Relevant bibliographies by topics / Quandles

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

Academic literature on the topic 'Quandles'

Author: Grafiati
Published: 4 June 2021
Last updated: 9 March 2023

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Journal articles on the topic "Quandles"

1

Crans, Alissa S., and Sam Nelson. "Hom quandles." Journal of Knot Theory and Its Ramifications 23, no. 02 (February 2014): 1450010. http://dx.doi.org/10.1142/s0218216514500102.

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If A is an abelian quandle and Q is a quandle, the hom set Hom (Q, A) of quandle homomorphisms from Q to A has a natural quandle structure. We exploit this fact to enhance the quandle counting invariant, providing an example of links with the same counting invariant values but distinguished by the hom quandle structure. We generalize the result to the case of biquandles, collect observations and results about abelian quandles and the hom quandle, and show that the category of abelian quandles is symmetric monoidal closed.
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Bae, Yongju, and Seongjeong Kim. "On quotient structure of Takasaki quandles II." Journal of Knot Theory and Its Ramifications 23, no. 07 (June 2014): 1460012. http://dx.doi.org/10.1142/s0218216514600128.

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A Takasaki quandle (T(G), *) is a quandle under the binary operation * defined by a*b = 2b-a for an abelian group (G, +). In this paper, we will show that if a subquandle X of a Takasaki quandle G is a image of subgroup of G under a quandle automorphism of T(G), then the set {X * g | g ∈ G} is a quandle under the binary operation *′ defined by (X * g) *′ (X * h) = X * (g * h). On the other hand, the quotient structure studied in [On quotients of quandles, J. Knot Theory Ramifications 19(9) (2010) 1145–1156] can be applied to the Takasaki quandles. In this paper, we will review the quotient structure studied in [On quotients of quandles, J. Knot Theory Ramifications 19(9) (2010) 1145–1156], and show that the quotient quandle coincides with the quotient quandle defined by Bunch, Lofgren, Rapp and Yetter in [On quotients of quandles, J. Knot Theory Ramifications 19(9) (2010) 1145–1156] for connected Takasaki quandles.
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Bae, Yongju, and Byeorhi Kim. "On sufficient conditions for being finite quandles." Journal of Knot Theory and Its Ramifications 28, no. 13 (November 2019): 1940011. http://dx.doi.org/10.1142/s021821651940011x.

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In 2006, Nelson and Wong introduced that every finite quandle can be decomposed into a disjoint union of some quandles and for given [Formula: see text] quandles satisfying certain conditions, there is a quandle which is a disjoint union of the [Formula: see text] quandles. In 2008, Ehrman, Gurpinar, Thibault and Yetter also introduced similar results about the decomposition of quandles. In this paper, we observe an operation table [Formula: see text] which consists of four sub-operation tables, where diagonal tables are quandle operations. We study which conditions make [Formula: see text] a quandle operation table.
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Singh, Mahender. "Classification of flat connected quandles." Journal of Knot Theory and Its Ramifications 25, no. 13 (November 2016): 1650071. http://dx.doi.org/10.1142/s0218216516500711.

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Let [Formula: see text] be an additive abelian group. Then the binary operation [Formula: see text] gives a quandle structure on [Formula: see text], denoted by [Formula: see text], and called the Takasaki quandle of [Formula: see text]. Viewing quandles as generalization of Riemannian symmetric spaces, Ishihara and Tamaru [Flat connected finite quandles, to appear in Proc. Amer. Math. Soc. (2016)] introduced flat quandles, and classified quandles which are finite, flat and connected. In this note, we classify all flat connected quandles. More precisely, we prove that a quandle [Formula: see text] is flat and connected if and only if [Formula: see text], where [Formula: see text] is a 2-divisible group.
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Bardakov, Valeriy, and Timur Nasybullov. "Embeddings of quandles into groups." Journal of Algebra and Its Applications 19, no. 07 (July 31, 2019): 2050136. http://dx.doi.org/10.1142/s0219498820501364.

