Relevant bibliographies by topics / Link-homotopy

Academic literature on the topic 'Link-homotopy'

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Published: 7 July 2024
Last updated: 7 July 2024

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Contents

  1. Journal articles
  2. Dissertations / Theses
  3. Book chapters
  4. Conference papers

Journal articles on the topic "Link-homotopy":

1

Koschorke, U. "Link homotopy." Proceedings of the National Academy of Sciences 88, no. 1 (January 1, 1991): 268–70. http://dx.doi.org/10.1073/pnas.88.1.268.

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Kaiser, Uwe. "Link homotopy in ℝ3andS3." Pacific Journal of Mathematics 151, no. 2 (December 1, 1991): 257–64. http://dx.doi.org/10.2140/pjm.1991.151.257.

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HUGHES, JAMES R. "LINK HOMOTOPY INVARIANT QUANDLES." Journal of Knot Theory and Its Ramifications 20, no. 05 (May 2011): 763–73. http://dx.doi.org/10.1142/s0218216511008930.

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We consider several approaches to defining a link homotopy version of the fundamental quandle Q(L) of a link L in S3. We first define the reduced fundamental quandle RQ(L) as a quotient of Q(L). We show that RQ(L) is a link homotopy invariant that carries at least as much information as the meridian-preserving isomorphism class of Milnor's reduced group RG(L). We then show that operator reduction, a plausible alternative approach to defining RQ(L), fails to yield a link homotopy invariant. Finally, we give a geometric characterization of RQ(L), and offer a caveat regarding a seemingly simpler approach.
4

MELLOR, BLAKE. "FINITE TYPE LINK HOMOTOPY INVARIANTS." Journal of Knot Theory and Its Ramifications 08, no. 06 (September 1999): 773–87. http://dx.doi.org/10.1142/s0218216599000481.

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In [2], Bar-Natan used unitrivalent diagrams to show that finite type invariants classify string links up to homotopy. In this paper, I will construct the correct spaces of chord diagrams and unitrivalent diagrams for links up to homotopy. I will use these spaces to show that, far from classifying links up to homotopy, the only rational finite type invariants of link homotopy are the linking numbers of the components.
5

HUGHES, JAMES R. "STRUCTURED GROUPS AND LINK-HOMOTOPY." Journal of Knot Theory and Its Ramifications 02, no. 01 (March 1993): 37–63. http://dx.doi.org/10.1142/s0218216593000040.

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We study link-homotopy classes of links in the three sphere using reduced groups endowed with peripheral structures derived from meridian-longitude pairs. Two types of peripheral structures are considered — Milnor’s original version (called “pre-peripheral structures” in Levine’s terminology) and Levine’s refinement (called simply “peripheral structures”). We show here that pre-peripheral structures are not strong enough to classify links up to link-homotopy, and that Levine’s peripheral structures, although strong enough to distinguish those classes not distinguished by pre-peripheral structures, are also in all likelihood not strong enough to distinguish all link-homotopy classes. Following Levine’s classification program, we compare structure-preserving and realizable automorphisms, using an obstruction-theoretic approach suggested by work of Habegger and Lin. We find that these automorphism groups are in general different, so that a more complex program for classification by structured groups is required.
6

BAR-NATAN, DROR. "VASSILIEV HOMOTOPY STRING LINK INVARIANTS." Journal of Knot Theory and Its Ramifications 04, no. 01 (March 1995): 13–32. http://dx.doi.org/10.1142/s021821659500003x.

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We investigate Vassiliev homotopy invariants of string links, and find that in this particular case, most of the questions left unanswered in [3] can be answered affirmatively, In particular, Vassiliev invariants classify string links up to homotopy, and all Vassiliev homotopy string link invariants come from marked surfaces as in [3], using the same construction that in the case of knots gives the HOMFLY and Kauffman polynomials. In addition, the Milnor μ invariants of string links are shown to be Vassiliev invariants, and it is re-proven, by elementary means, that Vassiliev invariants classify braids.
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Koschorke, Ulrich. "Link homotopy with many components." Topology 30, no. 2 (1991): 267–81. http://dx.doi.org/10.1016/0040-9383(91)90013-t.

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Elhamdadi, Mohamed, Minghui Liu, and Sam Nelson. "Quasi-trivial quandles and biquandles, cocycle enhancements and link-homotopy of pretzel links." Journal of Knot Theory and Its Ramifications 27, no. 11 (October 2018): 1843007. http://dx.doi.org/10.1142/s0218216518430071.

