64,213 research outputs found
On balanced planar graphs, following W. Thurston
Let be an orientation-preserving branched covering map of
degree , and let be an oriented Jordan curve passing through
the critical values of . Then is an oriented graph
on the sphere. In a group email discussion in Fall 2010, W. Thurston introduced
balanced planar graphs and showed that they combinatorially characterize all
such , where has distinct critical values. We give a
detailed account of this discussion, along with some examples and an appendix
about Hurwitz numbers.Comment: 17 page
Lawrence-Krammer-Bigelow representations and dual Garside length of braids
We show that the span of the variable in the Lawrence-Krammer-Bigelow
representation matrix of a braid is equal to the twice of the dual Garside
length of the braid, as was conjectured by Krammer. Our proof is close in
spirit to Bigelow's geometric approach. The key observation is that the dual
Garside length of a braid can be read off a certain labeling of its curve
diagram
Quantum computation with Turaev-Viro codes
The Turaev-Viro invariant for a closed 3-manifold is defined as the
contraction of a certain tensor network. The tensors correspond to tetrahedra
in a triangulation of the manifold, with values determined by a fixed spherical
category. For a manifold with boundary, the tensor network has free indices
that can be associated to qudits, and its contraction gives the coefficients of
a quantum error-correcting code. The code has local stabilizers determined by
Levin and Wen. For example, applied to the genus-one handlebody using the Z_2
category, this construction yields the well-known toric code.
For other categories, such as the Fibonacci category, the construction
realizes a non-abelian anyon model over a discrete lattice. By studying braid
group representations acting on equivalence classes of colored ribbon graphs
embedded in a punctured sphere, we identify the anyons, and give a simple
recipe for mapping fusion basis states of the doubled category to ribbon
graphs. We explain how suitable initial states can be prepared efficiently, how
to implement braids, by successively changing the triangulation using a fixed
five-qudit local unitary gate, and how to measure the topological charge.
Combined with known universality results for anyonic systems, this provides a
large family of schemes for quantum computation based on local deformations of
stabilizer codes. These schemes may serve as a starting point for developing
fault-tolerance schemes using continuous stabilizer measurements and active
error-correction.Comment: 53 pages, LaTeX + 199 eps figure
Harmonic measures for distributions with finite support on the mapping class group are singular
Kaimanovich and Masur showed that a random walk on the mapping class group
for an initial distribution with finite first moment and whose support
generates a non-elementary subgroup, converges almost surely to a point in the
space PMF of projective measured foliations on the surface. This defines a
harmonic measure on PMF. Here, we show that when the initial distribution has
finite support, the corresponding harmonic measure is singular with respect to
the natural Lebesgue measure on PMF.Comment: 43 pages, 16 figures. Minor improvements overall, specifically
Section 12. Added reference
Combinatorial approach to detection of fixed points, periodic orbits, and symbolic dynamics
We present a combinatorial approach to rigorously show the existence of fixed
points, periodic orbits, and symbolic dynamics in discrete-time dynamical
systems, as well as to find numerical approximations of such objects. Our
approach relies on the method of `correctly aligned windows'. We subdivide the
`windows' into cubical complexes, and we assign to the vertices of the cubes
labels determined by the dynamics. In this way we encode the dynamics
information into a combinatorial structure. We use a version of the Sperner
Lemma saying that if the labeling satisfies certain conditions, then there
exist fixed points/periodic orbits/orbits with prescribed itineraries. Our
arguments are elementary
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