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Journal articles on the topic 'Quandles'

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Published: 4 June 2021

Last updated: 9 March 2023

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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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2

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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3

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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4

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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5

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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6

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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7

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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8

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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9

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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10

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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11

Bonatto, Marco, and Petr Vojtěchovský. "Simply connected latin quandles." Journal of Knot Theory and Its Ramifications 27, no. 11 (October 2018): 1843006. http://dx.doi.org/10.1142/s021821651843006x.

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A (left) quandle is connected if its left translations generate a group that acts transitively on the underlying set. In 2014, Eisermann introduced the concept of quandle coverings, corresponding to constant quandle cocycles of Andruskiewitsch and Graña. A connected quandle is simply connected if it has no nontrivial coverings, or, equivalently, if all its second constant cohomology sets with coefficients in symmetric groups are trivial. In this paper, we develop a combinatorial approach to constant cohomology. We prove that connected quandles that are affine over cyclic groups are simply connected (extending a result of Graña for quandles of prime size) and that finite doubly transitive quandles of order different from [Formula: see text] are simply connected. We also consider constant cohomology with coefficients in arbitrary groups.

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12

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

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

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13

ELHAMDADI, MOHAMED, JENNIFER MACQUARRIE, and RICARDO RESTREPO. "AUTOMORPHISM GROUPS OF QUANDLES." Journal of Algebra and Its Applications 11, no. 01 (February 2012): 1250008. http://dx.doi.org/10.1142/s0219498812500089.

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We prove that the automorphism group of the dihedral quandle with n elements is isomorphic to the affine group of the integers mod n, and also obtain the inner automorphism group of this quandle. In [B. Ho and S. Nelson, Matrices and finite quandles, Homology Homotopy Appl.7(1) (2005) 197–208.], automorphism groups of quandles (up to isomorphisms) of order less than or equal to 5 were given. With the help of the software Maple, we compute the inner and automorphism groups of all seventy three quandles of order six listed in the appendix of [S. Carter, S. Kamada and M. Saito, Surfaces in 4-Space, Encyclopaedia of Mathematical Sciences, Vol. 142, Low-Dimensional Topology, III (Springer-Verlag, Berlin, 2004)]. Since computations of automorphisms of quandles relate to the problem of classification of quandles, we also describe an algorithm implemented in C for computing all quandles (up to isomorphism) of order less than or equal to nine.

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14

NELSON, SAM, and CHAU-YIM WONG. "ON THE ORBIT DECOMPOSITION OF FINITE QUANDLES." Journal of Knot Theory and Its Ramifications 15, no. 06 (August 2006): 761–72. http://dx.doi.org/10.1142/s0218216506004701.

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We study the structure of finite quandles in terms of subquandles. Every finite quandle Q decomposes in a natural way as a union of disjoint Q-complemented subquandles; this decomposition coincides with the usual orbit decomposition of Q. Conversely, the structure of a finite quandle with a given orbit decomposition is determined by its structure maps. We describe an algorithm for finding quandle structures on a disjoint union of quandles.

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15

ZABLOW, JOEL. "SOME CONSEQUENCES OF SMALL INTERSECTION NUMBERS IN THE DEHN QUANDLE AND BEYOND." Journal of Knot Theory and Its Ramifications 20, no. 12 (December 2011): 1741–68. http://dx.doi.org/10.1142/s0218216511009480.

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We extend the notion of geometric intersection numbers 0 and 1, for circles in the Dehn quandle, to general quandles, assuming existence of elements having certain algebraic properties in the latter. These properties enable the construction of many useful 2 and 3 cycles in Dehn quandle homology. Particularly, we construct a special 2-cycle homology representative and an operation which promotes it to higher dimensional cycles of the same type, in the Dehn quandle, and show how to do so in any quandles admitting such elements. For such quandles, this gives some non-triviality results in the quandle homology. We look at a secondary rack formed by tuples of elements of the Dehn quandle and using the ideas above, obtain some basic non-triviality results in its homology as well. We also give some initial possibilities for its application in representing monodromy of singularities of elliptic fibrations.

