Relevant bibliographies by topics / Invariants of knots and 3-manifolds

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

Academic literature on the topic 'Invariants of knots and 3-manifolds'

Author: Grafiati
Published: 4 June 2021
Last updated: 5 February 2022

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Journal articles on the topic "Invariants of knots and 3-manifolds"

1

Morton, H. R. "QUANTUM INVARIANTS OF KNOTS AND 3‐MANIFOLDS." Bulletin of the London Mathematical Society 28, no. 6 (November 1996): 669–70. http://dx.doi.org/10.1112/blms/28.6.669.

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CHERNOV TCHERNOV, VLADIMIR. "FRAMED KNOTS IN 3-MANIFOLDS AND AFFINE SELF-LINKING NUMBERS." Journal of Knot Theory and Its Ramifications 14, no. 06 (September 2005): 791–818. http://dx.doi.org/10.1142/s0218216505004056.

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The number |K| of non-isotopic framed knots that correspond to a given unframed knot K ⊂ S3 is infinite. This follows from the existence of the self-linking number slk of a zero homologous framed knot. We use the approach of Vassiliev–Goussarov invariants to construct "affine self-linking numbers" that are extensions of slk to the case of nonzero homologous framed knots in 3-manifolds. As a corollary we get that |K| = ∞ for all knots in an oriented (not necessarily compact) 3-manifold M that is not realizable as a connected sum (S1 × S2)# M′. This result for compact manifolds was first stated by Hoste and Przytycki. They referred to the works of McCullough for the idea of the proof, however to the best of our knowledge prior to this work the proof of this fundamental fact was not given in literature or in a preprint form. Our proof is based on different ideas. For M = (S1 × S2)# M′ we construct K in M such that |K| = 2 ≠ ∞.
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Kalfagianni, Efstratia. "Finite type invariants for knots in 3-manifolds." Topology 37, no. 3 (May 1998): 673–707. http://dx.doi.org/10.1016/s0040-9383(97)00034-7.

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KAWAUCHI, AKIO. "ON LINKING SIGNATURE INVARIANTS OF SURFACE-KNOTS." Journal of Knot Theory and Its Ramifications 11, no. 03 (May 2002): 369–85. http://dx.doi.org/10.1142/s0218216502001688.

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We show that the linking signature of a closed oriented 4-manifold with infinite cyclic first homology is twice the Rochlin invariant of an exact leaf with a spin support if such a leaf exists. In particular, the linking signature of a surface-knot in the 4-sphere is twice the Rochlin invariant of an exact leaf of an associated closed spin 4-manifold with infinite cyclic first homology. As an application, we characterize a difference between the spin structures on a homology quaternion space in terms of closed oriented 4-manifolds with infinite cyclic first homology, so that we can obtain examples showing that some different punctured embeddings into S4 produce different Rochlin invariants for some homology quaternion spaces.
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MIZUSAWA, ATSUHIKO, and JUN MURAKAMI. "INVARIANTS OF HANDLEBODY-KNOTS VIA YOKOTA'S INVARIANTS." Journal of Knot Theory and Its Ramifications 22, no. 11 (October 2013): 1350068. http://dx.doi.org/10.1142/s0218216513500685.

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We construct quantum [Formula: see text] type invariants for handlebody-knots in the 3-sphere S3. A handlebody-knot is an embedding of a handlebody in a 3-manifold. These invariants are linear sums of Yokota's invariants for colored spatial graphs which are defined by using the Kauffman bracket. We give a table of calculations of our invariants for genus 2 handlebody-knots up to six crossings. We also show our invariants are identified with special cases of the Witten–Reshetikhin–Turaev invariants.
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Kauffman, Louis Hirsch, and Vassily Olegovich Manturov. "Graphical constructions for the sl(3), C2 and G2 invariants for virtual knots, virtual braids and free knots." Journal of Knot Theory and Its Ramifications 24, no. 06 (May 2015): 1550031. http://dx.doi.org/10.1142/s0218216515500315.

