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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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2

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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3

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.
7

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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8

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].
11

Krushkal, Vyacheslav S., and Peter Teichner. "Alexander duality, gropes and link homotopy." Geometry & Topology 1, no. 1 (October 26, 1997): 51–69. http://dx.doi.org/10.2140/gt.1997.1.51.

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12

Cochran, Tim D. "Link concordance invariants and homotopy theory." Inventiones Mathematicae 90, no. 3 (October 1987): 635–45. http://dx.doi.org/10.1007/bf01389182.

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13

INOUE, AYUMU. "QUASI-TRIVIALITY OF QUANDLES FOR LINK-HOMOTOPY." Journal of Knot Theory and Its Ramifications 22, no. 06 (May 2013): 1350026. http://dx.doi.org/10.1142/s0218216513500260.

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We introduce the notion of quasi-triviality of quandles and define homology of quasi-trivial quandles. Quandle cocycle invariants are invariant under link-homotopy if they are associated with 2-cocycles of quasi-trivial quandles. We thus obtain a lot of numerical link-homotopy invariants.
14

KOTORII, YUKA. "THE MILNOR $\bar{\mu}$ INVARIANTS AND NANOPHRASES." Journal of Knot Theory and Its Ramifications 22, no. 02 (February 2013): 1250142. http://dx.doi.org/10.1142/s0218216512501428.

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Two link diagrams are link homotopic if one can be transformed into the other by a sequence of Reidemeister moves and self-crossing changes. Milnor introduced invariants under link homotopy called [Formula: see text]. Nanophrases, introduced by Turaev, generalize links. In this paper, we extend the notion of link homotopy to nanophrases. We also generalize [Formula: see text] to the set of those nanophrases that correspond to virtual links.
15

NAKANISHI, YASUTAKA, and YOSHIYUKI OHYAMA. "DELTA LINK HOMOTOPY FOR TWO COMPONENT LINKS, II." Journal of Knot Theory and Its Ramifications 11, no. 03 (May 2002): 353–62. http://dx.doi.org/10.1142/s0218216502001664.

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In this note, we will study Δ link homotopy (or self Δ-equivalence), which is an equivalence relation of ordered and oriented link types. Previously, a necessary condition is given in the terms of Conway polynomials for two link types to be Δ link homotopic. A pair of numerical invariants δ1 and δ2 classifies all (ordered and oriented) prime 2-component link types with seven crossings or less up to Δ link homotopy. We will show here that for any pair of integers n1 and n2 there exists a 2-component link κ such that δ1(κ) = n1 and δ2(κ) = n2 provided that at least one of n1 and n2 is even.
16

Dhingra, A. K., J. C. Cheng, and D. Kohli. "Synthesis of Six-link, Slider-crank and Four-link Mechanisms for Function, Path and Motion Generation Using Homotopy with m-homogenization." Journal of Mechanical Design 116, no. 4 (December 1, 1994): 1122–31. http://dx.doi.org/10.1115/1.2919496.

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This paper presents 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 auxiliary equations in addition to the synthesis equations, the number of homotopy paths to be tracked is drastically reduced. To synthesize a Watt’s six-link function generator for 6 through 11 precision positions, the number of homotopy paths to be tracked to obtain 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. It is shown that slider-crank path generation problems with 6, 7 and 8 prescribed positions require 320, 3840 and 17,920 paths to be tracked, respectively, whereas for four-link path generators with 6 through 8 specified positions, these numbers range from 640 to 71, 680. The number of homotopy paths to be tracked to body guidance problems of slider-crank and four-link mechanisms is exactly the same as the maximum number of possible solutions given by Burmester-Ball theories. Numerical examples dealing with the 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.
17

Koschorke, Ulrich, and Dale Rolfsen. "Higher-dimensional link operations and stable homotopy." Pacific Journal of Mathematics 139, no. 1 (September 1, 1989): 87–106. http://dx.doi.org/10.2140/pjm.1989.139.87.

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18

Kirk, Paul A. "Link homotopy with one codimension two component." Transactions of the American Mathematical Society 319, no. 2 (February 1, 1990): 663–88. http://dx.doi.org/10.1090/s0002-9947-1990-0970268-x.