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In this paper, we introduce the new construction of quandles. For a group [Formula: see text] and a subset [Formula: see text] of [Formula: see text] we construct a quandle [Formula: see text] which is called the [Formula: see text]-quandle and study properties of this quandle. In particular, we prove that if [Formula: see text] is a quandle such that the natural map [Formula: see text] from [Formula: see text] to the enveloping group [Formula: see text] of [Formula: see text] is injective, then [Formula: see text] is the [Formula: see text]-quandle for an appropriate group [Formula: see text] and a subset [Formula: see text] of [Formula: see text]. Also we introduce the free product of quandles and study this construction for [Formula: see text]-quandles. In addition, we classify all finite quandles with enveloping group [Formula: see text].
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INOUE, AYUMU. "QUANDLE HOMOMORPHISMS OF KNOT QUANDLES TO ALEXANDER QUANDLES." Journal of Knot Theory and Its Ramifications 10, no. 06 (September 2001): 813–21. http://dx.doi.org/10.1142/s0218216501001177.

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A quandle is a set with a binary operation satisfying some properties. A quandle homomorphism is a map between quandles preserving the structure of their binary operations. A knot determines a quandle called a knot quandle. We show that the number of all quandle homomorphisms of a knot quandle of a knot to an Alexander quandle is completely determined by Alexander polynomials of the knot. Further we show that the set of all quandle homomorphisms of a knot quandle to an Alexander quandle has a module structure.
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Bardakov, Valeriy G., Inder Bir S. Passi, and Mahender Singh. "Quandle rings." Journal of Algebra and Its Applications 18, no. 08 (July 5, 2019): 1950157. http://dx.doi.org/10.1142/s0219498819501573.

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In this paper, a theory of quandle rings is proposed for quandles analogous to the classical theory of group rings for groups, and interconnections between quandles and associated quandle rings are explored.
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Clark, W. Edwin, Mohamed Elhamdadi, Masahico Saito, and Timothy Yeatman. "Quandle colorings of knots and applications." Journal of Knot Theory and Its Ramifications 23, no. 06 (May 2014): 1450035. http://dx.doi.org/10.1142/s0218216514500357.

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We present a set of 26 finite quandles that distinguish (up to reversal and mirror image) by number of colorings, all of the 2977 prime oriented knots with up to 12 crossings. We also show that 1058 of these knots can be distinguished from their mirror images by the number of colorings by quandles from a certain set of 23 finite quandles. We study the colorings of these 2977 knots by all of the 431 connected quandles of order at most 35 found by Vendramin. Among other things, we collect information about quandles that have the same number of colorings for all of the 2977 knots. For example, we prove that if Q is a simple quandle of prime power order then Q and the dual quandle Q* of Q have the same number of colorings for all knots and conjecture that this holds for all Alexander quandles Q. We study a knot invariant based on a quandle homomorphism f : Q1 → Q0. We also apply the quandle colorings we have computed to obtain some new results for the bridge index, the Nakanishi index, the tunnel number, and the unknotting number. In an appendix we discuss various properties of the quandles in Vendramin's list. Links to the data computed and various programs in C, GAP and Maple are provided.
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AMEUR, KHEIRA, and MASAHICO SAITO. "POLYNOMIAL COCYCLES OF ALEXANDER QUANDLES AND APPLICATIONS." Journal of Knot Theory and Its Ramifications 18, no. 02 (February 2009): 151–65. http://dx.doi.org/10.1142/s0218216509006938.

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Cocycles are constructed by polynomial expressions for Alexander quandles. As applications, non-triviality of some quandle homology groups are proved, and quandle cocycle invariants of knots are studied. In particular, for an infinite family of quandles, the non-triviality of quandle homology groups is proved for all odd dimensions.
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Clark, W. Edwin, Masahico Saito, and Leandro Vendramin. "Quandle coloring and cocycle invariants of composite knots and abelian extensions." Journal of Knot Theory and Its Ramifications 25, no. 05 (April 2016): 1650024. http://dx.doi.org/10.1142/s0218216516500243.

