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Journal articles on the topic 'Topological invariants'

Author: Grafiati

Published: 4 June 2021

Last updated: 6 September 2023

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1

HATAKENAKA, ERI. "Invariants of 3-manifolds derived from covering presentations." Mathematical Proceedings of the Cambridge Philosophical Society 149, no. 2 (May 10, 2010): 263–95. http://dx.doi.org/10.1017/s0305004110000198.

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AbstractBy a covering presentation of a 3-manifold, we mean a labelled link (i.e., a link with a monodromy representation), which presents the 3-manifold as the simple 4-fold covering space of the 3-sphere branched along the link with the given monodromy. It is known that two labelled links present a homeomorphic 3-manifold if and only if they are related by a finite sequence of some local moves. This paper presents a method for constructing topological invariants of 3-manifolds based on their covering presentations. The proof of the topological invariance is shown by verifying the invariance under the local moves. As an example of such invariants, we present the Dijkgraaf–Witten invariant of 3-manifolds.

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2

Kaufmann, Ralph M., Dan Li, and Birgit Wehefritz-Kaufmann. "Notes on topological insulators." Reviews in Mathematical Physics 28, no. 10 (November 2016): 1630003. http://dx.doi.org/10.1142/s0129055x1630003x.

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This paper is a survey of the [Formula: see text]-valued invariant of topological insulators used in condensed matter physics. The [Formula: see text]-valued topological invariant, which was originally called the TKNN invariant in physics, has now been fully understood as the first Chern number. The [Formula: see text] invariant is more mysterious; we will explain its equivalent descriptions from different points of view and provide the relations between them. These invariants provide the classification of topological insulators with different symmetries in which K-theory plays an important role. Moreover, we establish that both invariants are realizations of index theorems which can also be understood in terms of condensed matter physics.

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3

Ohtsuki, Tomotada. "Invariants of 3-manifolds derived from universal invariants of framed links." Mathematical Proceedings of the Cambridge Philosophical Society 117, no. 2 (March 1995): 259–73. http://dx.doi.org/10.1017/s0305004100073102.

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Reshetikhin and Turaev [10] gave a method to construct a topological invariant of compact oriented 3-manifolds from a ribbon Hopf algebra (e.g. a quantum group Uq(sl2)) using finite-dimensional representations of it. In this paper we give another independent method to construct a topological invariant of compact oriented 3-manifolds from a ribbon Hopf algebra via universal invariants of framed links without using representations of the algebra. For Uq(sl2) these two methods give different invariants of 3-manifolds.

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4

Meilhan, Jean-Baptiste, and Sakie Suzuki. "The universal sl2 invariant and Milnor invariants." International Journal of Mathematics 27, no. 11 (October 2016): 1650090. http://dx.doi.org/10.1142/s0129167x16500907.

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The universal [Formula: see text] invariant of string links has a universality property for the colored Jones polynomial of links, and takes values in the [Formula: see text]-adic completed tensor powers of the quantized enveloping algebra of [Formula: see text]. In this paper, we exhibit explicit relationships between the universal [Formula: see text] invariant and Milnor invariants, which are classical invariants generalizing the linking number, providing some new topological insight into quantum invariants. More precisely, we define a reduction of the universal [Formula: see text] invariant, and show how it is captured by Milnor concordance invariants. We also show how a stronger reduction corresponds to Milnor link-homotopy invariants. As a byproduct, we give explicit criterions for invariance under concordance and link-homotopy of the universal [Formula: see text] invariant, and in particular for sliceness. Our results also provide partial constructions for the still-unknown weight system of the universal [Formula: see text] invariant.

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5

Tur, A. V., and V. V. Yanovsky. "Invariants in dissipationless hydrodynamic media." Journal of Fluid Mechanics 248 (March 1993): 67–106. http://dx.doi.org/10.1017/s0022112093000692.