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16

BUNCH, E., P. LOFGREN, A. RAPP, and D. N. YETTER. "ON QUOTIENTS OF QUANDLES." Journal of Knot Theory and Its Ramifications 19, no. 09 (September 2010): 1145–56. http://dx.doi.org/10.1142/s021821651000839x.

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This paper summarizes substantive new results derived by a student team (the first three authors) under the direction of the fourth author at the 2008 session of the KSU REU "Brainstorming and Barnstorming". The main results show that the construction of the inner automorphism group of a quandle gives rise to a functor from the category of quandles and surjective quandle homomorphisms to the category of groups, characterize quotient maps of quandles which do not change the group of inner automorphims, and characterize those normal subgroups of the inner automorphism group which arise as kernels of homomorphisms induced by quandle surjections.

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17

Clark, W. Edwin, and Masahico Saito. "Algebraic properties of quandle extensions and values of cocycle knot invariants." Journal of Knot Theory and Its Ramifications 25, no. 14 (December 2016): 1650080. http://dx.doi.org/10.1142/s0218216516500802.

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Quandle 2-cocycles define invariants of classical and virtual knots, and extensions of quandles. We show that the quandle 2-cocycle invariant with respect to a non-trivial [Formula: see text]-cocycle is constant, or takes some other restricted form, for classical knots when the corresponding extensions satisfy certain algebraic conditions. In particular, if an abelian extension is a conjugation quandle, then the corresponding cocycle invariant is constant. Specific examples are presented from the list of connected quandles of order less than 48. Relations among various quandle epimorphisms involved are also examined.

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18

Bae, Yongju, and Seonmi Choi. "On rack homology groups of finite quandles via permutations." Journal of Knot Theory and Its Ramifications 28, no. 11 (October 2019): 1940004. http://dx.doi.org/10.1142/s0218216519400042.

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A quandle is a set equipped with a binary operation satisfying three quandle axioms. It also can be expressed as a sequence of permutations of the underlying set satisfying certain conditions. In this paper, we will calculate the second rack homology group of the disjoint union of two finite quandles and the second and third rack homology groups of certain type of finite quandles.

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19

OSHIRO, KANAKO. "HOMOLOGY GROUPS OF TRIVIAL QUANDLES WITH GOOD INVOLUTIONS AND TRIPLE LINKING NUMBERS OF SURFACE-LINKS." Journal of Knot Theory and Its Ramifications 20, no. 04 (April 2011): 595–608. http://dx.doi.org/10.1142/s0218216511009340.

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The purpose of this paper is to determine the homology groups of trivial quandles with good involutions. We also show that the quandle cocycle invariants of surface-links obtained from trivial quandles with good involutions are equivalent to the triple linking numbers.

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20

Ishikawa, Katsumi. "Knot quandles vs. knot biquandles." International Journal of Mathematics 31, no. 02 (January 8, 2020): 2050015. http://dx.doi.org/10.1142/s0129167x20500159.

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As a generalization of quandles, biquandles have given many invariants of classical/surface/virtual links. In this paper, we show that the fundamental quandle [Formula: see text] of any classical/surface link [Formula: see text] detects the fundamental biquandle [Formula: see text]; more precisely, there exists a functor [Formula: see text] from the category of quandles to that of biquandles such that [Formula: see text]. Then, we can expect invariants from biquandles to be reduced to those from quandles. In fact, we introduce a right-adjoint functor [Formula: see text] of [Formula: see text], which implies that the coloring number of a biquandle [Formula: see text] is equal to that of the quandle [Formula: see text].

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21

ASHIHARA, SOSUKE. "FUNDAMENTAL BIQUANDLES OF RIBBON 2-KNOTS AND RIBBON TORUS-KNOTS WITH ISOMORPHIC FUNDAMENTAL QUANDLES." Journal of Knot Theory and Its Ramifications 23, no. 01 (January 2014): 1450001. http://dx.doi.org/10.1142/s0218216514500011.

