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Dissertations / Theses on the topic 'Knot polynomials'

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Published: 4 June 2021

Last updated: 25 July 2025

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1

Woodard, Mary Kay. "Conway's Link Polynomial: a Generalization of the Classic Alexander's Knot Polynomial." Thesis, North Texas State University, 1986. https://digital.library.unt.edu/ark:/67531/metadc501096/.

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The problem under consideration is that of determining a simple and effective invariant of knots. To this end, the Conway polynomial is defined as a generalization of Alexander's original knot polynomial. It is noted, however, that the Conway polynomial is not a complete invariant. If two knots are equivalent, as defined in this investigation, then they receive identical polynomials. Yet, if two knots have identical polynomials, no information about their equivalence may be obtained. To define the Conway polynomial, the Axioms for Computation are given and many examples of their use are includ

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2

Sacdalan, Alvin Mendoza. "Aspects of the Jones polynomial." CSUSB ScholarWorks, 2006. https://scholarworks.lib.csusb.edu/etd-project/2872.

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A knot invariant called the Jones polynomial will be defined in two ways, as the Kauffman Bracket polynomial and the Tutte polynomial. Three properties of the Jones polynomial are discussed. We also see how mutant knots share the same Jones polynomial.

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3

Keever, Robert Dudley. "Some problems in knot theory." Thesis, University of Edinburgh, 1989. http://hdl.handle.net/1842/12219.

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4

Noble, Steven D. "The complexity of graph polynomials." Thesis, University of Oxford, 1997. http://ora.ox.ac.uk/objects/uuid:c84702b4-b371-474b-a003-4d24f25e5a12.

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This thesis examines graph polynomials and particularly their complexity. We give short proofs of two results from Gessel and Sagan (1996) which present new evaluations of the Tutte polynomial concerning orientations. A theorem of Massey et al (1997) gives an expression concerning the average size of a forest in a graph. We generalise this result to any simplicial complex. We answer a question posed by Kleinschmidt and Onn (1995) by showing that the language of partitionable simplicial complexes is in NP. We prove the following result concerning the complexity of the Tutte polynomial: Theorem

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5

Roberts, Sharleen Adrienne. "Knots Not for Naught." Diss., CLICK HERE for online access, 2006. http://contentdm.lib.byu.edu/ETD/image/etd1446.pdf.

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6

Jacobsson, Magnus. "Khovanov homology and link cobordisms /." Uppsala : Matematiska institutionen, Univ. [distributör], 2003. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-3765.

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7

Lorton, Cody. "On the Breadth of the Jones Polynomial for Certain Classes of Knots and Links." TopSCHOLAR®, 2009. http://digitalcommons.wku.edu/theses/86.

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The problem of finding the crossing number of an arbitrary knot or link is a hard problem in general. Only for very special classes of knots and links can we solve this problem. Often we can only hope to find a lower bound on the crossing number Cr(K) of a knot or a link K by computing the Jones polynomial of K, V(K). The crossing number Cr(K) is bounded from below by the difference between the greatest degree and the smallest degree of the polynomial V(K). However the computation of the Jones polynomial of an arbitrary knot or link is also difficult in general. The goal of this thesis is to f

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8

Petersen, David Alan. "Tutte polynomial in knot theory." CSUSB ScholarWorks, 2007. https://scholarworks.lib.csusb.edu/etd-project/3128.

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This thesis reviews the history of knot theory with an emphasis on the diagrammatic approach to studying knots. Also covered are the basic concepts and notions of graph theory and how these two fields are related with an example of a knot diagram and how to associate it to a graph.

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9

Ameur, Kheira. "Polynomial quandle cocycles, their knot invariants and applications." [Tampa, Fla] : University of South Florida, 2006. http://purl.fcla.edu/usf/dc/et/SFE0001813.

