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Dušan Repovš - One of the best experts on this subject based on the ideXlab platform.
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n-QUASI-Isotopy: II. COMPARISON
Journal of Knot Theory and Its Ramifications, 2005Co-Authors: Sergey A. Melikhov, Dušan RepovšAbstract:Geometric aspects of the filtration on classical links by k-quasi-Isotopy are discussed, including the effect of Whitehead doubling, relations with Smythe's n-splitting and Kobayashi's k-contractibility. One observation is: ω-quasi-Isotopy is equivalent to PL Isotopy for links in a homotopy 3-sphere (respectively, contractible open 3-manifold) M if and only if M is homeomorphic to S3 (respectively, ℝ3). As a byproduct of the proof of the "if" part, we obtain that every compact subset of an acyclic open set in a compact orientable 3-manifold M is contained in a PL homology 3-ball in M. We show that k-quasi-Isotopy implies (k + 1)-cobordism of Cochran and Orr. If zm-1 (c0+c1z2+⋯+cnz2n) denotes the Conway polynomial of an m-component link, it follows that the residue class of ck modulo gcd(c0,…,ck-1) is invariant under k-quasi-Isotopy. Another corollary is that each Cochran's derived invariant βk is also invariant under k-quasi-Isotopy, and therefore assumes the same value on all PL links, sufficiently C0-close to a given topological link. This overcomes an algebraic obstacle encountered by Kojima and Yamasaki, who "became aware of impossibility to define" for wild links what for PL links is equivalent to the formal power series ∑βnzn by a change of variable.
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n-QUASI-Isotopy I: QUESTIONS OF NILPOTENCE
Journal of Knot Theory and Its Ramifications, 2005Co-Authors: Sergey A. Melikhov, Dušan RepovšAbstract:It is well-known that no knot can be cancelled in a connected sum with another knot, whereas every link can be cancelled up to link homotopy in a (componentwise) connected sum with another link. In this paper we address the question whether the noncancellation property of knots holds for (piecewise-linear) links up to some stronger analogue of link homotopy, which still does not distinguish between sufficiently close C0-approximations of a topological link. We introduce a sequence of such increasingly stronger equivalence relations under the name of k-quasi-Isotopy, k∈ℕ; all of them are weaker than Isotopy (in the sense of Milnor). We prove that every link can be cancelled up to peripheral structure preserving isomorphism of any quotient of the fundamental group, functorially invariant under k-quasi-Isotopy; functoriality means that the isomorphism between the quotients for links related by any allowable crossing change fits in the commutative diagram with the fundamental group of the complement to the intermediate singular link. The proof invokes Baer's theorem on the join of subnormal locally nilpotent subgroups. On the other hand, the integral generalized (lk ≠ 0) Sato–Levine invariant is invariant under 1-quasi-Isotopy, but is not determined by any quotient of the fundamental group (endowed with the peripheral structure), functorially invariant under 1-quasi-Isotopy — in contrast to Waldhausen's theorem. As a byproduct, we use to determine the image of the Kirk–Koschorke invariant of fibered link maps.
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n-Quasi-Isotopy: II. Comparison
arXiv: Geometric Topology, 2001Co-Authors: Sergey A. Melikhov, Dušan RepovšAbstract:Geometric aspects of the filtration on classical links by k-quasi-Isotopy are discussed, including the effect of Whitehead doubling, relations with Smythe's n-splitting and Kobayashi's k-contractibility. One observation is: \omega-quasi-Isotopy is equivalent to PL Isotopy for links in a homotopy 3-sphere (resp. contractible open 3-manifold) M if and only if M is homeomorphic to S^3 (resp. R^3). As a byproduct of the proof of the "if" part, we obtain that every compact subset of an acyclic open set in a compact orientable 3-manifold M is contained in a PL homology 3-ball in M. We show that k-quasi-Isotopy implies (k+1)-cobordism of Cochran and Orr. If z^{m-1}(c_0 + c_1 z^2 + ... + c_n z^{2n}) denotes the Conway polynomial of an m-component link, it follows that the residue class of c_k modulo gcd(c_0,..,c_{k-1}) is invariant under k-quasi-Isotopy. Another corollary is that each Cochran's derived invariant \beta^k is also invariant under k-quasi-Isotopy, and therefore assumes the same value on all PL links, sufficiently C^0-close to a given topological link. This overcomes an algebraic obstacle encountered by Kojima and Yamasaki, who "became aware of impossibility to define" for wild links what for PL links is equivalent to the formal power series \sum \beta^n z^n by a change of variable.