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The fundamental quandles and biquandles are invariants of classical knots and surface knots. It is unknown whether there exist classical or surface knots which have isomorphic fundamental quandles and distinct fundamental biquandles. We show that ribbon 2-knots or ribbon torus-knots with isomorphic fundamental quandles have isomorphic fundamental biquandles. For this purpose, we give a method for obtaining a presentation of the fundamental biquandle of a ribbon 2-knot/torus-knot from its fundamental quandle.

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22

Kamada, Seiichi, Jieon Kim, and Sang Youl Lee. "Computations of quandle cocycle invariants of surface-links using marked graph diagrams." Journal of Knot Theory and Its Ramifications 24, no. 10 (September 2015): 1540010. http://dx.doi.org/10.1142/s0218216515400106.

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By using the cohomology theory of quandles, quandle cocycle invariants and shadow quandle cocycle invariants are defined for oriented links and surface-links via broken surface diagrams. By using symmetric quandles, symmetric quandle cocycle invariants are also defined for unoriented links and surface-links via broken surface diagrams. A marked graph diagram is a link diagram possibly with 4-valent vertices equipped with markers. Lomonaco, Jr. and Yoshikawa introduced a method of describing surface-links by using marked graph diagrams. In this paper, we give interpretations of these quandle cocycle invariants in terms of marked graph diagrams, and introduce a method of computing them from marked graph diagrams.

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23

Clark, W. Edwin, Larry A. Dunning, and Masahico Saito. "Computations of quandle 2-cocycle knot invariants without explicit 2-cocycles." Journal of Knot Theory and Its Ramifications 26, no. 07 (February 24, 2017): 1750035. http://dx.doi.org/10.1142/s0218216517500353.

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We explore a knot invariant derived from colorings of corresponding [Formula: see text]-tangles with arbitrary connected quandles. When the quandle is an abelian extension of a certain type the invariant is equivalent to the quandle [Formula: see text]-cocycle invariant. We construct many such abelian extensions using generalized Alexander quandles without explicitly finding [Formula: see text]-cocycles. This permits the construction of many [Formula: see text]-cocycle invariants without exhibiting explicit [Formula: see text]-cocycles. We show that for connected generalized Alexander quandles the invariant is equivalent to Eisermann’s knot coloring polynomial. Computations using this technique show that the [Formula: see text]-cocycle invariant distinguishes all of the oriented prime knots up to 11 crossings and most oriented prime knots with 12 crossings including classification by symmetry: mirror images, reversals, and reversed mirrors.

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24

Ryder, Hayley. "An algebraic condition to determine whether a knot is prime." Mathematical Proceedings of the Cambridge Philosophical Society 120, no. 3 (October 1996): 385–89. http://dx.doi.org/10.1017/s0305004100074958.

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A quandle is the algebraic distillation of the second and third Reidemeister moves, on unframed links. Quandles have been studied by many people including Kauffman[K], Fenn and Rourke[F-R] and Joyce [J]. Joyce defines the fundamental quandle, which is a classifying invariant of irreducible, unframed links. Fenn and Rourke, who study generalised quandles or racks in [F—R], show that the natural map from the fundamental quandle of a knot K to the knot group is never injective if K is a connected sum. We now prove that the natural map from the fundamental quandle to the fundamental group of the complement of a link in S3 is injective if and only if the link is prime.

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25

JANG, YEONHEE, and KANAKO OSHIRO. "SYMMETRIC QUANDLE COLORINGS FOR SPATIAL GRAPHS AND HANDLEBODY-LINKS." Journal of Knot Theory and Its Ramifications 21, no. 04 (April 2012): 1250050. http://dx.doi.org/10.1142/s0218216511010024.

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In this paper, colorings by symmetric quandles for spatial graphs and handlebody-links are introduced. We also introduce colorings by LH-quandles for LH-links. LH-links are handlebody-links, some of whose circle components are specified, which are related to Heegaard splittings of link exteriors. We also discuss quandle (co)homology groups and cocycle invariants.