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Quandle colorings and cocycle invariants are studied for composite knots, and applied to chirality and abelian extensions. The square and granny knots, for example, can be distinguished by quandle colorings, so that a trefoil and its mirror can be distinguished by quandle coloring of composite knots. We investigate this and related phenomena. Quandle cocycle invariants are studied in relation to quandle coloring of the connected sum, and formulas are given for computing the cocycle invariant from the number of colorings of composite knots. Relations to corresponding abelian extensions of quandles are studied, and extensions are examined for the table of small connected quandles, called Rig quandles. Computer calculations are presented, and summaries of outputs are discussed.
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Dissertations / Theses on the topic "Quandles"

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Churchill, Indu Rasika U. "Contributions to Quandle Theory: A Study of f-Quandles, Extensions, and Cohomology." Scholar Commons, 2017. http://scholarcommons.usf.edu/etd/6814.

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Quandles are distributive algebraic structures that were introduced by David Joyce [24] in his Ph.D. dissertation in 1979 and at the same time in separate work by Matveev [34]. Quandles can be used to construct invariants of the knots in the 3-dimensional space and knotted surfaces in 4-dimensional space. Quandles can also be studied on their own right as any non-associative algebraic structures. In this dissertation, we introduce f-quandles which are a generalization of usual quandles. In the first part of this dissertation, we present the definitions of f-quandles together with examples, and properties. Also, we provide a method of producing a new f-quandle from a given f-quandle together with a given homomorphism. Extensions of f-quandles with both dynamical and constant cocycles theory are discussed. In Chapter 4, we provide cohomology theory of f-quandles in Theorem 4.1.1 and briefly discuss the relationship between Knot Theory and f-quandles. In the second part of this dissertation, we provide generalized 2,3, and 4- cocycles for Alexander f-quandles with a few examples. Considering “Hom-algebraic Structures” as our nutrient enriched soil, we planted “quandle” seeds to get f-quandles. Over the last couple of years, this f- quandle plant grew into a tree. We believe this tree will continue to grow into a larger tree that will provide future fruit and contributions.
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Budden, Stephen Mark. "Knots and quandles." Thesis, University of Auckland, 2009. http://hdl.handle.net/2292/5292.

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Quandles were introduced to Knot Theory in the 1980s as an almost complete algebraic invariant for knots and links. Like their more basic siblings, groups, they are difficult to distinguish so a major challenge is to devise means for determining when two quandles having different presentations are really different. This thesis addresses this point by studying algebraic aspects of quandles. Following what is mainly a recapitulation of existing work on quandles, we firstly investigate how a link quandle is related to the quandles of the individual components of the link. Next we investigate coset quandles. These are motivated by the transitive action of the operator, associated and automorphism group actions on a given quandle, allowing techniques of permutation group theory to be used. We will show that the class of all coset quandles includes the class of all Alexander quandles; indeed all group quandles. Coset quandles are used in two ways: to give representations of connected quandles, which include knot quandles; and to provide target quandles for homomorphism invariants which may be useful in enabling one to distinguish quandles by counting homomorphisms onto target quandles. Following an investigation of the information loss in going from the fundamental quandle of a link to the fundamental group, we apply our techniques to calculations for the figure eight knot and braid index two knots and involving lower triangular matrix groups. The thesis is rounded out by two appendices, one giving a short table of knot quandles for knots up to six crossings and the other a computer program for computing the homomorphism invariants.
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Macquarrie, Jennifer. "Automorphism Groups of Quandles." Scholar Commons, 2011. http://scholarcommons.usf.edu/etd/3226.

