A combinatorial algorithm for visualizing representatives with minimal self-intersection

Previously we defined an operation [Formula: see text] that generalizes Turaev’s cobracket for loops on a surface. We showed that, in contrast to the cobracket, this operation gives a formula for the minimum number of self-intersections of a loop in a given free homotopy class. In this paper, we consider the corresponding question for virtual strings, and conjecture that [Formula: see text] gives a formula for the minimum number of self-intersection points of a virtual string in a given virtual homotopy class. To support the conjecture, we show that [Formula: see text] gives a bound on the minimal self-intersection number of a virtual string which is stronger than a bound given by Turaev’s virtual string cobracket. We also use Turaev’s based matrices to describe a large set of strings [Formula: see text] such that [Formula: see text] gives a formula for the minimal self-intersection number [Formula: see text]. Finally, we compare the bound given by [Formula: see text] to a bound given by Turaev’s based matrix invariant [Formula: see text], and construct an example that shows the bound on the minimal self-intersection number given by [Formula: see text] is sometimes stronger than the bound [Formula: see text].

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Representing Homology Classes on Surfaces

Canadian Mathematical Bulletin ◽

10.4153/cmb-1976-058-4 ◽

1976 ◽

Vol 19 (3) ◽

pp. 373-374 ◽

Cited By ~ 2

Author(s):

James A. Schafer

Keyword(s):

Unit Circle ◽

Homotopy Class ◽

Simple Closed Curve ◽

Closed Curve ◽

Integral Basis

Let T2 = S1×S1, where S1 is the unit circle, and let {α, β} be the integral basis of H1(T2) induced by the 2 S1-factors. It is well known that 0 ≠ X = pα + qβ is represented by a simple closed curve (i.e. the homotopy class αppq contains a simple closed curve) if and only if gcd(p, q) = 1. It is the purpose of this note to extend this theorem to oriented surfaces of genus g.

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Symplectic deformations of Floer homology and non-contractible periodic orbits in twisted disc bundles

Communications in Contemporary Mathematics ◽

10.1142/s0219199719500846 ◽

2019 ◽

Vol 23 (01) ◽

pp. 1950084

Author(s):

Wenmin Gong

Keyword(s):

Magnetic Field ◽

Periodic Orbits ◽

Homotopy Class ◽

Linear Growth ◽

Floer Homology ◽

Hamiltonian Function ◽

Universal Cover ◽

Zero Section ◽

Free Homotopy Class ◽

The Given

In this paper, we establish the existence of periodic orbits belonging to any [Formula: see text]-atoroidal free homotopy class for Hamiltonian systems in the twisted disc bundle, provided that the compactly supported time-dependent Hamiltonian function is sufficiently large over the zero section and the magnitude of the weakly exact [Formula: see text]-form [Formula: see text] admitting a primitive with at most linear growth on the universal cover is sufficiently small. The proof relies on showing the invariance of Floer homology under symplectic deformations and on the computation of Floer homology for the cotangent bundle endowed with its canonical symplectic form. As a consequence, we also prove that, for any non-trivial atoroidal free homotopy class and any positive finite interval, if the magnitude of a magnetic field admitting a primitive with at most linear growth on the universal cover is sufficiently small, the twisted geodesic flow associated to the magnetic field has a periodic orbit on almost every energy level in the given interval whose projection to the underlying manifold represents the given free homotopy class. This application is carried out by showing the finiteness of the restricted Biran–Polterovich–Salamon capacity.

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Typical representatives of free homotopy classes in multi-punctured plane

Journal of Topology and Analysis ◽

10.1142/s1793525319500262 ◽

2019 ◽

Vol 11 (03) ◽

pp. 623-659

Author(s):

Maxim Arnold ◽

Yuliy Baryshnikov ◽

Yuriy Mileyko

Keyword(s):

Monte Carlo ◽

Markov Chain ◽

Markov Chain Monte Carlo ◽

Probability Measure ◽

Homotopy Class ◽

Piecewise Linear ◽

Homotopy Classes ◽

Simple Method ◽

Monte Carlo Techniques ◽

Free Homotopy Class

We show that a uniform probability measure supported on a specific set of piecewise linear loops in a nontrivial free homotopy class in a multi-punctured plane is overwhelmingly concentrated around loops of minimal lengths. Our approach is based on extending Mogulskii’s theorem to closed paths, which is a useful result of independent interest. In addition, we show that the above measure can be sampled using standard Markov Chain Monte Carlo techniques, thus providing a simple method for approximating shortest loops.

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ON LENGTH DISTORTIONS WITH RESPECT TO QUADRATIC DIFFERENTIAL METRICS

Glasgow Mathematical Journal ◽

10.1017/s001708951400010x ◽

2014 ◽

Vol 56 (3) ◽

pp. 681-689

Author(s):

ZONGLIANG SUN

Keyword(s):

Quasiconformal Mapping ◽

Homotopy Class ◽

Quadratic Differential ◽

Riemann Surfaces ◽

Simple Closed Curve ◽

Quasiconformal Mappings ◽

Closed Curve

AbstractIn this paper, we consider the question about length distortions under quasiconformal mappings with respect to quadratic differential metrics. More precisely, let X and Y be closed Riemann surfaces with genus at least 2, and f: X → Y being a K-quasiconformal mapping. Given two quadratic differential metrics |q1| and |q2| with unit areas on X and Y respectively, whether there exists a constant $\mathcal C$ depending only on K such that $\frac{1}{\mathcal C} l_{q_1} (\gamma) \leq l_{q_2} (f(\gamma)) \leq \mathcal C l_{q_1} (\gamma)$ holds for any simple closed curve γ ⊂ X. Here lqi(α) denotes the infimum of the lengths of curves in the homotopy class of α with respect to the metric |qi|, i = 1, 2. We give positive answers to this question, including the aspects that the desired constant ${\mathcal C}$ explicitly depends on q1, q2 and K, and that the constant $\mathcal C$ is universal for all the quantities involved.