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26

NELSON, SAM. "A POLYNOMIAL INVARIANT OF FINITE QUANDLES." Journal of Algebra and Its Applications 07, no. 02 (April 2008): 263–73. http://dx.doi.org/10.1142/s0219498808002801.

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We define a two-variable polynomial invariant of finite quandles. In many cases this invariant completely determines the algebraic structure of the quandle up to isomorphism. We use this polynomial to define a family of link invariants which generalize the quandle counting invariant.

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27

Bae, Yongju, and Seonmi Choi. "On properties of commutative Alexander quandles." Journal of Knot Theory and Its Ramifications 23, no. 07 (June 2014): 1460013. http://dx.doi.org/10.1142/s021821651460013x.

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An Alexander quandle Mt is an abelian group M with a quandle operation a * b = ta + (1 - t)b where t is a group automorphism of the abelian group M. In this paper, we will study the commutativity of an Alexander quandle and introduce the relationship between Alexander quandles Mt and M1-t determined by group automorphisms t and 1 - t, respectively.

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28

HARRELL, NATASHA, and SAM NELSON. "QUANDLES AND LINKING NUMBER." Journal of Knot Theory and Its Ramifications 16, no. 10 (December 2007): 1283–93. http://dx.doi.org/10.1142/s0218216507005804.

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We study the quandle counting invariant for a certain family of finite quandles with trivial orbit subquandles. We show how these invariants determine the linking number of classical two-component links up to sign.

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29

Carter, J. Scott, Mohamed Elhamdadi, Marina Appiou Nikiforou, and Masahico Saito. "Extensions of Quandles and Cocycle Knot Invariants." Journal of Knot Theory and Its Ramifications 12, no. 06 (September 2003): 725–38. http://dx.doi.org/10.1142/s0218216503002718.

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Quandle cocycles are constructed from extensions of quandles. The theory is parallel to that of group cohomology and group extensions. An interpretation of quandle cocycle invariants as obstructions to extending knot colorings is given, and is extended to links component-wise.

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30

Elhamdadi, Mohamed, Minghui Liu, and Sam Nelson. "Quasi-trivial quandles and biquandles, cocycle enhancements and link-homotopy of pretzel links." Journal of Knot Theory and Its Ramifications 27, no. 11 (October 2018): 1843007. http://dx.doi.org/10.1142/s0218216518430071.

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We investigate some algebraic structures called quasi-trivial quandles and we use them to study link-homotopy of pretzel links. Precisely, a necessary and sufficient condition for a pretzel link with at least two components being trivial under link-homotopy is given. We also generalize the quasi-trivial quandle idea to the case of biquandles and consider enhancement of the quasi-trivial biquandle cocycle counting invariant by quasi-trivial biquandle cocycles, obtaining invariants of link-homotopy type of links analogous to the quasi-trivial quandle cocycle invariants in Inoue’s paper [A. Inoue, Quasi-triviality of quandles for link-homotopy, J. Knot Theory Ramifications 22(6) (2013) 1350026, doi:10.1142/S0218216513500260, MR3070837].

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31

BAE, YONGJU. "COLORING LINK DIAGRAMS BY ALEXANDER QUANDLES." Journal of Knot Theory and Its Ramifications 21, no. 10 (July 11, 2012): 1250094. http://dx.doi.org/10.1142/s0218216512500940.

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In this paper, we study the colorability of link diagrams by the Alexander quandles. We show that if the reduced Alexander polynomial ΔL(t) is vanishing, then L admits a non-trivial coloring by any non-trivial Alexander quandle Q, and that if ΔL(t) = 1, then L admits only the trivial coloring by any Alexander quandle Q, also show that if ΔL(t) ≠ 0, 1, then L admits a non-trivial coloring by the Alexander quandle Λ/(ΔL(t)).

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32

Churchill, Indu R. U., Mohamed Elhamdadi, Matthew Green, and Abdenacer Makhlouf. "f-Racks, f-quandles, their extensions and cohomology." Journal of Algebra and Its Applications 16, no. 11 (October 4, 2017): 1750215. http://dx.doi.org/10.1142/s0219498817502152.