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We construct graph-valued analogues of the Kuperberg sl(3) and G2 invariants for virtual knots. The restriction of the sl(3) and G2 invariants for classical knots coincides with the usual Homflypt sl(3) invariant and G2 invariants. For virtual knots and graphs these invariants provide new graphical information that allows one to prove minimality theorems and to construct new invariants for free knots (unoriented and unlabeled Gauss codes taken up to abstract Reidemeister moves). A novel feature of this approach is that some knots are of sufficient complexity that they evaluate themselves in the sense that the invariant is the knot itself seen as a combinatorial structure. The paper generalizes these structures to virtual braids and discusses the relationship with the original Penrose bracket for graph colorings.
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Kuperberg, Greg. "Book Review: Quantum invariants of knots and 3-manifolds." Bulletin of the American Mathematical Society 33, no. 01 (January 1, 1996): 107–11. http://dx.doi.org/10.1090/s0273-0979-96-00621-0.

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Nikshych, Dmitri, Vladimir Turaev, and Leonid Vainerman. "Invariants of knots and 3-manifolds from quantum groupoids." Topology and its Applications 127, no. 1-2 (January 2003): 91–123. http://dx.doi.org/10.1016/s0166-8641(02)00055-x.

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Cahn, Patricia, Vladimir Chernov, and Rustam Sadykov. "The number of framings of a knot in a 3-manifold." Journal of Knot Theory and Its Ramifications 23, no. 13 (November 2014): 1450072. http://dx.doi.org/10.1142/s0218216514500722.

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In view of the self-linking invariant, the number |K| of framed knots in S3 with given underlying knot K is infinite. In fact, the second author previously defined affine self-linking invariants and used them to show that |K| is infinite for every knot in an orientable manifold unless the manifold contains a connected sum factor of S1 × S2; the knot K need not be zero-homologous and the manifold is not required to be compact. We show that when M is orientable, the number |K| is infinite unless K intersects a nonseparating sphere at exactly one point, in which case |K| = 2; the existence of a nonseparating sphere implies that M contains a connected sum factor of S1 × S2. For knots in nonorientable manifolds we show that if |K| is finite, then K is disorienting, or there is an orientation-preserving isotopy of the knot to itself which changes the orientation of its normal bundle, or it intersects some embedded S2 or ℝP2 at exactly one point, or it intersects some embedded S2 at exactly two points in such a way that a closed curve consisting of an arc in K between the intersection points and an arc in S2 is disorienting.
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Gukov, Sergei, Du Pei, Pavel Putrov, and Cumrun Vafa. "BPS spectra and 3-manifold invariants." Journal of Knot Theory and Its Ramifications 29, no. 02 (February 2020): 2040003. http://dx.doi.org/10.1142/s0218216520400039.

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We provide a physical definition of new homological invariants [Formula: see text] of 3-manifolds (possibly, with knots) labeled by abelian flat connections. The physical system in question involves a 6d fivebrane theory on [Formula: see text] times a 2-disk, [Formula: see text], whose Hilbert space of BPS states plays the role of a basic building block in categorification of various partition functions of 3d [Formula: see text] theory [Formula: see text]: [Formula: see text] half-index, [Formula: see text] superconformal index, and [Formula: see text] topologically twisted index. The first partition function is labeled by a choice of boundary condition and provides a refinement of Chern–Simons (WRT) invariant. A linear combination of them in the unrefined limit gives the analytically continued WRT invariant of [Formula: see text]. The last two can be factorized into the product of half-indices. We show how this works explicitly for many examples, including Lens spaces, circle fibrations over Riemann surfaces, and plumbed 3-manifolds.
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Dissertations / Theses on the topic "Invariants of knots and 3-manifolds"

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Tosun, Bulent. "Legendrian and transverse knots and their invariants." Diss., Georgia Institute of Technology, 2012. http://hdl.handle.net/1853/44880.

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In this thesis, we study Legendrian and transverse isotopy problem for cabled knot types. We give two structural theorems to describe when the (r,s)- cable of a Legendrian simple knot type K is also Legendrian simple. We then study the same problem for cables of the positive trefoil knot. We give a complete classification of Legendrian and transverse cables of the positive trefoil. Our results exhibit many new phenomena in the structural understanding of Legendrian and transverse knots. we then extend these results to the other positive torus knots. The key ingredient in these results is to find necessary and sufficient conditions on maximally thickened contact neighborhoods of the positive torus knots in three sphere.
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Cho, Karina Elle. "Enhancing the Quandle Coloring Invariant for Knots and Links." Scholarship @ Claremont, 2019. https://scholarship.claremont.edu/hmc_theses/228.