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n-Quasi-Isotopy: I. Questions of nilpotence
arXiv: Geometric Topology, 2001Co-Authors: Sergey A. Melikhov, Dušan RepovšAbstract:It is well-known that no knot can be cancelled in a connected sum with another knot, whereas every link can be cancelled up to link homotopy in a (componentwise) connected sum with another link. In this paper we address the question whether the noncancellation property of knots holds for some (piecewise-linear) links up to some stronger analogue of link homotopy, which still does not distinguish between sufficiently close C^0-approximations of a topological link. We introduce a sequence of such increasingly stronger equivalence relations under the name of k-quasi-Isotopy, k=1,2,...; all of them are weaker than Isotopy (in the sense of Milnor). We prove that every link can be cancelled up to peripheral structure preserving isomorphism of any quotient of the fundamental group, functorially invariant under k-quasi-Isotopy; functoriality means that the isomorphism between the quotients for links related by an allowable crossing change fits in the commutative diagram with the fundamental group of the complement to the intermediate singular link. The proof invokes Baer's theorem on the join of subnormal locally nilpotent subgroups. On the other hand, the integral generalized (lk\ne 0) Sato-Levine invariant \tilde\beta is invariant under 1-quasi-Isotopy, but is not determined by any quotient of the fundamental group (endowed with the peripheral structure), functorially invariant under 1-quasi-Isotopy - in contrast to Waldhausen's theorem. As a byproduct, we use \tilde\beta to determine the image of the Kirk-Koschorke invariant \tilde\sigma of fibered link maps.
Sergey A. Melikhov - One of the best experts on this subject based on the ideXlab platform.
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n-QUASI-Isotopy: II. COMPARISON
Journal of Knot Theory and Its Ramifications, 2005Co-Authors: Sergey A. Melikhov, Dušan RepovšAbstract:Geometric aspects of the filtration on classical links by k-quasi-Isotopy are discussed, including the effect of Whitehead doubling, relations with Smythe's n-splitting and Kobayashi's k-contractibility. One observation is: ω-quasi-Isotopy is equivalent to PL Isotopy for links in a homotopy 3-sphere (respectively, contractible open 3-manifold) M if and only if M is homeomorphic to S3 (respectively, ℝ3). As a byproduct of the proof of the "if" part, we obtain that every compact subset of an acyclic open set in a compact orientable 3-manifold M is contained in a PL homology 3-ball in M. We show that k-quasi-Isotopy implies (k + 1)-cobordism of Cochran and Orr. If zm-1 (c0+c1z2+⋯+cnz2n) denotes the Conway polynomial of an m-component link, it follows that the residue class of ck modulo gcd(c0,…,ck-1) is invariant under k-quasi-Isotopy. Another corollary is that each Cochran's derived invariant βk is also invariant under k-quasi-Isotopy, and therefore assumes the same value on all PL links, sufficiently C0-close to a given topological link. This overcomes an algebraic obstacle encountered by Kojima and Yamasaki, who "became aware of impossibility to define" for wild links what for PL links is equivalent to the formal power series ∑βnzn by a change of variable.