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We investigate some algebraic structures called quasi-trivial quandles and we use them to study link-homotopy of pretzel links. Precisely, a necessary and sufficient condition for a pretzel link with at least two components being trivial under link-homotopy is given. We also generalize the quasi-trivial quandle idea to the case of biquandles and consider enhancement of the quasi-trivial biquandle cocycle counting invariant by quasi-trivial biquandle cocycles, obtaining invariants of link-homotopy type of links analogous to the quasi-trivial quandle cocycle invariants in Inoue’s paper [A. Inoue, Quasi-triviality of quandles for link-homotopy, J. Knot Theory Ramifications 22(6) (2013) 1350026, doi:10.1142/S0218216513500260, MR3070837].
9

Lightfoot, Ash. "On invariants of link maps in dimension four." Journal of Knot Theory and Its Ramifications 25, no. 11 (October 2016): 1650060. http://dx.doi.org/10.1142/s0218216516500607.

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We fill a gap in the proof that the proposed link homotopy invariant [Formula: see text] of Li is well defined. It is also shown that if the homotopy invariant [Formula: see text] of Schneiderman–Teichner is to be adapted to a link homotopy invariant of link maps, the result coincides with [Formula: see text].
10

Lightfoot, Ash. "Detecting Whitney disks for link maps in the four-sphere." Journal of Knot Theory and Its Ramifications 26, no. 12 (October 2017): 1750077. http://dx.doi.org/10.1142/s0218216517500778.

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It is an open problem whether Kirk’s [Formula: see text]-invariant is the complete obstruction to a link map [Formula: see text] being link homotopic to the trivial link. The link homotopy invariant associates to such a link map [Formula: see text] a pair [Formula: see text], and we write [Formula: see text]. With the objective of constructing counterexamples, Li proposed a link homotopy invariant [Formula: see text] such that [Formula: see text] is defined on the kernel of [Formula: see text] and which also obstructs link null-homotopy. We show that, when defined, the invariant [Formula: see text] is determined by [Formula: see text], and is strictly weaker. In particular, this implies that if a link map [Formula: see text] has [Formula: see text], then after a link homotopy the self-intersections of [Formula: see text] may be equipped with framed, immersed Whitney disks in [Formula: see text] whose interiors are disjoint from [Formula: see text].
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Journal articles

Dissertations / Theses on the topic "Link-homotopy":

1

Fleming, Thomas R. "Generalized link homotopy invariants." Connect to a 24 p. preview or request complete full text in PDF format. Access restricted to UC campuses, 2006. http://wwwlib.umi.com/cr/ucsd/fullcit?p3208096.

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Thesis (Ph. D.)--University of California, San Diego, 2006.
Title from first page of PDF file (viewed June 2, 2006). Available via ProQuest Digital Dissertations. Vita. Includes bibliographical references (p. 72-75).
2

Bartels, Arthur C. "Link homotopy in codimension two /." Diss., Connect to a 24 p. preview or request complete full text in PDF format. Access restricted to UC campuses, 1999. http://wwwlib.umi.com/cr/ucsd/fullcit?p9936836.

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3

Graff, Emmanuel. ""Link-homotopy" in low dimensional topology." Electronic Thesis or Diss., Normandie, 2023. http://www.theses.fr/2023NORMC244.

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Ce mémoire explore la topologie de basse dimension, en mettant l'accent sur la théorie des nœuds. Une théorie consacrée à l'étude des nœuds tels qu'ils sont communément compris : des morceaux de ficelle enroulés dans l'espace ou, de manière plus générale, des entrelacs formés en prenant plusieurs bouts de ficelle. Les nœuds et les entrelacs sont étudiés à déformation près, par exemple, à isotopie près, ce qui implique des manipulations sans couper ni faire passer la ficelle à travers elle-même. Cette thèse explore la link-homotopie, une relation d'équivalence plus souple où des composantes distinctes demeurent séparées, mais où une composante donnée peut s'auto-intersecter. La théorie des claspers, des puissants outils de chirurgie, est développée à link-homotopie près. Leur utilisation permet une démonstration géométrique de la classification des entrelacs avec 4 composantes ou moins à link-homotopie près. Une attention particulière est ensuite accordée aux tresses, des objets mathématiques apparentés aux nœuds et aux entrelacs. Il est montré que le groupe de tresses homotopiques est linéaire, c'est-à-dire isomorphe à un sous-groupe de matrices. De nouvelles présentations de ce groupe sont également proposées. Enfin, il est établi que le groupe de tresse homotopique est sans torsion, quel que soit le nombre de composantes. Ce dernier résultat s'appuie sur le contexte plus large de la théorie des nœuds soudés
This thesis explores low-dimensional topology, with a focus on knot theory. Knot theory is dedicated to the study of knots as commonly understood: a piece of string tied in space or, more generally, links formed by taking several pieces of string. Knots and links are studied up to deformation, for example, up to isotopy, which involves manipulations that do not require cutting or passing the string through itself. This thesis explores link-homotopy, a more flexible equivalence relation where distinct components remain disjoint, but a single component can self-intersect. The theory of claspers, powerful tools of surgery, is developed up to link-homotopy. Their use allows for a geometric proof of the classification of links with 4 components or less up to link-homotopy. Special attention is then given to braids, mathematical objects related to knots and links. It is shown that the homotopy braid group is linear, meaning it is faithfully represented by a subgroup of matrices. New group presentations are also proposed. Finally, it is established that the homotopy braid group is torsion-free for any number of components. This last result draws upon the broader context of welded knot theory
4

Zhang, Melissa. "Localization for Khovanov homologies:." Thesis, Boston College, 2019. http://hdl.handle.net/2345/bc-ir:108470.