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This thesis arose from a desire to better understand the structures of automorphism groups and inner automorphism groups of quandles. We compute and give the structure of the automorphism groups of all dihedral quandles. In their paper Matrices and Finite Quandles, Ho and Nelson found all quandles (up to isomorphism) of orders 3, 4, and 5 and determined their automorphism groups. Here we find the automorphism groups of all quandles of orders 6 and 7. There are, up to isomoprhism, 73 quandles of order 6 and 289 quandles of order 7.
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Tamagawa, Sherilyn K. "Quandles of Virtual Knots." Scholarship @ Claremont, 2014. http://scholarship.claremont.edu/scripps_theses/358.

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Knot theory is an important branch of mathematics with applications in other branches of science. In this paper, we explore invariants on a special class of knots, known as virtual knots. We find new invariants by taking quotients of quandles, and introducing the fundamental Latin Alexander quandle and its Grobner basis. We also demonstrate examples of computations of these invariants.
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Paterson, Douglas James. "Finite representations of link quandles." Thesis, University of Warwick, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.272508.

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Jackson, Nicholas James. "Homological algebra of racks and quandles." Thesis, University of Warwick, 2004. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.406781.

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Green, Matthew J. "Generalizations of Quandles and their cohomologies." Scholar Commons, 2018. https://scholarcommons.usf.edu/etd/7299.

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Quandles are distributive algebraic structures originally introduced independently by David Joyce and Sergei Matveev in 1979, motivated by the study of knots. In this dissertation, we discuss a number of generalizations of the notion of quandles. In the first part of this dissertation we discuss biquandles, in the context of augmented biquandles, a representation of biquandles in terms of actions of a set by an augmentation group. Using this representation we are able to develop a homology and cohomology theory for these structures. We then introduce an n-ary generalization of the notion of quandles. We discuss a number of properties of these structures and provide a number of examples. Also discussed are methods of obtaining n-ary quandles through iteration of binary quandles, and obtaining binary quandles from n-ary quandles, along with a classification of low order ternary quandles. We build upon this generalization, introducing n-ary f-quandles, and similarly discuss examples, properties, and relations between the n-ary structures and their binary counter parts, as well as low order classification of ternary f-quandles. Finally we present cohomology theory for general n-ary f-quandles.
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Appiou, Nikiforou Marina. "Extensions of Quandles and Cocycle Knot Invariants." [Tampa, Fla.] : University of South Florida, 2002. http://purl.fcla.edu/fcla/etd/SFE0000125.

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Rehman, Naqeeb Ur [Verfasser], and I. [Akademischer Betreuer] Heckenberger. "Quandles and Hurwitz Orbits / Naqeeb ur Rehman ; Betreuer: I. Heckenberger." Marburg : Philipps-Universität Marburg, 2016. http://d-nb.info/1114394920/34.

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Rehman, Naqeeb ur [Verfasser], and I. [Akademischer Betreuer] Heckenberger. "Quandles and Hurwitz Orbits / Naqeeb ur Rehman ; Betreuer: I. Heckenberger." Marburg : Philipps-Universität Marburg, 2016. http://d-nb.info/1114394920/34.

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Books on the topic "Quandles"

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Nosaka, Takefumi. Quandles and Topological Pairs. Singapore: Springer Singapore, 2017. http://dx.doi.org/10.1007/978-981-10-6793-8.

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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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Der Aufstieg der Quandts: Eine deutsche Unternehmerdynastie. München: Beck, 2011.

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Caring Hands: The Quandries and Chronicles of a Body Removal Technician. [United States]: Hazel, 2017.

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Nomination of Paul A. Quander, Jr.: Hearing before the Committee on Governmental Affairs, United States Senate, One Hundred Seventh Congress, second session on the nomination of Paul A. Quander, Jr. to be Director, District of Columbia Court Services and Offender Supervision Agency, April 11, 2002. Washington: U.S. G.P.O., 2002.

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Nosaka, Takefumi. Quandles and Topological Pairs: Symmetry, Knots, and Cohomology. Springer, 2017.

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Die Quandts. Luebbe Verlagsgruppe, 2004.

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Jungbluth, Rüdiger. Die Quandts. Campus Sachbuch, 2002.

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Basing, Linda. Spectrum of Quands. Independently Published, 2019.