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n-QUASI-Isotopy I: QUESTIONS OF NILPOTENCE
Journal of Knot Theory and Its Ramifications, 2005Co-Authors: Sergey A. Melikhov, Dušan RepovšAbstract:It is well-known that no knot can be cancelled in a connected sum with another knot, whereas every link can be cancelled up to link homotopy in a (componentwise) connected sum with another link. In this paper we address the question whether the noncancellation property of knots holds for (piecewise-linear) links up to some stronger analogue of link homotopy, which still does not distinguish between sufficiently close C0-approximations of a topological link. We introduce a sequence of such increasingly stronger equivalence relations under the name of k-quasi-Isotopy, k∈ℕ; all of them are weaker than Isotopy (in the sense of Milnor). We prove that every link can be cancelled up to peripheral structure preserving isomorphism of any quotient of the fundamental group, functorially invariant under k-quasi-Isotopy; functoriality means that the isomorphism between the quotients for links related by any allowable crossing change fits in the commutative diagram with the fundamental group of the complement to the intermediate singular link. The proof invokes Baer's theorem on the join of subnormal locally nilpotent subgroups. On the other hand, the integral generalized (lk ≠ 0) Sato–Levine invariant is invariant under 1-quasi-Isotopy, but is not determined by any quotient of the fundamental group (endowed with the peripheral structure), functorially invariant under 1-quasi-Isotopy — in contrast to Waldhausen's theorem. As a byproduct, we use to determine the image of the Kirk–Koschorke invariant of fibered link maps.
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n-QUASI-Isotopy: III. ENGEL CONDITIONS
arXiv: Geometric Topology, 2003Co-Authors: Sergey A. Melikhov, Roman MikhailovAbstract:In part I it was shown that for each k � 1 the generalized Sato-Levine invariant detects a gap between k-quasi-Isotopy of link and peripheral structure pre- serving isomorphism of the finest quotient Gk of its fundamental group, 'functorially' invariant under k-quasi-Isotopy. Here we show that Cochran's derived invariantk , provided k � 3, and a series of ¯-invariants, starting with ¯(111112122) for k = 3, also fall in this gap. In fact, all ¯ µ-invariants where each index occurs at most k + 1 times, except perhaps for one occuring k + 2 times, can be extracted from Gk, and if they vanish, Gk is the same as that of the unlink. We also study the equivalence relation on links (called 'fine k-quasi-Isotopy') gen- erated by ambient Isotopy and the operation of interior connected sum with the second component of the (k + 1) th Milnor's link, where the complement to its first component is embedded into the link complement. We show that the finest quo- tient of the fundamental group, functorially invariant under fine k-quasi-Isotopy, is obtained from the fundamental group by forcing all meridians to be (k + 2)-Engel elements. We prove that any group generated by two 3-Engel elements has lower central series of length � 5.
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n-Quasi-Isotopy: II. Comparison
arXiv: Geometric Topology, 2001Co-Authors: Sergey A. Melikhov, Dušan RepovšAbstract:Geometric aspects of the filtration on classical links by k-quasi-Isotopy are discussed, including the effect of Whitehead doubling, relations with Smythe's n-splitting and Kobayashi's k-contractibility. One observation is: \omega-quasi-Isotopy is equivalent to PL Isotopy for links in a homotopy 3-sphere (resp. contractible open 3-manifold) M if and only if M is homeomorphic to S^3 (resp. R^3). As a byproduct of the proof of the "if" part, we obtain that every compact subset of an acyclic open set in a compact orientable 3-manifold M is contained in a PL homology 3-ball in M. We show that k-quasi-Isotopy implies (k+1)-cobordism of Cochran and Orr. If z^{m-1}(c_0 + c_1 z^2 + ... + c_n z^{2n}) denotes the Conway polynomial of an m-component link, it follows that the residue class of c_k modulo gcd(c_0,..,c_{k-1}) is invariant under k-quasi-Isotopy. Another corollary is that each Cochran's derived invariant \beta^k is also invariant under k-quasi-Isotopy, and therefore assumes the same value on all PL links, sufficiently C^0-close to a given topological link. This overcomes an algebraic obstacle encountered by Kojima and Yamasaki, who "became aware of impossibility to define" for wild links what for PL links is equivalent to the formal power series \sum \beta^n z^n by a change of variable.