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We propose a general geometric method of derivation of invariant relations for hydrodynamic dissipationless media. New dynamic invariants are obtained. General relations between the following three types of invariants are established, valid in all models: Lagrangian invariants, frozen-in vector fields and frozen-in co-vector fields. It is shown that frozen-in integrals form a Lie algebra with respect to the commutator of the frozen fields. The relation between frozen-in integrals derived here can be considered as the Backlund transformation for hydrodynamic-type systems of equations. We derive an infinite family of integral invariants which have either dynamic or topological nature. In particular, we obtain a new type of topological invariant which arises in all hydrodynamic dissipationless models when the well-known Moffatt invariant vanishes.

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6

KRIEGER, WOLFGANG. "On a syntactically defined invariant of symbolic dynamics." Ergodic Theory and Dynamical Systems 20, no. 2 (April 2000): 501–16. http://dx.doi.org/10.1017/s0143385700000249.

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A partially ordered set that is invariantly associated to a subshift is constructed. A property of subshifts, also an invariant of topological conjugacy, is described. If this property is present in a subshift then the constructed partially ordered set is a partially ordered semigroup (with zero). In the description of these invariants the notion of context is instrumental.

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7

ZHANG, R. B. "QUANTUM SUPERGROUPS AND TOPOLOGICAL INVARIANTS OF THREE-MANIFOLDS." Reviews in Mathematical Physics 07, no. 05 (July 1995): 809–31. http://dx.doi.org/10.1142/s0129055x95000311.

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The Reshetikhin-Turaev approach to topological invariants of three-manifolds is generalized to quantum supergroups. A general method for constructing three-manifold invariant is developed, which requires only the study of the eigenvalues of certain central elements of the quantum supergroup in irreducible representations. To illustrate how the method works, Uq(gl(2|1)) at odd roots of unity is studied in detail, and the corresponding topological invariants are obtained.

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8

Wang, Zhong, Xiao-Liang Qi, and Shou-Cheng Zhang. "Equivalent topological invariants of topological insulators." New Journal of Physics 12, no. 6 (June 17, 2010): 065007. http://dx.doi.org/10.1088/1367-2630/12/6/065007.

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9

CHO, YONG SEUNG. "GENERATING SERIES FOR SYMMETRIC PRODUCT SPACES." International Journal of Geometric Methods in Modern Physics 09, no. 05 (July 3, 2012): 1250045. http://dx.doi.org/10.1142/s0219887812500454.

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We consider the symmetric product spaces of closed manifolds. We introduce some geometric invariants and the topological properties of symmetric product spaces via the symmetric invariant ones of product spaces and apply to Gromov–Witten invariants. We examine the symmetric product spaces of the complex projective line, their Gromov–Witten invariants and compute the generating series induced by their Gromov–Witten invariants.

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10

Pattabiraman, K., M. Kameswari, and M. Seenivasan. "Generalized Version of <i>ISI</i> Invariant for some Molecular Structures." Materials Science Forum 1048 (January 4, 2022): 221–26. http://dx.doi.org/10.4028/www.scientific.net/msf.1048.221.

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Degree related topological invariants are the bygone and most victorioustype of graph invariants so far. In this article, we are interested in finding the generalized inverse indeg invariant of the nanostar dendrimers D[r],fullerene dendrimerNS4[r], and polymer dendrimerNS5[r]. Keywords: nanotubes; inverse indeg invariant; nanostar dendrimers; fullerene dendrimer; polymer dendrimer

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11

MALIK, R. P. "BRST COHOMOLOGY AND HODGE DECOMPOSITION THEOREM IN ABELIAN GAUGE THEORY." International Journal of Modern Physics A 15, no. 11 (April 30, 2000): 1685–705. http://dx.doi.org/10.1142/s0217751x00000756.