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19

Habegger, N., and U. Kaiser. "Link homotopy in the 2-metastable range." Topology 37, no. 1 (January 1998): 75–94. http://dx.doi.org/10.1016/s0040-9383(97)00010-4.

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20

Koschorke, Ulrich. "On link maps and their homotopy classification." Mathematische Annalen 286, no. 1-3 (March 1990): 753–82. http://dx.doi.org/10.1007/bf01453601.

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21

Nakanishi, Yasutaka. "Delta link homotopy for two component links." Topology and its Applications 121, no. 1-2 (June 2002): 169–82. http://dx.doi.org/10.1016/s0166-8641(01)00116-x.

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22

Yang, Seung Yeop. "Extended quandle spaces and shadow homotopy invariants of classical links." Journal of Knot Theory and Its Ramifications 26, no. 03 (March 2017): 1741010. http://dx.doi.org/10.1142/s0218216517410103.

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In 1993, Fenn, Rourke and Sanderson introduced rack spaces and rack homotopy invariants, and modifications to quandle spaces and quandle homotopy invariants were introduced by Nosaka in 2011. In this paper, we define the Cayley-type graph and the extended quandle space of a quandle in analogy to rack and quandle spaces. Moreover, we construct the shadow homotopy invariant of a classical link and prove that the shadow homotopy invariant is equal to the quandle homotopy invariant multiplied by the order of a quandle.
23

Friedl, Stefan, and Mark Powell. "Homotopy ribbon concordance and Alexander polynomials." Archiv der Mathematik 115, no. 6 (October 20, 2020): 717–25. http://dx.doi.org/10.1007/s00013-020-01517-5.

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24

Kauffman, Louis H. "Simplicial homotopy theory, link homology and Khovanov homology." Journal of Knot Theory and Its Ramifications 27, no. 07 (June 2018): 1841002. http://dx.doi.org/10.1142/s021821651841002x.

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This paper shows how, in principle, simplicial methods, including the well-known Dold–Kan construction can be applied to convert link homology theories into homotopy theories. The paper studies particularly the case of Khovanov homology and shows how simplicial structures are implicit in the construction of the Khovanov complex from a link diagram and how the homology of the Khovanov category, with coefficients in an appropriate Frobenius algebra, is related to Khovanov homology. This Khovanov category leads to simplicial groups satisfying the Kan condition that are relevant to a homotopy theory for Khovanov homology.
25

NAKANISHI, Yasutaka, and Yoshiyuki OHYAMA. "Delta link homotopy for two component links, III." Journal of the Mathematical Society of Japan 55, no. 3 (July 2003): 641–54. http://dx.doi.org/10.2969/jmsj/1191418994.

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26

HUGHES, JAMES R. "DISTINGUISHING LINK-HOMOTOPY CLASSES BY PRE-PERIPHERAL STRUCTURES." Journal of Knot Theory and Its Ramifications 07, no. 07 (November 1998): 925–44. http://dx.doi.org/10.1142/s0218216598000498.

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An open problem in link-homotopy of links in S3 is classification using peripheral invariants, analogous to that of Waldhausen for links up to ambient isotopy. An approach to such a classification was outlined by Levine, but shown not to be feasible by the author. Here, we develop an approach to finding classification counterexamples. The approach requires non-injectivity of a group homomorphism that is completely determined by minimal-weight commutator numbers (equivalent to the first non-vanishing [Formula: see text] invariants of Milnor). For non-injectivity, the minimal-weight commutator numbers must all be non-zero, and satisfy a certain system of polynomials, which we compute for 4- and 5-component links.
27

Audoux, Benjamin, Jean‐Baptiste Meilhan, and Emmanuel Wagner. "On codimension two embeddings up to link‐homotopy." Journal of Topology 10, no. 4 (November 17, 2017): 1107–23. http://dx.doi.org/10.1112/topo.12041.

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28

Gui-Song, Li. "An invariant of link homotopy in dimension four." Topology 36, no. 4 (July 1997): 881–97. http://dx.doi.org/10.1016/s0040-9383(96)00034-1.