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Quandles, which are algebraic structures related to knots, can be used to color knot diagrams, and the number of these colorings is called the quandle coloring invariant. We strengthen the quandle coloring invariant by considering a graph structure on the space of quandle colorings of a knot, and we call our graph the quandle coloring quiver. This structure is a categorification of the quandle coloring invariant. Then, we strengthen the quiver by decorating it with Boltzmann weights. Explicit examples of links that show that our enhancements are proper are provided, as well as background information in quandle theory.
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Kuhlmann, Sally Malinda. "Geodesic knots in hyperbolic 3 manifolds." Connect to thesis, 2005. http://repository.unimelb.edu.au/10187/916.

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This thesis is an investigation of simple closed geodesics, or geodesic knots, in hyperbolic 3-manifolds.
Adams, Hass and Scott have shown that every orientable finite volume hyperbolic 3-manifold contains at least one geodesic knot. The first part of this thesis is devoted to extending this result. We show that all cusped and many closed orientable finite volume hyperbolic 3-manifolds contain infinitely many geodesic knots. This is achieved by studying infinite families of closed geodesics limiting to an infinite length geodesic in the manifold. In the cusped manifold case the limiting geodesic runs cusp-to-cusp, while in the closed manifold case its ends spiral around a short geodesic in the manifold. We show that in the above manifolds infinitely many of the closed geodesics in these families are embedded.
The second part of the thesis is an investigation into the topology of geodesic knots, and is motivated by Thurston’s Geometrization Conjecture relating the topology and geometry of 3-manifolds.We ask whether the isotopy class of a geodesic knot can be distinguished topologically within its homotopy class. We derive a purely topological description for infinite subfamilies of the closed geodesics studied previously in cusped manifolds, and draw explicit projection diagrams for these geodesics in the figure-eight knot complement. This leads to the result that the figure-eight knot complement contains geodesics of infinitely many different knot types in the3-sphere when the figure-eight cusp is filled trivially.
We conclude with a more direct investigation into geodesic knots in the figure-eight knot complement. We discuss methods of locating closed geodesics in this manifold including ways of identifying their isotopy class within a free homotopy class of closed curves. We also investigate a specially chosen class of knots in the figure-eight knot complement, namely those arising as closed orbits in its suspension flow. Interesting examples uncovered here indicate that geodesics of small tube radii may be difficult to distinguish topologically in their free homotopy class.
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López, Neumann Daniel. "Kuperberg invariants for sutured 3-manifolds." Thesis, Université de Paris (2019-....), 2020. http://www.theses.fr/2020UNIP7036.

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Dans cette thèse, on étudie les invariants quantiques des 3-variétés de Kuperberg, qui sont basées sur les algèbres de Hopf. On montre que, pour les super-algèbres de Hopf involutives, les invariants de Kuperberg s’étendent à la classe, plus générale, des 3-variétés suturées balancées et en particulier aux compléments d’entrelacs. Pour accomplir ceci, on relève plusieurs aspects de la théorie des torsions de Reidemeister au monde des invariants quantiques, tels que la procédure pour tordre des invariants, le calcul de Fox et les structures Spin^c, et on clarifie les aspects de la théorie des algèbres de Hopf auxquels ils correspondent. Quand notre construction est spécialisée au cas d’une algèbre extérieure, on montre qu’elle calcule la torsion de Reidemeister tordue des 3-variétés suturées
In this thesis, we study Kuperberg's Hopf algebra approach to quantum invariants of closed 3-manifolds. We show that, for involutive Hopf superalgebras, Kuperberg invariants extend to the more general class of balanced sutured 3-manifolds, and in particular, to link complements. To achieve this, we bring many aspects of Reidemeister torsion theory into the realm of quantum invar-iants, such as twisting, Fox calculus and Spin^c structures and we make clear to which aspects of Hopf algebra theory these correspond. When our construction is specialized to an exterior algebra, we show that it recovers the twisted Reidemeister torsion of sutured 3-manifolds
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Calimici, Giulio. "State sum invariants of combed 3-manifolds." Thesis, Lille 1, 2019. http://www.theses.fr/2019LIL1I018/document.