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10

Yokota, Yoshiyuki. "Polynomial invariants of periodic knots /." Electronic version of summary, 1992. http://www.wul.waseda.ac.jp/gakui/gaiyo/1852.pdf.

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Thesis (Sci. D.)--Waseda University, 1992.<br>Accompanied by summary (5 p. : ill. ; 26 cm.) in Japanese. Includes bibliographical references (leaves 58-59). "A list of papers by Yoshiyuki Yokota": leaf 60.

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11

Gaebler, Robert. "Alexander Polynomials of Tunnel Number One Knots." Scholarship @ Claremont, 2004. https://scholarship.claremont.edu/hmc_theses/162.

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Every two-bridge knot or link is characterized by a rational number p/q, and has a fundamental group which has a simple presentation with only two generators and one relator. The relator has a form that gives rise to a formula for the Alexander polynomial of the knot or link in terms of p and q [15]. Every two-bridge knot or link also has a corresponding “up down” graph in terms of p and q. This graph is analyzed combinatorially to prove several properties of the Alexander polynomial. The number of two-bridge knots and links of a given crossing number are also counted.

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12

Tram, Heather. "Khovanov Homology as an Generalization of the Jones Polynomial in Kauffman Terms." OpenSIUC, 2016. https://opensiuc.lib.siu.edu/theses/1987.

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This paper explains the construction of Khovanov homology of which begins by un derstanding how Louis Kauﬀman generalizes the Jones polynomial using a state sum model of the bracket polynomial for an unoriented knot or link and in turn recovers the Jones polynomial, a knot invariant for an oriented knot or link. Kauﬀman associates the unknot by the polynomial (−A2 − A−2) whereas Khovanov associates the unknot by (q + q−1) through a change of variables. As an oriented knot or link K with n crossings produces 2n smoothings, Khovanov builds a commutative cube {0,1}n and associates a graded vector

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13

Lipson, Andrew Solomon. "Polynomial invariants of knots and links." Thesis, University of Cambridge, 1989. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.303206.

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14

Morse, Jennifer. "Explicit expansions for Knop-Sahi and Macdonald polynomials /." Diss., Connect to a 24 p. preview or request complete full text in PDF format. Access restricted to UC campuses, 1999. http://wwwlib.umi.com/cr/ucsd/fullcit?p9935465.

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15

Ahlquist, Mari. "On Knots and DNA." Thesis, Linköpings universitet, Matematiska institutionen, 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-144294.

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Knot theory is the mathematical study of knots. In this thesis we study knots and one of its applications in DNA. Knot theory sits in the mathematical field of topology and naturally this is where the work begins. Topological concepts such as topological spaces, homeomorphisms, and homology are considered. Thereafter knot theory, and in particular, knot theoretical invariants are examined, aiming to provide insights into why it is difficult to answer the question "How can we tell knots appart?". In knot theory invariants such as the bracket polynomial, the Jones polynomial and tricolorability

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16

Collins, Julia. "On the concordance orders of knots." Thesis, University of Edinburgh, 2011. http://hdl.handle.net/1842/5034.

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This thesis develops some general calculational techniques for finding the orders of knots in the topological concordance group C . The techniques currently available in the literature are either too theoretical, applying to only a small number of knots, or are designed to only deal with a specific knot. The thesis builds on the results of Herald, Kirk and Livingston [HKL10] and Tamulis [Tam02] to give a series of criteria, using twisted Alexander polynomials, for determining whether a knot is of infinite order in C. There are two immediate applications of these theorems. The first is to give

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17

Moon, Hyeyoung. "Calculating knot distances and solving tangle equations involving Montesinos links." Diss., University of Iowa, 2010. https://ir.uiowa.edu/etd/859.

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My research area is applications of topology to biology, especially DNA topology. DNA topology studies the shape and path of DNA in three dimensional space. My thesis relates to the study of DNA topology in a protein-DNA complex by solving tangle equations and calculating distances between DNA knots.

