) h^k at t=-1, instead of t=1 as usual. We obtain a
family of link invariants d_k(L), starting with the link determinant d_0(L) =
det(L) obtained from a Seifert surface S spanning L. The invariants d_k(L) are
not of finite type with respect to crossing changes of L, but they turn out to
be of finite type with respect to band crossing changes of S. This discovery is
the starting point of a theory of surface invariants of finite type, which
promises to reconcile quantum invariants with the theory of Seifert surfaces,
or more generally ribbon surfaces.Comment: 38 pages, reformatted in G&T style; minor changes suggested by the
refere
Linking number and Milnor invariants
This is a concise overview of the definitions and properties of the linking
number and its higher-order generalization, Milnor invariants.Comment: 7 pages. Expository article written for an upcoming concise
encyclopaedia of knot theor
Spectral geometry, link complements and surgery diagrams
We provide an upper bound on the Cheeger constant and first eigenvalue of the
Laplacian of a finite-volume hyperbolic 3-manifold M, in terms of data from any
surgery diagram for M. This has several consequences. We prove that a family of
hyperbolic alternating link complements is expanding if and only if they have
bounded volume. We also provide examples of hyperbolic 3-manifolds which
require 'complicated' surgery diagrams, thereby proving that a recent theorem
of Constantino and Thurston is sharp. Along the way, we find a new upper bound
on the bridge number of a knot that is not tangle composite, in terms of the
twist number of any diagram of the knot. The proofs rely on a theorem of Lipton
and Tarjan on planar graphs, and also the relationship between many different
notions of width for knots and 3-manifolds.Comment: 20 pages, 11 figure
Driven Tunneling: Chaos and Decoherence
Chaotic tunneling in a driven double-well system is investigated in absence
as well as in the presence of dissipation. As the constitutive mechanism of
chaos-assisted tunneling, we focus on the dynamics in the vicinity of
three-level crossings in the quasienergy spectrum. The coherent quantum
dynamics near the crossing is described satisfactorily by a three-state model.
It fails, however, for the corresponding dissipative dynamics, because
incoherent transitions due to the interaction with the environment indirectly
couple the three states in the crossing to the remaining quasienergy states.
The asymptotic state of the driven dissipative quantum dynamics partially
resembles the, possibly strange, attractor of the corresponding damped driven
classical dynamics, but also exhibits characteristic quantum effects.Comment: 32 pages, 35 figures, lamuphys.st
Higher-dimensional Algebra and Topological Quantum Field Theory
The study of topological quantum field theories increasingly relies upon
concepts from higher-dimensional algebra such as n-categories and n-vector
spaces. We review progress towards a definition of n-category suited for this
purpose, and outline a program in which n-dimensional TQFTs are to be described
as n-category representations. First we describe a "suspension" operation on
n-categories, and hypothesize that the k-fold suspension of a weak n-category
stabilizes for k >= n+2. We give evidence for this hypothesis and describe its
relation to stable homotopy theory. We then propose a description of
n-dimensional unitary extended TQFTs as weak n-functors from the "free stable
weak n-category with duals on one object" to the n-category of "n-Hilbert
spaces". We conclude by describing n-categorical generalizations of deformation
quantization and the quantum double construction.Comment: 36 pages, LaTeX; this version includes all 36 figure
The largest crossing number of tanglegrams
A tanglegram consists of two rooted binary trees with the same
number of leaves, and a perfect matching between the two leaf sets. In a
layout, the tanglegrams is drawn with the leaves on two parallel lines, the
trees on either side of the strip created by these lines are drawn as plane
trees, and the perfect matching is drawn in straight line segments inside the
strip. The tanglegram crossing number of is the
smallest number of crossings of pairs of matching edges, over all possible
layouts of . The size of the tanglegram is the number of matching
edges, say . An earlier paper showed that the maximum of the tanglegram
crossing number of size tanglegrams is ; but is
at least for infinitely many .
Now we make better bounds: the maximum crossing number of a size tanglegram
is at most , but for infinitely many ,
at least . The problem shows
analogy with the Unbalancing Lights Problem of Gale and Berlekamp
Deformation of Crystals: Connections with Statistical Physics
We give a bird's-eye view of the plastic deformation of crystals aimed at the statistical physics community, as well as a broad introduction to the statistical theories of forced rigid systems aimed at the plasticity community. Memory effects in magnets, spin glasses, charge density waves, and dilute colloidal suspensions are discussed in relation to the onset of plastic yielding in crystals. Dislocation avalanches and complex dislocation tangles are discussed via a brief introduction to the renormalization group and scaling. Analogies to emergent scale invariance in fracture, jamming, coarsening, and a variety of depinning transitions are explored. Dislocation dynamics in crystals challenge nonequilibrium statistical physics. Statistical physics provides both cautionary tales of subtle memory effects in nonequilibrium systems and systematic tools designed to address complex scale-invariant behavior on multiple length scales and timescales
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