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We discuss the Becchi–Rouet–Stora–Tyutin (BRST) cohomology and Hodge decomposition theorem for the two-dimensional free U(1) gauge theory. In addition to the usual BRST charge, we derive a local, conserved and nilpotent co(dual)-BRST charge under which the gauge-fixing term remains invariant. We express the Hodge decomposition theorem in terms of these charges and the Laplacian operator. We take a single photon state in the quantum Hilbert space and demonstrate the notion of gauge invariance, no-(anti)ghost theorem, transversality of photon and establish the topological nature of this theory by exploiting the concepts of BRST cohomology and Hodge decomposition theorem. In fact, the topological nature of this theory is encoded in the vanishing of the Laplacian operator when equations of motion are exploited. On the two-dimensional compact manifold, we derive two sets of topological invariants with respect to the conserved and nilpotent BRST- and co-BRST charges and express the Lagrangian density of the theory as the sum of terms that are BRST- and co-BRST invariants. Mathematically, this theory captures together some of the key features of both Witten- and Schwarz-type of topological field theories.

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12

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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13

Kolobyanina, A. E., E. V. Nozdrinova, and O. V. Pochinka. "Classification of rough transformations of a circle from a modern point of view." Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva 20, no. 4 (December 30, 2018): 408–18. http://dx.doi.org/10.15507/2079-6900.20.201804.408-418.

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In this paper the authors use modern methods and approaches to present a solution to the problem of the topological classification of circle’s rough transformations in canonical formulation. In the modern theory of dynamical systems such problems are understood as the complete topological classification: finding topological invariants, proving the completeness of the set of invariants found and constructing a standard representative from a given set of topological invariants. Namely, in the first theorem of this paper the type of periodic data of circle’s rough transformations is established. In the second theorem necessary and sufficient conditions of their conjugacy are proved. These conditions mean coincidence of periodic data and rotation numbers. In the third theorem the admissible set of parameters is implemented by a rough transformation of a circle. While proving the theorems, we assume that the results on the local topological classification of hyperbolic periodic points, as well as the results on the global representation of the ambient manifold as a union of invariant manifolds of periodic points, are known.

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14

Lukova-Chuiko, Nataliia. "Minimal function on 3-manifolds with boundary." Proceedings of the International Geometry Center 8, no. 3-4 (March 1, 2020): 46–52. http://dx.doi.org/10.15673/tmgc.v8i3-4.1619.

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We construct the complete topological invariant of minimal functions on the three-dimensional manifolds and proved the theorem about the implementation of this invariant feature. Thus, it is received the minimal topological classification of functions. Efficiency of constructed invariants is demonstrated by examples. We describe all the functions, the complexity of which does not exceed three.

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15

Yildirim, Tuna. "Topologically massive Yang–Mills theory and link invariants." International Journal of Modern Physics A 30, no. 07 (March 5, 2015): 1550034. http://dx.doi.org/10.1142/s0217751x15500347.

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Topologically massive Yang–Mills theory is studied in the framework of geometric quantization. Since this theory has a mass gap proportional to the topological mass m, Yang–Mills contribution decays exponentially at very large distances compared to 1/m, leaving a pure Chern–Simons theory with level number k. In this paper, the near Chern–Simons limit is studied where the distance is large enough to give an almost topological theory, with a small contribution from the Yang–Mills term. It is shown that this almost topological theory consists of two copies of Chern–Simons with level number k/2, very similar to the Chern–Simons splitting of topologically massive AdS gravity. Also, gauge invariance of these half-Chern–Simons theories is discussed. As m approaches to infinity, the split parts add up to give the original Chern–Simons term with level k. Reduction of the phase space is discussed in this limit. Finally, a relation between the observables of topologically massive Yang–Mills theory and Chern–Simons theory is shown. One of the two split Chern–Simons pieces is shown to be associated with Wilson loops while the other with 't Hooft loops. This allows one to use skein relations to calculate topologically massive Yang–Mills theory observables in the near Chern–Simons limit.

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16

Clementini, E., and P. Di Felice. "Topological invariants for lines." IEEE Transactions on Knowledge and Data Engineering 10, no. 1 (1998): 38–54. http://dx.doi.org/10.1109/69.667085.