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29

Habegger, Nathan, and Xiao-Song Lin. "The classification of links up to link-homotopy." Journal of the American Mathematical Society 3, no. 2 (May 1, 1990): 389. http://dx.doi.org/10.1090/s0894-0347-1990-1026062-0.

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30

MELLOR, BLAKE, and DYLAN THURSTON. "On the Existence of Finite Type Link Homotopy Invariants." Journal of Knot Theory and Its Ramifications 10, no. 07 (November 2001): 1025–39. http://dx.doi.org/10.1142/s0218216501001323.

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We show that for links with at most 5 components, the only finite type homotopy invariants are products of the linking numbers. In contrast, we show that for links with at least 9 components, there must exist finite type homotopy invariants which are not products of the linking numbers. This corrects the errors of the first author in [11, 12].
31

Abel, Michael, and Lev Rozansky. "Virtual crossings and a filtration of the triply graded link homology of a link diagram." Journal of Knot Theory and Its Ramifications 26, no. 10 (September 2017): 1750052. http://dx.doi.org/10.1142/s0218216517500523.

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A filtration of Soergel bimodules by virtual crossing bimodules extends to Rouquier’s complexes associated with braid words. We show that these complexes are invariant up to filtered homotopy with respect to the second Reidemeister move, and the filtration of the triply graded link diagram homology, constructed by Khovanov through the application of the Hochschild homology, is invariant under Markov moves. We also prove that the homotopy equivalence of the complexes of braid words related by the third Reidemeister move violates filtration by at most two units.
32

Gallot, L., E. Pilon, and F. Thuillier. "Topological gauge fixing II: A homotopy formulation." Modern Physics Letters A 30, no. 20 (June 10, 2015): 1550102. http://dx.doi.org/10.1142/s0217732315501023.

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We revisit the implementation of the metric-independent Fock–Schwinger gauge in the Abelian Chern–Simons field theory defined in ℝ3 by means of a homotopy condition. This leads to the Lagrangian [Formula: see text] in terms of curvatures F and of the Poincaré homotopy operator h. The corresponding field theory provides the same link invariants as the Abelian Chern–Simons theory. Incidentally the part of the gauge field propagator which yields the link invariants of the Chern–Simons theory in the Fock–Schwinger gauge is recovered without any computation.
33

HUGHES, JAMES R. "FINITE TYPE LINK HOMOTOPY INVARIANTS OF k-TRIVIAL LINKS." Journal of Knot Theory and Its Ramifications 12, no. 03 (May 2003): 375–93. http://dx.doi.org/10.1142/s0218216503002524.

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In a recent paper [8], Xiao-Song Lin gave an example of a finite type invariant of links up to link homotopy that is not simply a polynomial in the pairwise linking numbers. Here we present a reformulation of the problem of finding such polynomials using the primary geometric obstruction homomorphism, previously used to study realizability of link group automorphisms by link homotopies. Using this reformulation, we generalize Lin's results to k-trivial links (links that become homotopically trivial when any k components are deleted). Our approach also gives a method for finding torsion finite type link homotopy invariants within "linking classes," generalizing an idea explored earlier in [1] and [10], and yielding torsion invariants within linking classes that are different from Milnor's invariants in their original indeterminacy.
34

Wiest, Bert. "RACK SPACES AND LOOP SPACES." Journal of Knot Theory and Its Ramifications 08, no. 01 (February 1999): 99–114. http://dx.doi.org/10.1142/s0218216599000080.

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We prove that the rack and quandle spaces of links in 3-manifolds, considered only as topological spaces (disregarding their cubical structure), are closely related to certain subspaces of the loop spaces on the 3-manifold, which we call the vertical and the straight loop space of the link. Using these models we prove that the homotopy type of the non-augmented rack and quandle spaces of a link L in a 3-manifold M depends essentially only on the homotopy type of the pair (M,M -L).
35

Songhafouo Tsopméné, Paul Arnaud, and Victor Turchin. "Rational homology and homotopy of high-dimensional string links." Forum Mathematicum 30, no. 5 (September 1, 2018): 1209–35. http://dx.doi.org/10.1515/forum-2016-0192.