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Cette thèse concerne la topologie quantique, une branche des mathématiques née dans les années 1980 suite aux travaux de Jones, Drinfeld et Witten. Un exemple fondamental d'invariant quantique des 3-variétés est due à Turaev-Viro en 1992. Leur approche, dans sa forme générale due à Barrett et Westbury, utilise une catégorie de fusion sphérique comme ingrédient principal et consiste en une somme d'états sur un squelette de la 3-variété dont les sommets sont coloriés par les 6j-symboles de la catégorie.Le résultat principal de la thèse est la construction d'un invariant topologique des 3-variétés peignées (c'est-à-dire des 3-variétés munies d'un champ de vecteurs jamais nuls) qui généralise celui de Turaev-Viro. Ce nouvel invariant est défini au moyen d'une catégorie de fusion pivotale et consiste en une somme d'états sur un squelette ramifié représentant la 3-variété peignée.Lorsque la catégorie de fusion pivotale n'est pas sphérique, l'invariant permet en général de distinguer des champs de vecteurs non homotopes sur une même 3-variété. Ceci est montré en considérant une catégorie de fusion pivotale associée à un caractère d'un groupe fini. Pour cette catégorie, l'invariant correspond à l'évaluation par le caractère de la classe d'Euler d'un certain fibré vectoriel de rang 2 associé au champ de vecteurs
This thesis concerns quantum topology, a branch of mathematics born in the 1980s after the work of Jones, Drinfeld and Witten. A fundamental example of a quantum invariant of 3-manifolds is due to Turaev-Viro in 1992. Their approach, in its general form due to Barrett and Westbury, uses a spherical fusion category as the main ingredient and consists in a state sum on a skeleton of the 3-manifold whose vertices are colored by the 6j-symbols of the category. The main result of the thesis is the construction of a topological invariant of combed 3-manifolds (that is, of 3-manifolds endowed with a nowhere-zero vector field) which generalizes that of Turaev-Viro. This new invariant is defined by means of a pivotal fusion category and consists in a state sum on a branched skeleton representing the combed 3-manifold. When the pivotal fusion category is not spherical, the invariant allows in general to distinguish non homotopic vector fields on the same 3-manifold. This is proved by considering a pivotal fusion category associated with a character of a finite group. For this category, the invariant corresponds to the evaluation by the character of the Euler class of a certain vector bundle of rank 2 associated to the vector field
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Alsaeed, Suliman. "Local invariants of fronts in 3-manifolds." Thesis, University of Liverpool, 2014. http://livrepository.liverpool.ac.uk/2006756/.

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An invariant is a quantity which remains unchanged under certain classes of transformations. A wave front (or a front) in a 3-manifold is the image of a surface under a Legendrian map. The aim of this thesis is the description of all local invariants of fronts in 3-manifolds. The front invariants under consideration are those whose increments in generic homotopies are determined entirely by diffeomorphism types of local bifurcations of the fronts. Such invariants are dual to trivial codimension 1 cycles supported on the discriminant in the space of corresponding Legendrian maps. We describe the spaces of the discriminantal cycles (possibly non-trivial) for various orientation and co-orientation settings of the fronts in an arbitrary oriented 3-manifold, both for the integer and mod2 coefficients. For the majority of these cycles we find homotopy-independent interpretations which guarantee the triviality required. In particular, in the case of framed fronts we show that all integer local invariants of Legendrian maps without corank 2 points are essentially exhausted by the numbers of points of isolated singularity types of the fronts.
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Millichap, Christian R. "Mutations and Geometric Invariants of Hyperbolic 3-Manifolds." Diss., Temple University Libraries, 2015. http://cdm16002.contentdm.oclc.org/cdm/ref/collection/p245801coll10/id/321918.