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17

Esperet, Louis. "Boxicity and topological invariants." European Journal of Combinatorics 51 (January 2016): 495–99. http://dx.doi.org/10.1016/j.ejc.2015.07.020.

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18

Martins, Nuno, Ricardo Severino, and J. Sousa Ramos. "Isentropic Real Cubic Maps." International Journal of Bifurcation and Chaos 13, no. 07 (July 2003): 1701–9. http://dx.doi.org/10.1142/s0218127403007552.

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Given a family of bimodal maps on the interval, we need to consider a second topological invariant, other than the usual topological entropy, in order to classify it. With this work, we want to understand how to use this second invariant to distinguish bimodal maps with the same topological entropy and, in particular, how this second invariant changes within a given type of topological entropy level set. In order to do that, we use the kneading theory framework and introduce a symbolic product * between kneading invariants of maps from the same topological entropy level set, for which we show that the second invariant is preserved. Finally, we also show that the change of the second invariant follows closely the symbolic order between bimodal kneading sequences.

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19

Bataineh, Khaled. "On the skein theory of dichromatic links and invariants of finite type." Journal of Knot Theory and Its Ramifications 26, no. 13 (November 2017): 1750092. http://dx.doi.org/10.1142/s0218216517500924.

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In [Dichromatic link invariants, Trans. Amer. Math. Soc. 321(1) (1990) 197–229], Hoste and Kidwell investigated the skein theory of oriented dichromatic links in [Formula: see text]. They introduced a multi-variable polynomial invariant [Formula: see text]. We use special substitutions for some of the parameters of the invariant [Formula: see text] to show how to deduce invariants of finite type from [Formula: see text] using partial derivatives. Then we consider the 2-component 1-trivial dichromatic links. We study the Vassiliev invariants of the 2-component in the complement of the 1-component, which is equivalent to studying Vassiliev invariants for knots in [Formula: see text] We give combinatorial formulas for the type-zero and type-one invariants and we connect these invariants to existing invariants such as Aicardi's invariant. This provides us with a topological meaning of the first partial derivative, which is also shown to be universal as a type-one invariant.

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20

Idrees, Nazeran, Muhammad Saif, and Tehmina Anwar. "Eccentricity-Based Topological Invariants of Some Chemical Graphs." Atoms 7, no. 1 (February 6, 2019): 21. http://dx.doi.org/10.3390/atoms7010021.

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Topological index is an invariant of molecular graphs which correlates the structure with different physical and chemical invariants of the compound like boiling point, chemical reactivity, stability, Kovat’s constant etc. Eccentricity-based topological indices, like eccentric connectivity index, connective eccentric index, first Zagreb eccentricity index, and second Zagreb eccentricity index were analyzed and computed for families of Dutch windmill graphs and circulant graphs.

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21

Li, Xinfei, and Xin Liu. "Topological invariants for superconducting cosmic strings." International Journal of Modern Physics A 33, no. 27 (September 27, 2018): 1850156. http://dx.doi.org/10.1142/s0217751x18501567.

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Superconducting cosmic strings (SCSs) have received revived interests recently. In this paper we treat closed SCSs as oriented knotted line defects, and concentrate on their topology by studying the Hopf topological invariant. This invariant is an Abelian Chern–Simons action, from which the HOMFLYPT knot polynomial can be constructed. It is shown that the two independent parameters of the polynomial correspond to the writhe and twist contributions, separately. This new method is topologically stronger than the traditional (self-)linking number method, which fails to detect essential topology of knots sometimes, shedding new light upon the study of physical intercommunications of superconducting cosmic strings as a complex system.

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22

Aicardi, Francesca. "Mod2 local invariants of maps between 3-manifolds." Journal of Knot Theory and Its Ramifications 27, no. 03 (March 2018): 1840012. http://dx.doi.org/10.1142/s0218216518400126.