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Jungbluth, Rüdiger. Die Quandts: Deutschlands erfolgreichste Unternehmerfamilie. Campus Verlag GmbH, 2015.

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Book chapters on the topic "Quandles"

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Kamada, Seiichi. "Quandles." In Springer Monographs in Mathematics, 125–55. Singapore: Springer Singapore, 2017. http://dx.doi.org/10.1007/978-981-10-4091-7_8.

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Nosaka, Takefumi. "Basics of Quandles." In SpringerBriefs in Mathematics, 5–18. Singapore: Springer Singapore, 2017. http://dx.doi.org/10.1007/978-981-10-6793-8_2.

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Elhamdadi, Mohamed. "Distributivity in Quandles and Quasigroups." In Springer Proceedings in Mathematics & Statistics, 325–40. Berlin, Heidelberg: Springer Berlin Heidelberg, 2014. http://dx.doi.org/10.1007/978-3-642-55361-5_19.

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Churchill, Indu Rasika, Mohamed Elhamdadi, and Nicolas Van Kempen. "On the Classification of f-Quandles." In Springer Proceedings in Mathematics & Statistics, 359–69. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-41850-2_14.

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Belk, James, and Robert W. McGrail. "The Word Problem for Finitely Presented Quandles is Undecidable." In Logic, Language, Information, and Computation, 1–13. Berlin, Heidelberg: Springer Berlin Heidelberg, 2015. http://dx.doi.org/10.1007/978-3-662-47709-0_1.

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Carter, Scott, Seiichi Kamada, and Masahico Saito. "Quandle Cocycle Invariants." In Surfaces in 4-Space, 123–66. Berlin, Heidelberg: Springer Berlin Heidelberg, 2004. http://dx.doi.org/10.1007/978-3-662-10162-9_4.

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Kamada, Seiichi. "Quandle Homology Groups and Invariants." In Springer Monographs in Mathematics, 157–72. Singapore: Springer Singapore, 2017. http://dx.doi.org/10.1007/978-981-10-4091-7_9.

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Nosaka, Takefumi. "Topology on the Quandle Homotopy Invariant." In SpringerBriefs in Mathematics, 59–70. Singapore: Springer Singapore, 2017. http://dx.doi.org/10.1007/978-981-10-6793-8_6.

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Nosaka, Takefumi. "Some of Quandle Cocycle Invariants of Links." In SpringerBriefs in Mathematics, 33–44. Singapore: Springer Singapore, 2017. http://dx.doi.org/10.1007/978-981-10-6793-8_4.

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Clark, W. Edwin, and Masahico Saito. "Algebraic and Computational Aspects of Quandle 2-Cocycle Invariant." In Knots, Low-Dimensional Topology and Applications, 147–62. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-16031-9_6.

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Conference papers on the topic "Quandles"

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McCarron, James. "Small homogeneous quandles." In the 37th International Symposium. New York, New York, USA: ACM Press, 2012. http://dx.doi.org/10.1145/2442829.2442867.

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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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KAMADA, Seiichi. "QUANDLES WITH GOOD INVOLUTIONS, THEIR HOMOLOGIES AND KNOT INVARIANTS." In Intelligence of Low Dimensional Topology 2006 - The International Conference. WORLD SCIENTIFIC, 2007. http://dx.doi.org/10.1142/9789812770967_0013.

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McGrail, Robert W., Thuy Trang Nguyen, Thanh Thuy Trang Tran, and Arti Tripathi. "A Terminating and Confluent Term Rewriting System for the Pure Equational Theory of Quandles." In 2018 20th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC). IEEE, 2018. http://dx.doi.org/10.1109/synasc.2018.00035.

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IWAKIRI, Masahide. "QUANDLE COCYCLE INVARIANTS OF TORUS LINKS." In Intelligence of Low Dimensional Topology 2006 - The International Conference. WORLD SCIENTIFIC, 2007. http://dx.doi.org/10.1142/9789812770967_0008.

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