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Thesis advisor: Julia Elisenda Grigsby
Thesis advisor: David Treumann
In 2010, Seidel and Smith used their localization framework for Floer homologies to prove a Smith-type rank inequality for the symplectic Khovanov homology of 2-periodic links in the 3-sphere. Hendricks later used similar geometric techniques to prove analogous rank inequalities for the knot Floer homology of 2-periodic links. We use combinatorial and space-level techniques to prove analogous Smith-type inequalities for various flavors of Khovanov homology for periodic links in the 3-sphere of any prime periodicity. First, we prove a graded rank inequality for the annular Khovanov homology of 2-periodic links by showing grading obstructions to longer differentials in a localization spectral sequence. We remark that the same method can be extended to p-periodic links. Second, in joint work with Matthew Stoffregen, we construct a Z/p-equivariant stable homotopy type for odd and even, annular and non-annular Khovanov homologies, using Lawson, Lipshitz, and Sarkar's Burnside functor construction of a Khovanov stable homotopy type. Then, we identify the fixed-point sets and apply a version of the classical Smith inequality to obtain spectral sequences and rank inequalities relating the Khovanov homology of a periodic link with the annular Khovanov homology of the quotient link. As a corollary, we recover a rank inequality for Khovanov homology conjectured by Seidel and Smith's work on localization and symplectic Khovanov homology
Thesis (PhD) — Boston College, 2019
Submitted to: Boston College. Graduate School of Arts and Sciences
Discipline: Mathematics

Book chapters on the topic "Link-homotopy":

1

Koschorke, Ulrich. "Higher order homotopy invariants for higher dimensional link maps." In Lecture Notes in Mathematics, 116–29. Berlin, Heidelberg: Springer Berlin Heidelberg, 1985. http://dx.doi.org/10.1007/bfb0074427.

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Mobahi, Hossein, and John W. Fisher. "On the Link between Gaussian Homotopy Continuation and Convex Envelopes." In Lecture Notes in Computer Science, 43–56. Cham: Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-14612-6_4.

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Azizi, Tahmineh. "Using Homotopy Link Function with Lipschitz Threshold in Studying Synchronized Fluctuations in Hierarchical Models." In Springer Proceedings in Mathematics & Statistics, 75–95. Cham: Springer International Publishing, 2023. http://dx.doi.org/10.1007/978-3-031-25225-9_4.

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4

Grigoriev, D., and A. Slissenko. "Computing minimum-link path in a homotopy class amidst semi-algebraic obstacles in the plane." In Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, 114–29. Berlin, Heidelberg: Springer Berlin Heidelberg, 1997. http://dx.doi.org/10.1007/3-540-63163-1_9.

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5

"Link homotopy in simply connected 3-manifolds." In AMS/IP Studies in Advanced Mathematics, 118–22. Providence, Rhode Island: American Mathematical Society, 1996. http://dx.doi.org/10.1090/amsip/002.1/07.

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Waldhausen, Friedhelm, Bjørn Jahren, and John Rognes. "Introduction." In Spaces of PL Manifolds and Categories of Simple Maps (AM-186). Princeton University Press, 2013. http://dx.doi.org/10.23943/princeton/9780691157757.003.0001.

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This book presents a proof of the stable parametrized h-cobordism theorem, which deals with the existence of a natural homotopy equivalence for each compact CAT manifold. In this theorem, a stable CAT h-cobordism space is defined in terms of manifolds, whereas a CAT Whitehead space is defined in terms of algebraic K-theory. This is a stable range extension to parametrized families of the classical hand s-cobordism theorems first stated by A. E. Hatcher, but his proofs were incomplete. This book provides a full proof of this key result, which provides the link between the geometric topology of high-dimensional manifolds and their automorphisms, as well as the algebraic K-theory of spaces and structured ring spectra.

Conference papers on the topic "Link-homotopy":

1

Dhingra, Anoop K., Jyun-Cheng Cheng, and Dilip Kohli. "Complete Solutions to Synthesis of Six-Link, Slider-Crank and Four-Link Mechanisms for Function, Path and Motion Generation Using Homotopy With M-Homogenization." In ASME 1992 Design Technical Conferences. American Society of Mechanical Engineers, 1992. http://dx.doi.org/10.1115/detc1992-0308.

