64 research outputs found
On arc index and maximal Thurston-Bennequin number
We discuss the relation between arc index, maximal Thurston--Bennequin
number, and Khovanov homology for knots. As a consequence, we calculate the arc
index and maximal Thurston--Bennequin number for all knots with at most 11
crossings. For some of these knots, the calculation requires a consideration of
cables which also allows us to compute the maximal self-linking number for all
knots with at most 11 crossings.Comment: 10 pages, v4: corrected typo
A new algorithm for recognizing the unknot
The topological underpinnings are presented for a new algorithm which answers
the question: `Is a given knot the unknot?' The algorithm uses the braid
foliation technology of Bennequin and of Birman and Menasco. The approach is to
consider the knot as a closed braid, and to use the fact that a knot is
unknotted if and only if it is the boundary of a disc with a combinatorial
foliation. The main problems which are solved in this paper are: how to
systematically enumerate combinatorial braid foliations of a disc; how to
verify whether a combinatorial foliation can be realized by an embedded disc;
how to find a word in the the braid group whose conjugacy class represents the
boundary of the embedded disc; how to check whether the given knot is isotopic
to one of the enumerated examples; and finally, how to know when we can stop
checking and be sure that our example is not the unknot.Comment: 46 pages. Published copy, also available at
http://www.maths.warwick.ac.uk/gt/GTVol2/paper9.abs.htm
Assessing morphology and function of the semicircular duct system: Introducing new in-situ visualization and software toolbox
International audienceThe semicircular duct system is part of the sensory organ of balance and essential for navigation and spatial awareness in vertebrates. Its function in detecting head rotations has been modelled with increasing sophistication, but the biomechanics of actual semicircular duct systems has rarely been analyzed, foremost because the fragile membranous structures in the inner ear are hard to visualize undistorted and in full. Here we present a new, easy-to-apply and non-invasive method for three-dimensional in-situ visualization and quantification of the semicircular duct system, using X-ray micro tomography and tissue staining with phosphotungstic acid. Moreover, we introduce Ariadne, a software toolbox which provides comprehensive and improved morphological and functional analysis of any visualized duct system. We demonstrate the potential of these methods by presenting results for the duct system of humans, the squirrel monkey and the rhesus macaque, making comparisons with past results from neurophysiological, oculometric and biomechanical studies
The projective consciousness model and phenomenal selfhood
We summarize our recently introduced Projective Consciousness Model (PCM) (Rudrauf et al., 2017) and relate it to outstanding conceptual issues in the theory of consciousness. The PCM combines a projective geometrical model of the perspectival phenomenological structure of the field of consciousness with a variational Free Energy minimization model of active inference, yielding an account of the cybernetic function of consciousness, viz., the modulation of the field's cognitive and affective dynamics for the effective control of embodied agents. The geometrical and active inference components are linked via the concept of projective transformation, which is crucial to understanding how conscious organisms integrate perception, emotion, memory, reasoning, and perspectival imagination in order to control behavior, enhance resilience, and optimize preference satisfaction. The PCM makes substantive empirical predictions and fits well into a (neuro)computationalist framework. It also helps us to account for aspects of subjective character that are sometimes ignored or conflated: pre-reflective self-consciousness, the first-person point of view, the sense of minenness or ownership, and social self-consciousness. We argue that the PCM, though still in development, offers us the most complete theory to date of what Thomas Metzinger has called "phenomenal selfhood.
Overtwisted energy-minimizing curl eigenfields
We consider energy-minimizing divergence-free eigenfields of the curl
operator in dimension three from the perspective of contact topology. We give a
negative answer to a question of Etnyre and the first author by constructing
curl eigenfields which minimize energy on their co-adjoint orbit, yet are
orthogonal to an overtwisted contact structure. We conjecture that -contact
structures on -bundles always define tight minimizers, and prove a partial
result in this direction.Comment: published versio
Weak and strong fillability of higher dimensional contact manifolds
For contact manifolds in dimension three, the notions of weak and strong
symplectic fillability and tightness are all known to be inequivalent. We
extend these facts to higher dimensions: in particular, we define a natural
generalization of weak fillings and prove that it is indeed weaker (at least in
dimension five),while also being obstructed by all known manifestations of
"overtwistedness". We also find the first examples of contact manifolds in all
dimensions that are not symplectically fillable but also cannot be called
overtwisted in any reasonable sense. These depend on a higher-dimensional
analogue of Giroux torsion, which we define via the existence in all dimensions
of exact symplectic manifolds with disconnected contact boundary.Comment: 68 pages, 5 figures. v2: Some attributions clarified, and other minor
edits. v3: exposition improved using referee's comments. Published by Invent.
Mat
On recurrence and ergodicity for geodesic flows on noncompact periodic polygonal surfaces
We study the recurrence and ergodicity for the billiard on noncompact
polygonal surfaces with a free, cocompact action of or . In the
-periodic case, we establish criteria for recurrence. In the more difficult
-periodic case, we establish some general results. For a particular
family of -periodic polygonal surfaces, known in the physics literature
as the wind-tree model, assuming certain restrictions of geometric nature, we
obtain the ergodic decomposition of directional billiard dynamics for a dense,
countable set of directions. This is a consequence of our results on the
ergodicity of \ZZ-valued cocycles over irrational rotations.Comment: 48 pages, 12 figure
Tightness in contact metric 3-manifolds
This paper begins the study of relations between Riemannian geometry and
global properties of contact structures on 3-manifolds. In particular we prove
an analog of the sphere theorem from Riemannian geometry in the setting of
contact geometry. Specifically, if a given three dimensional contact manifold
(M,\xi) admits a complete compatible Riemannian metric of positive 4/9-pinched
curvature then the underlying contact structure \xi is tight; in particular,
the contact structure pulled back to the universal cover is the standard
contact structure on S^3. We also describe geometric conditions in dimension
three for \xi to be universally tight in the nonpositive curvature setting.Comment: 29 pages. Added the sphere theorem, removed high dimensional material
and an alternate approach to the three dimensional tightness radius estimate
Altered Perceptual Sensitivity to Kinematic Invariants in Parkinson's Disease
Ample evidence exists for coupling between action and perception in neurologically healthy individuals, yet the precise nature of the internal representations shared between these domains remains unclear. One experimentally derived view is that the invariant properties and constraints characterizing movement generation are also manifested during motion perception. One prominent motor invariant is the âtwo-third power law,â describing the strong relation between the kinematics of motion and the geometrical features of the path followed by the hand during planar drawing movements. The two-thirds power law not only characterizes various movement generation tasks but also seems to constrain visual perception of motion. The present study aimed to assess whether motor invariants, such as the two thirds power law also constrain motion perception in patients with Parkinson's disease (PD). Patients with PD and age-matched controls were asked to observe the movement of a light spot rotating on an elliptical path and to modify its velocity until it appeared to move most uniformly. As in previous reports controls tended to choose those movements close to obeying the two-thirds power law as most uniform. Patients with PD displayed a more variable behavior, choosing on average, movements closer but not equal to a constant velocity. Our results thus demonstrate impairments in how the two-thirds power law constrains motion perception in patients with PD, where this relationship between velocity and curvature appears to be preserved but scaled down. Recent hypotheses on the role of the basal ganglia in motor timing may explain these irregularities. Alternatively, these impairments in perception of movement may reflect similar deficits in motor production
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