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AbstractArone and the second author showed that when the dimensions are in the stable range, the rational homology and homotopy of the high-dimensional analogues of spaces of long knots can be calculated as the homology of a direct sum of finite graph-complexes that they described explicitly. They also showed that these homology and homotopy groups can be interpreted as the higher-order Hochschild homology, also called Hochschild–Pirashvili homology. In this paper, we generalize all these results to high-dimensional analogues of spaces of string links. The methods of our paper are applicable in the range when the ambient dimension is at least twice the maximal dimension of a link component plus two, which in particular guarantees that the spaces under study are connected. However, we conjecture that our homotopy graph-complex computes the rational homotopy groups of link spaces always when the codimension is greater than two, i.e. always when the Goodwillie–Weiss calculus is applicable. Using Haefliger’s approach to calculate the groups of isotopy classes of higher-dimensional links, we confirm our conjecture at the level of {\pi_{0}}.
36

SHIBUYA, TETSUO, and AKIRA YASUHARA. "Boundary links are self delta-equivalent to trivial links." Mathematical Proceedings of the Cambridge Philosophical Society 143, no. 2 (September 2007): 449–58. http://dx.doi.org/10.1017/s0305004107000254.

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AbstractSelf Δ-equivalence is an equivalence relation for links, which is stronger than link-homotopy defined by J. W. Milnor. It was shown that any boundary link is link-homotopic to a trivial link by L. Cervantes and R. A. Fenn and by D. Dimovski independently. In this paper we will show that any boundary link is self Δ-equivalent to a trivial link.
37

Grbić, Jelena, Stephen Theriault, and Hao Zhao. "Properties of Selick's filtration of the double suspension E2." Journal of Topology and Analysis 06, no. 03 (June 16, 2014): 421–40. http://dx.doi.org/10.1142/s1793525314500150.

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To help study the double suspension [Formula: see text] when localised at a prime p, Selick filtered Ω2S2n+1 by H-spaces which geometrically realise a natural Hopf algebra filtration of H*(Ω2S2n+1;ℤ/p). Later, Gray showed that the fiber Wn of E2 has an integral classifying space BWn and there is a homotopy fibration [Formula: see text]. In this paper we correspondingly filter BWn in a manner compatible with Selick's filtration and the homotopy fibration [Formula: see text], study the multiplicative properties and homotopy exponents of the spaces in the filtrations, and use the filtrations to filter exponent information for the homotopy groups of S2n+1. Our results link three seemingly different in nature classical homotopy fibrations given by Toda, Selick and Gray and make them special cases of a systematic whole. In addition we construct a spectral sequence which converges to the homotopy groups of BWn.
38

KOSCHORKE, ULRICH. "LINK HOMOTOPY IN Sn×ℝm-n AND HIGHER ORDER μ-INVARIANTS." Journal of Knot Theory and Its Ramifications 13, no. 07 (November 2004): 917–38. http://dx.doi.org/10.1142/s0218216504003573.

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Given a suitable link map f into a manifold M, we constructed, in a previous publication, link homotopy invariants κ(f) and μ(f). In the present paper we study the case M=Sn×ℝm-n in detail. Here μ(f) turns out to be the starting term of a whole sequence μ(s)(f), s=0, 1,…, of higher μ-invariants which together capture all the information contained in κ(f). We discuss the geometric significance of these new invariants. In several instances we obtain complete classification results. A central ingredient of our approach is the homotopy theory of wedges of spheres.
39

Hoste, Jim, and Jósef H. Przytycki. "Homotopy skein modules of orientable 3-manifolds." Mathematical Proceedings of the Cambridge Philosophical Society 108, no. 3 (November 1990): 475–88. http://dx.doi.org/10.1017/s0305004100069371.

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AbstractWe define the homotopy skein module of an arbitrary orientable 3-manifold M. This module is similar to the ordinary skein module defined by the second author but is more appropriate when considering oriented links in M up to link homotopy rather than isotopy. We compute the homotopy skein module of M = F × I for any orientable surface F and show that it is free. In the case where M = F × I the homotopy skein module may be given an algebra structure and we show that as an algebra it is isomorphic to the universal enveloping algebra of the Goldman–Wolpert Lie algebra of F. We show, also in this case, that the homotopy skein module is a quantization of the symmetric tensor algebra associated to the Goldman–Wolpert Lie algebra.
40

Akhmechet, Rostislav, Vyacheslav Krushkal, and Michael Willis. "Stable homotopy refinement of quantum annular homology." Compositio Mathematica 157, no. 4 (April 2021): 710–69. http://dx.doi.org/10.1112/s0010437x20007721.