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18

Chan, Shih-huang. "Polynomial spline regression with unknown knots and AR(1) errors." The Ohio State University, 1989. http://rave.ohiolink.edu/etdc/view?acc_num=osu1340986578.

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19

Seifert, Bastian [Verfasser], and Knut [Gutachter] Hüper. "Multivariate Chebyshev polynomials and FFT-like algorithms / Bastian Seifert ; Gutachter: Knut Hüper." Würzburg : Universität Würzburg, 2020. http://d-nb.info/1213659728/34.

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20

Watanabe, Tadayuki. "Configuration space integral for long n-knots and the Alexander polynomial." 京都大学 (Kyoto University), 2007. http://hdl.handle.net/2433/136744.

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21

Salazar-Torres, Dido Uvaldo. "The Khovanov homology of the jumping jack." Diss., University of Iowa, 2015. https://ir.uiowa.edu/etd/1745.

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We study the sl(3) web algebra via morphisms on foams. A pre-foam is a cobordism between two webs that contains singular arcs, which are sets of points whose neighborhoods are homeomorphic to the cross-product of the letter "Y'' and the unit interval. Pre-foams may have a distinguished point, and it can be moved around as long as it does not cross a singular arc. A foam is an isotopy class of pre-foams modulo a set of certain relations involving dots on the pre-foams. Composition in Foams is achieved by stacking pre-foams. We compute the cohomology ring of the sl(3) web algebra and apply a fun

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22

Williamson, Mathew. "Kauffman-Harary Conjecture for Virtual Knots." Scholar Commons, 2007. http://scholarcommons.usf.edu/etd/3916.

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In this paper, we examine Fox colorings of virtual knots, and moves called k-swap moves defined for virtual knot diagrams. The k-swap moves induce a one-to-one correspondence between colorings before and after the move, and can be used to reduce the number of virtual crossings. For the study of colorings, we characterize families of alternating virtual knots to generalize (2, n)-torus knots, alternating pretzel knots, and alternating 2-bridge knots. The k-swap moves are then applied to prove a "virtualization" of the Kauffman-Harary conjecture, originally stated for classical knot diagrams, fo

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23

Ahmadi, Abhari Seyed Hamed. "Quantum Algorithms for: Quantum Phase Estimation, Approximation of the Tutte Polynomial and Black-box Structures." Doctoral diss., University of Central Florida, 2012. http://digital.library.ucf.edu/cdm/ref/collection/ETD/id/5096.

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In this dissertation, we investigate three different problems in the field of Quantum computation. First, we discuss the quantum complexity of evaluating the Tutte polynomial of a planar graph. Furthermore, we devise a new quantum algorithm for approximating the phase of a unitary matrix. Finally, we provide quantum tools that can be utilized to extract the structure of black-box modules and algebras. While quantum phase estimation (QPE) is at the core of many quantum algorithms known to date, its physical implementation (algorithms based on quantum Fourier transform (QFT)) is highly con

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24

Tran, Anh Tuan. "The volume conjecture, the aj conjectures and skein modules." Diss., Georgia Institute of Technology, 2012. http://hdl.handle.net/1853/44811.

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This dissertation studies quantum invariants of knots and links, particularly the colored Jones polynomials, and their relationships with classical invariants like the hyperbolic volume and the A-polynomial. We consider the volume conjecture that relates the Kashaev invariant, a specialization of the colored Jones polynomial at a specific root of unity, and the hyperbolic volume of a link; and the AJ conjecture that relates the colored Jones polynomial and the A-polynomial of a knot. We establish the AJ conjecture for some big classes of two-bridge knots and pretzel knots, and confirm the volu

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25

AL-Hashimi, Ghazwan Mohammed. "A ZETA FUNCTION FOR FLOWS WITH L(−1,−1) TEMPLATE." OpenSIUC, 2016. https://opensiuc.lib.siu.edu/dissertations/1291.