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This paper refers to the work [V. Goryunov, Local invariants of maps between 3-manifolds, J. Topology 6 (2013) 757–776] on local invariants of maps between 3-manifolds. It is assumed that the manifolds have no boundary, and that the source is compact. In the case when the source and the target are oriented, Goryunov proved that every local order one invariant with integer values can be written as a linear combination of seven basic invariants, and gave a geometrical interpretation for them. When the target is the oriented [Formula: see text], there are further four basic mod2 invariants. One of the mod2 invariants has been provided with a topological interpretation, in terms of the number of components and of the self-linking of a framed link constructed from the cuspidal edge. Here, we show that two further independent linear combinations of the mod2 invariants have a topological interpretation, involving the self-linking number of two curves defined by all irregular points of the critical value set of a generic map from an oriented closed 3-manifold to [Formula: see text].

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23

Xiao, Xiao, J. K. Freericks, and A. F. Kemper. "Robust measurement of wave function topology on NISQ quantum computers." Quantum 7 (April 27, 2023): 987. http://dx.doi.org/10.22331/q-2023-04-27-987.

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Topological quantum phases of quantum materials are defined through their topological invariants. These topological invariants are quantities that characterize the global geometrical properties of the quantum wave functions and thus are immune to local noise. Here, we present a strategy to measure topological invariants on quantum computers. We show that our strategy can be easily integrated with the variational quantum eigensolver (VQE) so that the topological properties of generic quantum many-body states can be characterized on current quantum hardware. We demonstrate the robust nature of the method by measuring topological invariants for both non-interacting and interacting models, and map out interacting quantum phase diagrams on quantum simulators and IBM quantum hardware.

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24

Avron, J. E., L. Sadun, J. Segert, and B. Simon. "Topological Invariants in Fermi Systems with Time-Reversal Invariance." Physical Review Letters 61, no. 12 (September 19, 1988): 1329–32. http://dx.doi.org/10.1103/physrevlett.61.1329.

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25

Castro, Eduardo, Andrey Toropov, Alexandra Nesterova, and Ozad Nabiev. "QSPR modeling aqueous solubility of polychlorinated biphenyls by optimization of correlation weights of local and global graph invariants." Open Chemistry 2, no. 3 (September 1, 2004): 500–523. http://dx.doi.org/10.2478/bf02476204.

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AbstractAqueous solubilities of polychlorinated biphenyls have been correlated with topological molecular descriptors which are functions of local and global invariants of labeled hydrogen filled graphs. Morgan extended connectivity and nearest neighboring codes have been used as local graph invariants. The number of chlorine atoms in biphenyls has been employed as a global graph invariant. Present results show that taking into account correlation weights of global invariants gives quite reasonable improvement of statistical characteristics for the prediction of aqueous solubilities of polychlorinated biphenyls.

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26

OZAWA, TETSUYA. "FINITE ORDER TOPOLOGICAL INVARIANTS OF PLANE CURVES." Journal of Knot Theory and Its Ramifications 08, no. 01 (February 1999): 33–47. http://dx.doi.org/10.1142/s0218216599000055.

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We introduce three families of topological invariants of stable closed plane curves, which contain infinitely many mutually independent invariants among them. We study the order of these invariants in the sense of Vassiliev. As a consequence, we conclude that there exist infinitely many independent topological invariants for stable closed plane curves with order equal to 1.

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27

TRAPP, ROLLAND. "TWIST SEQUENCES AND VASSILIEV INVARIANTS." Journal of Knot Theory and Its Ramifications 03, no. 03 (September 1994): 391–405. http://dx.doi.org/10.1142/s0218216594000289.

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In this paper we describe a difference sequence technique, hereafter referred to as the twist sequence technique, for studying Vassiliev invariants. This technique is used to show that Vassiliev invariants have polynomial growth on certain sequences of knots. Restrictions of Vassiliev invariants to the sequence of (2, 2i + 1) torus knots are characterized. As a corollary it is shown that genus, crossing number, signature, and unknotting number are not Vassiliev invariants. This characterization also determines the topological information about (2, 2i + 1) torus knots encoded in finite-type invariants. The main result obtained is that the complement of the space of Vassiliev invariants is dense in the space of all numeric knot invariants. Finally, we show that the uniform limit of a sequence of Vassiliev invariants must be a Vassiliev invariant.