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Mathematics
Ph.D.
The main goal of this thesis is to examine the quality of geometric invariants of finite volume hyperbolic 3-manifolds. In particular, we examine how to construct large classes of hyperbolic 3-manifolds that are geometrically similar: they have a number of geometric invariants that are the same, but are non-isometric. Large classes of geometrically similar hyperbolic 3-manifolds provide examples where the minimal geometric data needed to determine M must be quite large. For our constructions, we will use a cut and paste operation known as mutation. Ruberman has shown that mutations of hyperelliptic surfaces inside hyperbolic 3-manifolds preserve volume. Here, we provide geometric and topological conditions under which such mutations also preserve the initial length spectrum. This work requires us to analyze when least area surfaces could intersect short geodesics in a hyperbolic 3-manifold. As a corollary of this result, we show that the number of hyperbolic knot complements with the same volume and the same initial length spectrum grows at least factorially fast with the volume and the number of twist regions; a similar statement holds for closed hyperbolic 3-manifolds, obtained via Dehn surgery. Furthermore, we show that the knot complements used for this construction are pairwise incommensurable by analyzing their cusp shapes.
Temple University--Theses
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Nosaka, Takefumi. "4-fold symmetric quandle invariants of 3-manifolds." 京都大学 (Kyoto University), 2012. http://hdl.handle.net/2433/157740.

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Sequin, Matthew James. "Comparing Invariants of 3-Manifolds Derived from Hopf Algebras." The Ohio State University, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=osu1338251228.

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Kroll, Jochen. "A Casson-Lin invariant for knots in homology 3-spheres." [S.l. : s.n.], 2003. http://deposit.ddb.de/cgi-bin/dokserv?idn=967833590.

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Books on the topic "Invariants of knots and 3-manifolds"

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Quantum invariants of knots and 3-manifolds. Berlin: W. de Gruyter, 1994.

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Quantum invariants of knots and 3-manifolds. 2nd ed. Berlin: De Gruyter, 2010.

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Quantum invariants: A study of knots, 3-manifolds, and their sets. Singapore: World Scientific, 2002.

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Kauffman, LouisH. Temperley-Lieb recoupling theory and invariants of 3-manifolds. Princeton, N.J: Princeton University Press, 1994.

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Kauffman, Louis H. Temperley-Lieb recoupling theory and invariants of 3-manifolds. Princeton, N.J: Princeton University Press, 1994.

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Prasolov, V. V. Knots, links, braids and 3-manifolds: An introduction to the new invariants in low-dimensional topology. Providence, R.I: American Mathematical Society, 1997.

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Knots, links, braids, and 3-manifolds: An introduction to the new invariants in low-dimensional topology. Providence, R.I: American Mathematical Society, 1997.

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András, Stipsicz, and Szabó Zoltán 1965-, eds. Grid homology for knots and links. Providence, Rhode Island: American Mathematical Society, 2015.

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1974-, Nelson Sam, ed. Quandles: An introduction to the algebra of knots. Providence, Rhode Island: American Mathematical Society, 2015.

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Kassel, Christian. Quantum groups and knot invariants. Paris: Société mathématique de France, 1997.

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Book chapters on the topic "Invariants of knots and 3-manifolds"

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Jackson, David M., and Iain Moffatt. "Vassiliev Invariants of Framed Knots." In CMS Books in Mathematics, 211–17. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-05213-3_12.

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Chekanov, Yuri. "New Invariants of Legendrian Knots." In European Congress of Mathematics, 525–34. Basel: Birkhäuser Basel, 2001. http://dx.doi.org/10.1007/978-3-0348-8266-8_45.

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der Veen, Roland van. "Introduction to Quantum Invariants of Knots." In MATRIX Book Series, 637–56. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-72299-3_27.

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Gabrovšek, Boštjan, and Eva Horvat. "Knot Invariants in Lens Spaces." In Knots, Low-Dimensional Topology and Applications, 347–61. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-16031-9_17.

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Morton, H. R. "Invariants of Links and 3-Manifolds From Skein Theory and From Quantum Groups." In Topics in Knot Theory, 107–55. Dordrecht: Springer Netherlands, 1993. http://dx.doi.org/10.1007/978-94-011-1695-4_8.