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We construct a stable homotopy refinement of quantum annular homology, a link homology theory introduced by Beliakova, Putyra and Wehrli. For each $r\geq ~2$ we associate to an annular link $L$ a naive $\mathbb {Z}/r\mathbb {Z}$ -equivariant spectrum whose cohomology is isomorphic to the quantum annular homology of $L$ as modules over $\mathbb {Z}[\mathbb {Z}/r\mathbb {Z}]$ . The construction relies on an equivariant version of the Burnside category approach of Lawson, Lipshitz and Sarkar. The quotient under the cyclic group action is shown to recover the stable homotopy refinement of annular Khovanov homology. We study spectrum level lifts of structural properties of quantum annular homology.
41

Tsai, Lung-Wen, and Jeong-Jang Lu. "Coupler-Point-Curve Synthesis Using Homotopy Methods." Journal of Mechanical Design 112, no. 3 (September 1, 1990): 384–89. http://dx.doi.org/10.1115/1.2912619.

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A numerical method called “Homotopy Method” (or Continuation Method) is applied to the problem of four-bar coupler-curve synthesis. We have shown that: for five precision points, the “General Homotopy Method” can be applied to find the link lengths of 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.
42

Rajeswari, V., and T. Nithiya. "CONSTRUCTING TRI-TOPOLOGICAL NETWORK SPACE MODEL USING CONNECTED COMPONENT GRAPH THEORY (T3-C2G) BASED ON HOMOTOPY ALGEBRAIC INVARIANCE MODEL." Advances in Mathematics: Scientific Journal 10, no. 5 (May 7, 2021): 2433–47. http://dx.doi.org/10.37418/amsj.10.5.11.

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The complex network contains non-deterministic topological spaces under an invariance structural approach to create failures on a continual link during communication. The non-lineardynamic topological structure leads to problematic threading links on network nodes due to a non-identical path to route the data. To resolve this problem, we propose atri-logical algebraic mathematical construction model called homotopy based tri-topological network spa- ce using connected component graph $(T^3-C^2G)$ under network nonlinear structure,The Algebraic Invariance Linear Queuing Theory (HA/I/LQT) is used to resolve the link failure route propagation to make improved communication performance. This homotopy reduction to reduce the complex nature to make continual link based on Quillen topological structure space under the covariance tri-topological structure. Further, this makes tri-logical structure resembles the sequence of triangle structured route space to make the nearest point of node adjustment from the nearest path. This balances the M/M/G-$T^3$-Max queuing theory on triangular weightage in routing schemes to specify the dynamic homotopy topological structure to make continuous routing links to reduce the complex nature of network routing.
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KOSCHORKE, U. "LINK MAPS IN ARBITRARY MANIFOLDS AND THEIR HOMOTOPY INVARIANTS." Journal of Knot Theory and Its Ramifications 12, no. 01 (February 2003): 79–104. http://dx.doi.org/10.1142/s0218216503002329.

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In this paper we generalize Milnor's μ-invariants of classical links to certain ("κ-Brunnian") higher dimensional link maps into fairly arbitrary manifolds. Our approach involves the homotopy theory of configuration spaces and of wedges of spheres. We discuss the strength of these invariants and their compatibilities e.g. with (Hilton decompositions of) linking coefficients. Our results suggest, in particular, a conjecture about possible new link homotopies.
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de Campos, José Eduardo Prado Pires. "Distinguishing links up to link-homotopy by algebraic methods." Topology and its Applications 157, no. 3 (February 2010): 605–14. http://dx.doi.org/10.1016/j.topol.2009.11.001.

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NAKANISHI, YASUTAKA, and TETSUO SHIBUYA. "LINK HOMOTOPY AND QUASI SELF DELTA-EQUIVALENCE FOR LINKS." Journal of Knot Theory and Its Ramifications 09, no. 05 (August 2000): 683–91. http://dx.doi.org/10.1142/s0218216500000372.