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28

Frolkova, Anastasia V. "Topological Invariants of Vapor–Liquid, Vapor–Liquid–Liquid and Liquid–Liquid Phase Diagrams." Entropy 23, no. 12 (December 10, 2021): 1666. http://dx.doi.org/10.3390/e23121666.

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The study of topological invariants of phase diagrams allows for the development of a qualitative theory of the processes being researched. Studies of the properties of objects in the same equivalence class may be carried out with the aim of predicting the properties of unexplored objects from this class, or predicting the behavior of a whole system. This paper describes a number of topological invariants in vapor–liquid, vapor–liquid–liquid and liquid–liquid equilibrium diagrams. The properties of some invariants are studied and illustrated. It is shown that the invariant of a diagram with a miscibility gap can be used to distinguish equivalence classes of phase diagrams, and that the balance equation of the singular-point indices, based on the Euler characteristic, may be used to analyze the binodal-surface structure of a quaternary system.

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Bedoya Giraldo, Juan Pablo, Aura Lucía Pérez Escobar, and Javier Guillermo Valdés Duque. "Tools for Calculating Topological Invariants." Sistemas y Telemática 2, no. 3 (July 28, 2006): 59. http://dx.doi.org/10.18046/syt.v2i3.931.

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30

Greene, Robert E., Kang-Tae Kim, and Nikolay V. Shcherbina. "Topological invariants and Holomorphic Mappings." Comptes Rendus. Mathématique 360, G8 (September 14, 2022): 829–44. http://dx.doi.org/10.5802/crmath.336.

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31

Gurski, Nick, Niles Johnson, and Angelica M. Osorno. "Topological Invariants from Higher Categories." Notices of the American Mathematical Society 66, no. 08 (September 1, 2019): 1. http://dx.doi.org/10.1090/noti1934.

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Guichard, Olivier, and Anna Wienhard. "Topological invariants of Anosov representations." Journal of Topology 3, no. 3 (2010): 578–642. http://dx.doi.org/10.1112/jtopol/jtq018.

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33

DA CRUZ, WELLINGTON. "TOPOLOGICAL INVARIANTS AND ANYONIC PROPAGATORS." Modern Physics Letters A 14, no. 28 (September 14, 1999): 1933–36. http://dx.doi.org/10.1142/s0217732399002005.

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We obtain the Hausdorff dimension, h=2-2s, for particles with fractional spins in the interval, 0≤ s ≤0.5, such that the manifold is characterized by a topological invariant given by, [Formula: see text]. This object is related to fractal properties of the path swept out by fractional spin particles, the spin of these particles, and the genus (number of anyons) of the manifold. We prove that the anyonic propagator can be put into a path integral representation which gives us a continuous family of Lagrangians in a convenient gauge. The formulas for, h and [Formula: see text], were obtained taking into account the anyon model as a particle-flux system and by a qualitative inference of the topology.

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34

Amann, Manuel. "Computational Complexity of Topological Invariants." Proceedings of the Edinburgh Mathematical Society 58, no. 1 (December 11, 2014): 27–32. http://dx.doi.org/10.1017/s0013091514000455.

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AbstractWe answer the following question posed by Lechuga: given a simply connected spaceXwith bothH*(X; ℚ) and π*(X) ⊗ ℚ being finite dimensional, what is the computational complexity of an algorithm computing the cup length and the rational Lusternik—Schnirelmann category ofX?Basically, by a reduction from the decision problem of whether a given graph isk-colourable fork≥ 3, we show that even stricter versions of the problems above are NP-hard.

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35

Cappell, Sylvain E., and Julius L. Shaneson. "Stratifiable maps and topological invariants." Journal of the American Mathematical Society 4, no. 3 (September 1, 1991): 521. http://dx.doi.org/10.1090/s0894-0347-1991-1102578-4.