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Smooth and Topological Almost Concordance

International Mathematics Research Notices ◽

10.1093/imrn/rnx338 ◽

2018 ◽

Vol 2019 (23) ◽

pp. 7324-7355 ◽

Cited By ~ 1

Author(s):

Matthias Nagel ◽

Patrick Orson ◽

JungHwan Park ◽

Mark Powell

Keyword(s):

Homotopy Class ◽

Infinite Family ◽

Lens Space ◽

Free Homotopy Class ◽

Concordance Classes

Abstract We investigate the disparity between smooth and topological almost concordance of knots in general 3-manifolds Y. Almost concordance is defined by considering knots in Y modulo concordance in Y × [0, 1] and the action of the concordance group of knots in S3 that ties in local knots. We prove that the trivial free homotopy class in every 3-manifold other than the 3-sphere contains an infinite family of knots, all topologically concordant, but not smoothly almost concordant to one another. Then, in every lens space and for every free homotopy class, we find a pair of topologically concordant but not smoothly almost concordant knots. Finally, as a topological counterpoint to these results, we show that in every lens space every free homotopy class contains infinitely many topological almost concordance classes.

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MAPS OF SURFACE GROUPS TO FINITE GROUPS WITH NO SIMPLE LOOPS IN THE KERNEL

Journal of Knot Theory and Its Ramifications ◽

10.1142/s021821650000061x ◽

2000 ◽

Vol 09 (08) ◽

pp. 1029-1036 ◽

Cited By ~ 1

Author(s):

CHARLES LIVINGSTON

Keyword(s):

Finite Group ◽

Finite Groups ◽

Upper Bound ◽

Simple Closed Curve ◽

Orientable Surface ◽

Closed Curve ◽

Surface Groups ◽

Closed Orientable Surface

Let Fg denote the closed orientable surface of genus g. What is the least order finite group, Gg, for which there is a homomorphism ψ:π1(Fg)→Gg so that nontrivial simple closed curve on Fg represents an element in Ker (ψ)? For the torus it is easily seen that G1=Z2×Z2 suffices. We prove here that G2 is a group of order 32 and that an upper bound for the order of Gg is given by g2g+1. The previously known upper bound was greater than 2g22g.

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Classification and rigidity of totally periodic pseudo-Anosov flows in graph manifolds

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2014.9 ◽

2014 ◽

Vol 35 (6) ◽

pp. 1681-1722 ◽

Cited By ~ 2

Author(s):

THIERRY BARBOT ◽

SÉRGIO R. FENLEY

Keyword(s):

Homotopy Class ◽

Closed Orbit ◽

Dehn Surgery ◽

Anosov Flow ◽

Free Homotopy Class ◽

Graph Manifold ◽

Finite Power ◽

Graph Manifolds ◽

Anosov Flows ◽

Regular Fibers

In this article we analyze totally periodic pseudo-Anosov flows in graph 3-manifolds. This means that in each Seifert fibered piece of the torus decomposition, the free homotopy class of regular fibers has a finite power which is also a finite power of the free homotopy class of a closed orbit of the flow. We show that each such flow is topologically equivalent to one of the model pseudo-Anosov flows which we previously constructed in Barbot and Fenley (Pseudo-Anosov flows in toroidal manifolds.Geom. Topol. 17(2013), 1877–1954). A model pseudo-Anosov flow is obtained by glueing standard neighborhoods of Birkhoff annuli and perhaps doing Dehn surgery on certain orbits. We also show that two model flows on the same graph manifold are isotopically equivalent (i.e. there is a isotopy of$M$mapping the oriented orbits of the first flow to the oriented orbits of the second flow) if and only if they have the same topological and dynamical data in the collection of standard neighborhoods of the Birkhoff annuli.

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Explicit solutions of certain orientable quadratic equations in free groups

International Journal of Algebra and Computation ◽

10.1142/s0218196719500589 ◽

2019 ◽

Vol 29 (08) ◽

pp. 1451-1466

Author(s):

D. Gonçalves ◽

T. Nasybullov

Keyword(s):

Free Group ◽

Homotopy Class ◽

Explicit Solution ◽

Orientable Surface ◽

Explicit Solutions ◽

Natural Homomorphism ◽

Quadratic Equations ◽

Wecken Property ◽

Number Formula ◽

Genus Formula

For [Formula: see text] denote by [Formula: see text] the free group on [Formula: see text] generators and let [Formula: see text]. For [Formula: see text] and elements [Formula: see text], we study orientable quadratic equations of the form [Formula: see text] with unknowns [Formula: see text] and provide explicit solutions for them for the minimal possible number [Formula: see text]. In the particular case when [Formula: see text], [Formula: see text] for [Formula: see text] and [Formula: see text] the minimal number which satisfies [Formula: see text], we provide two types of solutions depending on the image of the subgroup [Formula: see text] generated by the solution under the natural homomorphism [Formula: see text]: the first solution, which is called a primitive solution, satisfies [Formula: see text], the second solution satisfies [Formula: see text]. We also provide an explicit solution of the equation [Formula: see text] for [Formula: see text] in [Formula: see text], and prove that if [Formula: see text], then every solution of this equation is primitive. As a geometrical consequence, for every solution, we obtain a map [Formula: see text] from the orientable surface [Formula: see text] of genus [Formula: see text] to the torus [Formula: see text] which has the minimal number of roots among all maps from the homotopy class of [Formula: see text]. Depending on the number [Formula: see text], such maps have fundamentally different geometric properties: in some cases, they satisfy the Wecken property and in other cases not.

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