84 research outputs found
Tight Beltrami fields with symmetry
Let be a compact orientable Seifered fibered 3-manifold without a
boundary, and an -invariant contact form on . In a suitable
adapted Riemannian metric to , we provide a bound for the volume
and the curvature, which implies the universal tightness of the
contact structure .Comment: 26 page
Remarks on Legendrian Self-Linking
The Thurston-Bennequin invariant provides one notion of self-linking for any
homologically-trivial Legendrian curve in a contact three-manifold. Here we
discuss related analytic notions of self-linking for Legendrian knots in
Euclidean space. Our definition is based upon a reformulation of the elementary
Gauss linking integral and is motivated by ideas from supersymmetric gauge
theory. We recover the Thurston-Bennequin invariant as a special case.Comment: 42 pages, many figures; v2: minor revisions, published versio
Studying uniform thickness I: Legendrian simple iterated torus knots
We prove that the class of topological knot types that are both Legendrian
simple and satisfy the uniform thickness property (UTP) is closed under
cabling. An immediate application is that all iterated cabling knot types that
begin with negative torus knots are Legendrian simple. We also examine, for
arbitrary numbers of iterations, iterated cablings that begin with positive
torus knots, and establish the Legendrian simplicity of large classes of these
knot types, many of which also satisfy the UTP. In so doing we obtain new
necessary conditions for both the failure of the UTP and Legendrian
non-simplicity in the class of iterated torus knots, including specific
conditions on knot types.Comment: 21 pages, 5 figures; final version, to appear in Algebraic and
Geometric Topolog
The Influence of Pain Distribution on Walking Velocity and Horizontal Ground Reaction Forces in Patients with Low Back Pain
Objective. The primary purpose of this paper was to evaluate the influence of pain distribution on gait characteristics in subjects with low back problems (LBP) during walking at preferred and fastest speeds. Design. Cross-sectional, observational study. Setting. Gait analysis laboratory in a health professions university. Participants. A convenience age- and gender-matched sample of 20 subjects with back pain only (BPO), 20 with referred leg pain due to back problems (LGP), and 20 pain-free individuals (CON). Methods and Measures. Subjects completed standardized self-reports on pain and disability and were videotaped as they walked at their preferred and fastest speeds along a walkway embedded with a force plate. Temporal and spatial gait characteristics were measured at the midsection of the walkway, and peak medial, lateral, anterior, and posterior components of horizontal ground reaction forces (hGRFs) were measured during the stance phase. Results. Patients with leg pain had higher levels of pain intensity and affect compared to those with back pain only (t = 4.91, P < .001 and t = 5.80, P < 0.001, resp.) and walking had an analgesic effect in the BPO group. Gait velocity was highest in the control group followed by the BPO and LGP group and differed between groups at both walking speeds (F2.57 = 13.62, P < .001 and F2.57 = 9.09, P < .001, for preferred and fastest speed condition, resp.). When normalized against gait velocity, the LGP group generated significantly less lateral force at the fastest walking speed (P = .005) and significantly less posterior force at both walking speeds (P †.01) compared to the control group. Conclusions. Pain intensity and distribution differentially influence gait velocity and hGRFs during gait. Those with referred leg pain tend to utilize significantly altered gait strategies that are more apparent at faster walking speeds
Right-veering diffeomorphisms of compact surfaces with boundary II
We continue our study of the monoid of right-veering diffeomorphisms on a
compact oriented surface with nonempty boundary, introduced in [HKM2]. We
conduct a detailed study of the case when the surface is a punctured torus; in
particular, we exhibit the difference between the monoid of right-veering
diffeomorphisms and the monoid of products of positive Dehn twists, with the
help of the Rademacher function. We then generalize to the braid group B_n on n
strands by relating the signature and the Maslov index. Finally, we discuss the
symplectic fillability in the pseudo-Anosov case by comparing with the work of
Roberts [Ro1,Ro2].Comment: 25 pages, 5 figure
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
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
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
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