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In this dissertation, we study the flows on R3 associated with a nonlinear system differential equation introduced by Clark Robinson in [46]. The periodic orbits are modeled by a semi-flow on the L(−1,−1) template. It is known that these are positive knots, but need not have positive braid presentations. Here we prove that they are fibered. We investigate their linking and we construct a zeta-function that counts periodic orbits according to their twisting. This extends work by M. Sullivan in [55], and [57].

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26

Dehornoy, Pierre. "Invariants topologiques des orbites périodiques d'un champ de vecteurs." Phd thesis, Ecole normale supérieure de lyon - ENS LYON, 2011. http://tel.archives-ouvertes.fr/tel-00656900.

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Cette thèse se situe à l'interface entre théorie des nœuds et théorie des systèmes dynamiques. Le thème central consiste, étant donné un champ de vecteurs dans une variété de dimension 3, à considérer ses orbites périodiques, et à s'interroger sur les informations qu'elles donnent sur le champ de vecteurs et la variété initiaux.La première partie est consacrée au flot géodésique défini sur le fibré unitaire tangentd'une surface, ou d'une orbiface, à courbure constante. L'observation de certains exemples (sphère, tore, surface modulaire) suggère la conjecture suivante, due à Étienne Ghys : l'en

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27

Banks, Jessica E. "The Kakimizu complex of a link." Thesis, University of Oxford, 2012. http://ora.ox.ac.uk/objects/uuid:d89d46a3-03f0-4a71-a746-8f024f988f63.

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We study Seifert surfaces for links, and in particular the Kakimizu complex MS(L) of a link L, which is a simplicial complex that records the structure of the set of taut Seifert surfaces for L. First we study a connection between the reduced Alexander polynomial of a link and the uniqueness of taut Seifert surfaces. Specifically, we reprove and extend a particular case of a result of Juhasz, using very different methods, showing that if a non-split homogeneous link has a reduced Alexander polynomial whose constant term has modulus at most 3 then the link has a unique incompressible Seifert su

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28

Leturcq, David. "Compter des configurations spatiales en dimension impaire avec la torsion de Reidemeister." Thesis, Université Grenoble Alpes, 2020. http://www.theses.fr/2020GRALM025.

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Les nœuds longs étudiés dans cette thèse sont des plongements standard à l'infini de R^n dans un R^{n+2} asymptotique d'homologie entière, pour n impair. Pour ces nœuds, on définit des invariants (Z_k)_{k > 1} à difféomorphismes ambiants triviaux hors d'une boule près. Ces invariants généralisent des invariants (Z_k)_{k>1} définis par Bott, Cattaneo, et Rossi pour les nœuds longs de R^{n+2}, et on donne une définition plus souple de ces invariants. L'invariant Z_k est défini comme une combinaison linéaire d'intégrales de certaines formes différentielles sur des espaces de configurations

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29

Kohli, Ben-Michael. "Les invariants de Links-Gould comme généralisations du polynôme d’Alexander." Thesis, Dijon, 2016. http://www.theses.fr/2016DIJOS062/document.

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On s’intéresse dans cette thèse aux rapports qui existent entre deux invariants d’entrelacs. D’une part l’invariant d’Alexander ∆ qui est l’invariant de nœuds le plus classique, et le plus étudié avec le polynôme de Jones, et d’autre part la famille des invariants de Links-Gould LGn,m qui sont des invariants quantiques dérivés des super algèbres de Hopf Uqgl(n|m). On démontre en particulier un cas de la conjecture de De Wit-Ishii-Links : certaines spécialisa- tions des polynômes de Links-Gould fournissent des puissances du polynôme d’Alexander. Les polynômes LG sont donc des généralisations du

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30

Kucharski, Piotr. "Knots and BPS invariants." Doctoral thesis, 2017. https://depotuw.ceon.pl/handle/item/2213.