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Abstract This paper presents complete solutions to the function, motion and path generation problems of Watt’s and Stephenson six-link, slider-crank and four-link mechanisms using homotopy methods with m-homogenization. It is shown that using the matrix method for synthesis, applying m-homogeneous group theory, and by defining compatibility equations in addition to the synthesis equations, the number of homotopy paths to be tracked can be drastically reduced. For Watt’s six-link function generators with 6 thru 11 precision positions, the number of homotopy paths to be tracked in obtaining all possible solutions range from 640 to 55,050,240. For Stephenson-II and -III mechanisms these numbers vary from 640 to 412,876,800. For 6, 7 and 8 point slider-crank path generation problems, the number of paths to be tracked are 320, 3840 and 17,920, respectively, whereas for four-link path generators with 6 thru 8 positions these numbers range from 640 to 71,680. It is also shown that for body guidance problems of slider-crank and four-link mechanisms, the number of homotopy paths to be tracked is exactly same as the maximum number of possible solutions given by the Burmester-Ball theories. Numerical results of synthesis of slider-crank path generators for 8 precision positions and six-link Watt and Stephenson-III function generators for 9 prescribed positions are also presented.
2

Dhingra, A. K., and M. Zhang. "Multiply Separated Synthesis of Six-Link Mechanisms Using Parallel Homotopy With M-Homogenization." In ASME 1998 Design Engineering Technical Conferences. American Society of Mechanical Engineers, 1998. http://dx.doi.org/10.1115/detc98/mech-5929.

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Abstract This paper presents complete solutions to the function generation problem of six-link Watt and Stephenson mechanisms, with multiply separated precision positions (PP), using homotopy methods with m-homogenization. It is seen that using the matrix method for synthesis, applying m-homogeneous group theory and by defining auxiliary equations in addition to the synthesis equations, the number of homotopy paths to be tracked in obtaining all possible solutions to the synthesis problem can be drastically reduced. Numerical work dealing with the synthesis of Watt and Stephenson mechanisms for 6 and 9 multiply separated precision points is presented. For both mechanisms, it is seen that complete solutions for 6 and 9 precision points can be obtained by tracking 640 and 286,720 paths, respectively. A parallel implementation of homotopy methods on the Connection Machine on which several thousand homotopy paths can be tracked concurrently is also discussed.
3

Tsai, L. W., and J. J. Lu. "Coupler-Point-Curve Synthesis Using Homotopy Methods." In ASME 1989 Design Technical Conferences. American Society of Mechanical Engineers, 1989. http://dx.doi.org/10.1115/detc1989-0154.

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Abstract A numerical method called “Homotopy Method” (or Continuation Method) is applied to the problem of four-bar coupler-point-curve synthesis. We have shown that: for five precision points, the “General Homotopy method” can be applied to find the link lengths of a number of four-bar linkages, and for nine precision points, a heuristic “Cheater’s Homotopy” can be applied to find some four-bar linkages. The nine-coupler-points synthesis problem is highly non-linear and highly singular. We have found that Newton-Raphson’s method and Powell’s method tend to converge to the singular solutions or do not converge at all, while the Cheater’s Homotopy always finds some non-singular solutions although sometimes the solutions may be complex.
4

Baskar, Aravind, and Mark Plecnik. "Synthesis of Stephenson III Timed Curve Generators Using a Probabilistic Continuation Method." In ASME 2019 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2019. http://dx.doi.org/10.1115/detc2019-98136.

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Abstract The kinematic synthesis equations of fairly simple planar linkage topologies are vastly nonlinear. This indicates that a large number of solutions exist, and hence a large number of design candidates might be present. Recent algorithms based in polynomial homotopy continuation have enabled the computation of entire solution sets that were previously not possible. These algorithms are based on a technique that stochastically accumulates finite roots and guarantees the exclusion of infinite roots. Here we apply the Cyclic Coefficient Parameter Continuation (CCPC) method to obtain for the first time the complete solution of a Stephenson III six-bar that traces a path and coordinates the angle of its input link along that path. Linkages of this type, called timed curve generators, are particularly useful for controlling the motion of an end effector point and influencing its transmission properties from a rotary input. For a numerically general version of the synthesis equations, we computed an approximately complete set of 1,017,708 solutions that divides into subsets of four according to the Stephenson III cognate structure. This numerically generic solution set essentially represents a design tool. It can be used in conjunction with a parameter homotopy to efficiently obtain all isolated roots of other systems of this same structure that correspond to a specific synthesis task. This is demonstrated with two example synthesis tasks.

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