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n-Quasi-Isotopy: I. Questions of nilpotence
arXiv: Geometric Topology, 2001Co-Authors: Sergey A. Melikhov, Dušan RepovšAbstract:It is well-known that no knot can be cancelled in a connected sum with another knot, whereas every link can be cancelled up to link homotopy in a (componentwise) connected sum with another link. In this paper we address the question whether the noncancellation property of knots holds for some (piecewise-linear) links up to some stronger analogue of link homotopy, which still does not distinguish between sufficiently close C^0-approximations of a topological link. We introduce a sequence of such increasingly stronger equivalence relations under the name of k-quasi-Isotopy, k=1,2,...; all of them are weaker than Isotopy (in the sense of Milnor). We prove that every link can be cancelled up to peripheral structure preserving isomorphism of any quotient of the fundamental group, functorially invariant under k-quasi-Isotopy; functoriality means that the isomorphism between the quotients for links related by an allowable crossing change fits in the commutative diagram with the fundamental group of the complement to the intermediate singular link. The proof invokes Baer's theorem on the join of subnormal locally nilpotent subgroups. On the other hand, the integral generalized (lk\ne 0) Sato-Levine invariant \tilde\beta is invariant under 1-quasi-Isotopy, but is not determined by any quotient of the fundamental group (endowed with the peripheral structure), functorially invariant under 1-quasi-Isotopy - in contrast to Waldhausen's theorem. As a byproduct, we use \tilde\beta to determine the image of the Kirk-Koschorke invariant \tilde\sigma of fibered link maps.
E. D. Tymchatyn - One of the best experts on this subject based on the ideXlab platform.
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Extension of isotopies in the plane.
arXiv: General Topology, 2018Co-Authors: L. C. Hoehn, Lex G. Oversteegen, E. D. TymchatynAbstract:Let $A$ be any plane set. It is known that a holomorphic motion $h: A \times \mathbb{D} \to \mathbb{C}$ always extends to a holomorphic motion of the entire plane. It was recently shown that any Isotopy $h: X \times [0,1] \to \mathbb{C}$, starting at the identity, of a plane continuum $X$ also extends to an Isotopy of the entire plane. Easy examples show that this result does not generalize to all plane compacta. In this paper we will provide a characterization of isotopies of uniformly perfect plane compacta $X$ which extend to an Isotopy of the entire plane. Using this characterization, we prove that such an extension is always possible provided the diameters of all components of $X$ are uniformly bounded away from zero.
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Extending isotopies of planar continua
Annals of Mathematics, 2010Co-Authors: Lex G. Oversteegen, E. D. TymchatynAbstract:In this paper we solve the following problem in the affirmative: Let Z be a continuum in the plane C and suppose that h: Z x [0,1] → C is an Isotopy starting at the identity. Can h be extended to an Isotopy of the plane? We will provide a new characterization of an accessible point in a planar continuum Z and use it to show that accessibility of a point is preserved during the Isotopy. We show next that the Isotopy can be extended over small hyperbolic crosscuts which are shown to remain small under the Isotopy. The proof makes use of the notion of a metric external ray, which mimics the notion of a conformal external ray, but is easier to control during an Isotopy. It also relies on the existence of a partition of a hyperbolic, simply connected domain U in the sphere, into hyperbolically convex subsets, which have limited distortion under conformal maps to the unit disk.
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Extending Isotopies of Planar Continua
arXiv: Geometric Topology, 2008Co-Authors: Lex G. Oversteegen, E. D. TymchatynAbstract:In this paper we solve the following problem in the affirmative: Let $Z$ be a continuum in the plane $\complex$ and suppose that $h:Z\times [0,1]\to\complex$ is an Isotopy starting at the identity. Can $h$ be extended to an Isotopy of the plane? We will provide a new characterization of an accessible point in a planar continuum $Z$ and use it to show that an accessible point is preserved during the Isotopy. We show next that the Isotopy can be extended over hyperbolic crosscuts. The proof makes use of the notion of a metric external ray, which mimics the notion of a conformal external ray, but is easier to control during an Isotopy.
Jingzhi Yan - One of the best experts on this subject based on the ideXlab platform.