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Jednym z najważniejszych celów współczesnej fizyki teoretycznej jest znalezienie zunifikowanego opisu wszystkich oddziaływań fundamentalnych. Model Standardowy stanowi teorię oddziaływań elektromagnetycznych, słabych i silnych, natomiast grawitacja opisywana jest przez ogólną teorię względności. Do dziś nie udało się osiągnąć unifikacji potwierdzonej w eksperymentach, ale jedną z prób odnoszących najwięcej sukcesów jest teoria strun. Oprócz innych osiągnięć, wniosła ona bardzo wysublimowane narzędzia dla fizyki teoretycznej i matematyki. Szczególnie owocne okazały się badania łączące teorię st

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31

Archibald, Jana. "The Multivariable Alexander Polynomial on Tangles." Thesis, 2010. http://hdl.handle.net/1807/26151.

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The multivariable Alexander polynomial (MVA) is a classical invariant of knots and links. We give an extension to regular virtual knots which has simple versions of many of the relations known to hold for the classical invariant. By following the previous proofs that the MVA is of finite type we give a new definition for its weight system which can be computed as the determinant of a matrix created from local information. This is an improvement on previous definitions as it is directly computable (not defined recursively) and is computable in polynomial time. We also show that our e

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32

Ho, Chi Fai. "On Polynomial Invariants for Knots and Links." Thesis, 1986. https://thesis.library.caltech.edu/11454/2/Ho_CF_1986.pdf.

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<p>This thesis presents an investigation of many known polynomial invariants of knots and links. Following Alexander's original idea, we define another multi-indeterminant polynomial for links and show that it satisfies some of Torres' conditions. We conjecture that they are equivalent.</p> <p>Conway polynomials have been known since the sixties. In this paper, we show that the polynomials of various orientations of a link are related, at least in the first and second coefficients. The relationship can be expressed as a function of the Conway polynomials of all sublinks.</p> <p>A new inv

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33

Huynh, Vu Quang. "Reidemeister torsion, twisted Alexander polynomial, the A-polynomial, and the colored Jones polynomial of some classes of knots." 2005. http://wwwlib.umi.com/dissertations/fullcit/3174140.

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Thesis (Ph.D.)--State University of New York at Buffalo, 2005.<br>Title from PDF title page (viewed on Nov. 30, 2005) Available through UMI ProQuest Digital Dissertations. Thesis adviser: Thang Le. Includes bibliographical references.

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34

Long, Ligang. "Slice ribbon conjecture, pretzel knots and mutation." Thesis, 2014. http://hdl.handle.net/2152/27145.

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In this paper we explore the slice-ribbon conjecture for some families of pretzel knots. Donaldson's diagonalization theorem provides a powerful obstruction to sliceness via the union of the double branched cover W of B⁴ over a slicing disk and a plumbing manifold P([capital gamma]). Donaldson's theorem classifies all slice 4-strand pretzel knots up to mutation. The correction term is another 3-manifold invariant defined by Ozsváth and Szabó. For a slice knot K the number of vanishing correction terms of Y[subscript K] is at least the square root of the order of H₁(Y[subscript K];Z). Donalds

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35

Gutierrez, Quispe Robert Gerson. "Aspectos de la teoría de nudos." Bachelor's thesis, 2019. http://hdl.handle.net/11086/14649.

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Tesis (Lic. en Matemática)--Universidad Nacional de Córdoba, Facultad de Matemática, Astronomía, Física y Computación, 2019.<br>Los nudos, tal cual aparecen en nuestra vida cotidiana, son un objeto de estudio en la Matemática. La Teoría de Nudos es la rama de la Matemática que se encarga de su estudio. Un problema central es el de poder decir si dos nudos dados son equivalentes o no. Los matemáticos, en la búsqueda de responder esta pregunta, entre otras, han desarrollado diversas técnicas y herramientas en esta área de estudio. En este trabajo se hace un recorrido en el estudio de la Teoría d

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