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36

Ooguri, Hirosi, and Cumrun Vafa. "Knot invariants and topological strings." Nuclear Physics B 577, no. 3 (June 2000): 419–38. http://dx.doi.org/10.1016/s0550-3213(00)00118-8.

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37

Silver, Daniel S. "Knot invariants from topological entropy." Topology and its Applications 61, no. 2 (February 1995): 159–77. http://dx.doi.org/10.1016/0166-8641(94)00028-2.

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38

Gonçalves, Icaro, and Eduardo Longa. "Topological Invariants for Closed Hypersurfaces." Bulletin of the Brazilian Mathematical Society, New Series 50, no. 2 (September 26, 2018): 533–42. http://dx.doi.org/10.1007/s00574-018-0115-7.

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39

Hirayama, M., and N. Sugimasa. "Novel topological invariants and anomalies." Physical Review D 35, no. 2 (January 15, 1987): 600–608. http://dx.doi.org/10.1103/physrevd.35.600.

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40

Budach, Lothar. "Topological invariants of classification problems." Theoretical Computer Science 72, no. 1 (April 1990): 3–26. http://dx.doi.org/10.1016/0304-3975(90)90043-h.

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41

Mekhfi, M. "Invariants of topological quantum mechanics." International Journal of Theoretical Physics 35, no. 8 (August 1996): 1709–18. http://dx.doi.org/10.1007/bf02302264.

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42

KAWAUCHI, AKIO. "TOPOLOGICAL IMITATION, MUTATION AND THE QUANTUM SU(2) INVARIANTS." Journal of Knot Theory and Its Ramifications 03, no. 01 (March 1994): 25–39. http://dx.doi.org/10.1142/s0218216594000058.

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It is proved that any two mutative closed oriented 3-manifolds have the same quantum SU(2) invariant. By a constructive argument of topological imitation, we construct finitely many mutative hyperbolic imitations of any given closed oriented 3-manifold with certain arbitrariness of isometry groups whose quantum SU(2) invariants are close to the original one.

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43

WATANABE, TADAYUKI. "CLASPER-MOVES AMONG RIBBON 2-KNOTS CHARACTERIZING THEIR FINITE TYPE INVARIANTS." Journal of Knot Theory and Its Ramifications 15, no. 09 (November 2006): 1163–99. http://dx.doi.org/10.1142/s0218216506005056.

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Habiro found in his thesis a topological interpretation of finite type invariants of knots in terms of local moves called Habiro's Ck-moves. Ck-moves are defined by using his claspers. In this paper we define "oriented" claspers and RCk-moves among ribbon 2-knots as modifications of Habiro's notions to give a similar interpretation of Habiro–Kanenobu–Shima's finite type invariants of ribbon 2-knots. It works also for ribbon 1-knots. Furthermore, by using oriented claspers for ribbon 1-knots, we can prove Habiro–Shima's conjecture in the case of ℚ-valued invariants, saying that ℚ-valued Habiro–Kanenobu–Shima finite type invariant and ℚ-valued Vassiliev–Goussarov finite type invariant are the same thing.

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Sleptsov, Alexey. "QFT and unification of knot theories." Modern Physics Letters A 29, no. 24 (August 7, 2014): 1430025. http://dx.doi.org/10.1142/s0217732314300250.

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We discuss relation between knot theory and topological quantum field theory. Also it is considered a theory of superpolynomial invariants of knots which generalizes all other known theories of knot invariants. We discuss a possible generalization of topological quantum field theory with the help of superpolynomial invariants.

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AICARDI, F. "MULTIPLICATIVE INVARIANT POLYNOMIALS OF PLANE CURVES." Journal of Knot Theory and Its Ramifications 10, no. 06 (September 2001): 937–42. http://dx.doi.org/10.1142/s0218216501001256.