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Queffelec, Hoel, and Antonio Sartori. "A Note on $$\mathfrak {gl}_{m|n}$$ Link Invariants and the HOMFLY–PT Polynomial." In Knots, Low-Dimensional Topology and Applications, 277–86. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-16031-9_13.

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Millett, Kenneth C. "Knot Theory, Jones’ Polynomials, Invariants of 3-Manifolds, and the Topological Theory of Fluid Dynamics." In Topological Aspects of the Dynamics of Fluids and Plasmas, 29–64. Dordrecht: Springer Netherlands, 1992. http://dx.doi.org/10.1007/978-94-017-3550-6_2.

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Lickorish, W. B. Raymond. "3-Manifold Invariants from the Jones Polynomial." In An Introduction to Knot Theory, 133–45. New York, NY: Springer New York, 1997. http://dx.doi.org/10.1007/978-1-4612-0691-0_13.

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Tyurina, Svetlana, and Alexander Varchenko. "Finite-order Invariants for (n, 2)-Torus Knots and the Curve $${Y^2}={X^3}+{X^2}$$." In Notions of Positivity and the Geometry of Polynomials, 401–3. Basel: Springer Basel, 2011. http://dx.doi.org/10.1007/978-3-0348-0142-3_21.

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Bott, Raoul. "On Knot and Manifold Invariants." In NATO ASI Series, 37–52. Boston, MA: Springer US, 1992. http://dx.doi.org/10.1007/978-1-4615-3472-3_2.

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Conference papers on the topic "Invariants of knots and 3-manifolds"

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Jeong, Myeong-Ju, and Chan-Young Park. "Polynomial invariants and Vassiliev invariants." In Invariants of Knots and 3--manifolds. Mathematical Sciences Publishers, 2002. http://dx.doi.org/10.2140/gtm.2002.4.89.

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Turaev, Vladimir. "Torsions of 3–manifolds." In Invariants of Knots and 3--manifolds. Mathematical Sciences Publishers, 2002. http://dx.doi.org/10.2140/gtm.2002.4.295.

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Roberts, Justin, and Justin Sawon. "Generalisations of Rozansky–Witten invariants." In Invariants of Knots and 3--manifolds. Mathematical Sciences Publishers, 2002. http://dx.doi.org/10.2140/gtm.2002.4.263.

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Matveev, Sergei, and Michael Polyak. "Cubic complexes and finite type invariants." In Invariants of Knots and 3--manifolds. Mathematical Sciences Publishers, 2002. http://dx.doi.org/10.2140/gtm.2002.4.215.

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Garoufalidis, Stavros. "Periodicity of Goussarov–Vassiliev knot invariants." In Invariants of Knots and 3--manifolds. Mathematical Sciences Publishers, 2002. http://dx.doi.org/10.2140/gtm.2002.4.43.

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Kamada, Seiichi. "Knot invariants derived from quandles and racks." In Invariants of Knots and 3--manifolds. Mathematical Sciences Publishers, 2002. http://dx.doi.org/10.2140/gtm.2002.4.103.

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Bar-Natan, Dror. "Bracelets and the Goussarov Filtration of the Space of Knots." In Invariants of Knots and 3--manifolds. Mathematical Sciences Publishers, 2002. http://dx.doi.org/10.2140/gtm.2002.4.1.

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Kerler, Thomas. "p–Modular TQFT's, Milnor torsion and the Casson–Lescop invariant." In Invariants of Knots and 3--manifolds. Mathematical Sciences Publishers, 2002. http://dx.doi.org/10.2140/gtm.2002.4.119.

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Baseilhac, Stephane, and Riccardo Benedetti. "QHI, 3–manifolds scissors congruence classes and the volume conjecture." In Invariants of Knots and 3--manifolds. Mathematical Sciences Publishers, 2002. http://dx.doi.org/10.2140/gtm.2002.4.13.

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Kohno, Toshitake. "Loop spaces of configuration spaces and finite type invariants." In Invariants of Knots and 3--manifolds. Mathematical Sciences Publishers, 2002. http://dx.doi.org/10.2140/gtm.2002.4.143.

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