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Existence of torsion-low maximal identity isotopies for area preserving surface homeomorphisms
Discrete & Continuous Dynamical Systems - A, 2018Co-Authors: Jingzhi YanAbstract:The paper concerns area preserving homeomorphisms of surfaces that are isotopic to the identity. The purpose of the paper is to find a maximal Isotopy such that we can give a fine description of the dynamics of its transverse foliation. We will define a sort of identity isotopies: torsion-low isotopies. In particular, when \begin{document} $f$ \end{document} is a diffeomorphism with finitely many fixed points such that every fixed point is not degenerate, an identity Isotopy \begin{document} $I$ \end{document} of \begin{document} $f$ \end{document} is torsion-low if and only if for every point \begin{document} $z$ \end{document} fixed along the Isotopy, the (real) rotation number \begin{document} $ρ(I, z)$ \end{document} (which is well defined when one blows up \begin{document} $f$ \end{document} at \begin{document} $z$ \end{document} ) is contained in \begin{document} $(-1, 1)$ \end{document} . We will prove the existence of torsion-low maximal isotopies, and will deduce the local dynamics of the transverse foliations of any torsion-low maximal Isotopy near any isolated singularity.
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Existence of torsion-low maximal identity isotopies for area preserving surface homeomorphisms
2014Co-Authors: Jingzhi YanAbstract:The paper concerns area preserving homeomorphisms of surfaces that are isotopic to the identity. The purpose of the paper is to find a maximal identity Isotopy such that we can give a fine descriptions of the dynamics of its transverse foliation. We will define a kind of identity isotopies: torsion-low isotopies. In particular, when $f$ is a diffeomorphism with finitely many fixed points such that every fixed point is not degenerate, an identity Isotopy $I$ of $f$ is torsion-low if and only if for every point $z$ fixed along the Isotopy, the (real) rotation number $\rho(I,z)$, which is well defined when one blows-up $f$ at $z$, is contained in $(-1,1)$. We will prove the existence of torsion-low maximal identity isotopies, and we will deduce the local dynamics of the transverse foliations of any torsion-low maximal Isotopy near any isolated singularity.
Philip Boyland - One of the best experts on this subject based on the ideXlab platform.
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Isotopy Stability of Dynamics on Surfaces
arXiv: Dynamical Systems, 1999Co-Authors: Philip BoylandAbstract:This paper investigates dynamics that persist under Isotopy in classes of orientation-preserving homeomorphisms of orientable surfaces. The persistence of periodic points with respect to periodic and strong Nielsen equivalence is studied. The existence of a dynamically minimal representative with respect to these relations is proved and necessary and sufficient conditions for the Isotopy stability of an equivalence class are given. It is also shown that most the dynamics of the minimal representative persist under Isotopy in the sense that any isotopic map has an invariant set that is semiconjugate to it.
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topological methods in surface dynamics
Topology and its Applications, 1994Co-Authors: Philip BoylandAbstract:Abstract This paper surveys applications of low-dimensional topology to the study of the dynamics of iterated homeomorphisms on surfaces. A unifying theme in the paper is the analysis and application of Isotopy stable dynamics, i.e. dynamics that are present in the appropriate sense in every homeomorphism in an Isotopy class. The first step in developing this theme is to assign coordinates to periodic orbits. These coordinates record the Isotopy, homotopy, or homology class of the corresponding orbit in the suspension flow. The Isotopy stable coordinates are then characterized, and it is shown that there is a map in each Isotopy class that has just these periodic orbits and no others. Such maps are called dynamically minimal representatives, and they turn out to have strong global Isotopy stability properties as maps. The main tool used in these results is the Thurston-Nielsen theory of Isotopy classes of homeomorphisms of surfaces. This theory is outlined and then applications of Isotopy stability results are given. These results are applied to the class rel a periodic orbit to reach conclusions about the complexity of the dynamics of a given homeomorphism. Another application is via dynamical partial orders, in which a periodic orbit with a given coordinate is said to dominate another when it always implies the existence of the other. Applications to rotation sets are also surveyed.