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SHIBUYA, TETSUO, and AKIRA YASUHARA. "SELF Ck-MOVE, QUASI SELF Ck-MOVE AND THE CONWAY POTENTIAL FUNCTION FOR LINKS." Journal of Knot Theory and Its Ramifications 13, no. 07 (November 2004): 877–93. http://dx.doi.org/10.1142/s0218216504003500.

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Nakanishi and Shibuya gave a relation between link homotopy and quasi self delta-equivalence. And they also gave a necessary condition for two links to be self delta-equivalent by using the multivariable Alexander polynomial. Link homotopy and quasi self delta-equivalence are also called self C1-equivalence and quasi self C2-equivalence respectively. In this paper, we generalize their results. In Sec. 1, we give a relation between self Ck-equivalence and quasi self Ck+1-equivalence. In Secs. 2 and 3, we give necessary conditions for two links to be self Ck-equivalent by using the multivariable Conway potential function and the Conway polynomial respectively.
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MELLOR, BLAKE. "FINITE TYPE LINK CONCORDANCE INVARIANTS." Journal of Knot Theory and Its Ramifications 09, no. 03 (May 2000): 367–85. http://dx.doi.org/10.1142/s0218216500000177.

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This paper is a follow-up to [10], in which the author showed that the only real-valued finite type invariants of link homotopy are the linking numbers of the components. In this paper, we extend the methods used to show that the only real-valued finite type invariants of link concordance are, again, the linking numbers of the components.
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HATAKENAKA, ERI, and TAKEFUMI NOSAKA. "SOME TOPOLOGICAL ASPECTS OF 4-FOLD SYMMETRIC QUANDLE INVARIANTS OF 3-MANIFOLDS." International Journal of Mathematics 23, no. 07 (June 27, 2012): 1250064. http://dx.doi.org/10.1142/s0129167x12500644.

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The paper relates the 4-fold symmetric quandle homotopy (cocycle) invariants to topological objects. We show that the 4-fold symmetric quandle homotopy invariants are at least as powerful as the Dijkgraaf–Witten invariants. As an application, for an odd prime p, we show that the quandle cocycle invariant of a link in S3 constructed by the Mochizuki 3-cocycle is equivalent to the Dijkgraaf–Witten invariant with respect to ℤ/pℤ of the double covering of S3 branched along the link. We also reconstruct the Chern–Simons invariant of closed 3-manifolds as a quandle cocycle invariant via the extended Bloch group, in analogy to [A. Inoue and Y. Kabaya, Quandle homology and complex volume, preprint(2010), arXiv:math/1012.2923].
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Chernov, Vladimir, David Freund, and Rustam Sadykov. "Minimizing intersection points of curves under virtual homotopy." Journal of Knot Theory and Its Ramifications 29, no. 03 (March 2020): 2050007. http://dx.doi.org/10.1142/s0218216520500078.

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A flat virtual link is a finite collection of oriented closed curves [Formula: see text] on an oriented surface [Formula: see text] considered up to virtual homotopy, i.e., a composition of elementary stabilizations, destabilizations, and homotopies. Specializing to a pair of curves [Formula: see text], we show that the minimal number of intersection points of curves in the virtual homotopy class of [Formula: see text] equals to the number of terms of a generalization of the Anderson–Mattes–Reshetikhin Poisson bracket. Furthermore, considering a single curve, we show that the minimal number of self-intersections of a curve in its virtual homotopy class can be counted by a generalization of the Cahn cobracket.
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MELLOR, BLAKE. "FINITE TYPE LINK HOMOTOPY INVARIANTS II: Milnor's $\bar\mu$-Invariants." Journal of Knot Theory and Its Ramifications 09, no. 06 (September 2000): 735–58. http://dx.doi.org/10.1142/s0218216500000426.

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We define a notion of finite type invariants for links with a fixed linking matrix. We show that Milnor's link homotopy invariant [Formula: see text] is a finite type invariant, of type 1, in this sense. We also generalize this approach to Milnor's higher order [Formula: see text] invariants and show that they are also, in a sense, of finite type. Finally, we compare our approach to another approach for defining finite type invariants within linking classes.

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