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This note answer a question of V. F. R. Jones, whether there exist topological invariant polynomials of plane curves that, instead of additive as the Arnold invariants, are multiplicative with respect to the connected sum of two curves.

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SWANSON, RICHARD, and HANS VOLKMER. "Invariants of weak equivalence in primitive matrices." Ergodic Theory and Dynamical Systems 20, no. 2 (April 2000): 611–26. http://dx.doi.org/10.1017/s0143385700000316.

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Weak equivalence of primitive matrices is a known invariant arising naturally from the study of inverse limit spaces. Several new invariants for weak equivalence are described. It is proved that a positive dimension group isomorphism is a complete invariant for weak equivalence. For the transition matrices corresponding to periodic kneading sequences, the discriminant is proved to be an invariant when the characteristic polynomial is irreducible. The results have direct application to the topological classification of one-dimensional inverse limit spaces.

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Fock, Vladimir, Valdo Tatitscheff, and Alexander Thomas. "Topological quantum field theories from Hecke algebras." Representation Theory of the American Mathematical Society 27, no. 9 (May 22, 2023): 248–91. http://dx.doi.org/10.1090/ert/640.

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We construct one-dimensional non-commutative topological quantum field theories (TQFTs), one for each Hecke algebra corresponding to a finite Coxeter system. These TQFTs associate an invariant to each ciliated surface, which is a Laurent polynomial for punctured surfaces. There is a graphical way to compute the invariant using minimal colored graphs. We give explicit formulas in terms of the Schur elements of the Hecke algebra and prove positivity properties for the invariants when the Coxeter group is of classical type, or one of the exceptional types H 3 H_3 , E 6 E_6 and E 7 E_7 .

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NIEH, H. T. "A TORSIONAL TOPOLOGICAL INVARIANT." International Journal of Modern Physics A 22, no. 29 (November 20, 2007): 5237–44. http://dx.doi.org/10.1142/s0217751x07038414.

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Curvature and torsion are the two tensors characterizing a general Riemannian space–time. In Einstein's general theory of gravitation, with torsion postulated to vanish and the affine connection identified to the Christoffel symbol, only the curvature tensor plays the central role. For such a purely metric geometry, two well-known topological invariants, namely the Euler class and the Pontryagin class, are useful in characterizing the topological properties of the space–time. From a gauge theory point of view, and especially in the presence of spin, torsion naturally comes into play, and the underlying space–time is no longer purely metric. We describe a torsional topological invariant, discovered in 1982, that has now found increasing usefulness in recent developments.

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RUBINSZTEIN, RYSZARD L. "TOPOLOGICAL QUANDLES AND INVARIANTS OF LINKS." Journal of Knot Theory and Its Ramifications 16, no. 06 (August 2007): 789–808. http://dx.doi.org/10.1142/s0218216507005518.

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We introduce a notion of topological quandle. Given a topological quandle Q we associate to every classical link L in ℝ3 an invariant JQ(L) which is a topological space (defined up to a homeomorphism). The space JQ(L) can be interpreted as a space of colorings of a diagram of the link L with colors from the quandle Q.

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50

Murakami, Hitoshi. "QuantumSO(3)-invariants dominate theSU(2)-invariant of Casson and Walker." Mathematical Proceedings of the Cambridge Philosophical Society 117, no. 2 (March 1995): 237–49. http://dx.doi.org/10.1017/s0305004100073084.

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For a compact Lie groupG, E. Witten proposed topological invariants of a threemanifold (quantumG-invariants) in 1988 by using the Chern-Simons functional and the Feynman path integral [30]. See also [2]. N. Yu. Reshetikhin and V. G. Turaev gave a mathematical proof of existence of such invariants forG=SU(2) [28]. R. Kirby and P. Melvin found that the quantumSU(2)-invariantassociated toq= exp(2π √ − 1/r) withrodd splits into the product of the quantumSO(3)-invariantand[15]. For other approaches to these invariants, see [3, 4, 5, 16, 22, 27].

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