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The purpose of this paper is to introduce and study the notions of [Formula: see text]-rack and [Formula: see text]-quandle which are obtained by twisting the usual equational identities by a map. We provide some key constructions, examples and classification of low order [Formula: see text]-quandles. Moreover, we define modules over [Formula: see text]-racks, discuss extensions and define a cohomology theory for [Formula: see text]-quandles and give examples.

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33

Watanabe, Taisuke. "The structure of connected quandles with the profile {1,ℓ,ℓ} or {1,ℓ,ℓ,ℓ} and its inner group." Journal of Knot Theory and Its Ramifications 26, no. 01 (January 2017): 1750001. http://dx.doi.org/10.1142/s0218216517500018.

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In this paper, we study connected quandles with the profile [Formula: see text] or [Formula: see text]. For such a quandle, we prove that its inner group is a Frobenius group, its order is a prime power and it is an affine quandle.

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34

BROOKE-TAYLOR, ANDREW D., and SHEILA K. MILLER. "THE QUANDARY OF QUANDLES: A BOREL COMPLETE KNOT INVARIANT." Journal of the Australian Mathematical Society 108, no. 2 (October 30, 2019): 262–77. http://dx.doi.org/10.1017/s1446788719000399.

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We show that the isomorphism problems for left distributive algebras, racks, quandles and kei are as complex as possible in the sense of Borel reducibility. These algebraic structures are important for their connections with the theory of knots, links and braids. In particular, Joyce showed that a quandle can be associated with any knot, and this serves as a complete invariant for tame knots. However, such a classification of tame knots heuristically seemed to be unsatisfactory, due to the apparent difficulty of the quandle isomorphism problem. Our result confirms this view, showing that, from a set-theoretic perspective, classifying tame knots by quandles replaces one problem with (a special case of) a much harder problem.

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35

CARRELL, TIM, and SAM NELSON. "ON RACK POLYNOMIALS." Journal of Algebra and Its Applications 10, no. 06 (December 2011): 1221–32. http://dx.doi.org/10.1142/s0219498811005142.

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We study rack polynomials and the link invariants they define. We show that constant action racks are classified by their generalized rack polynomials and show that nsata-quandles are not classified by their generalized quandle polynomials. We use subrack polynomials to define enhanced rack counting invariants, generalizing the quandle polynomial invariants.

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36

NIEBRZYDOWSKI, MACIEJ, and JÓZEF H. PRZYTYCKI. "THE SECOND QUANDLE HOMOLOGY OF THE TAKASAKI QUANDLE OF AN ODD ABELIAN GROUP IS AN EXTERIOR SQUARE OF THE GROUP." Journal of Knot Theory and Its Ramifications 20, no. 01 (January 2011): 171–77. http://dx.doi.org/10.1142/s0218216511008693.

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We prove that if G is an abelian group of odd order then there is an isomorphism from the second quandle homology [Formula: see text] to G ∧ G, where ∧ is the exterior product. In particular, for [Formula: see text], k odd, we have [Formula: see text]. Nontrivial [Formula: see text] allows us to use 2-cocycles to construct new quandles from T(G), and to construct link invariants. Computation of [Formula: see text] is also the first, fundamental step in the direction of computing homology of Takasaki quandles in general.

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37

Przytycki, Józef H., and Seung Yeop Yang. "Annihilation of torsion in homology of finite m-AQ quandles." Journal of Knot Theory and Its Ramifications 25, no. 12 (October 2016): 1642012. http://dx.doi.org/10.1142/s0218216516420128.

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It is a classical result in reduced homology of finite groups that the order of a group annihilates its homology. Similarly, we have proved that the torsion subgroup of rack and quandle homology of a finite quasigroup quandle is annihilated by its order. However, it does not hold for connected quandles in general. In this paper, we define an [Formula: see text]-almost quasigroup ([Formula: see text]-AQ) quandle which is a generalization of a quasigroup quandle and study annihilation of torsion in its rack and quandle homology groups.

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DIONÍSIO, F. MIGUEL, and PEDRO LOPES. "QUANDLES AT FINITE TEMPERATURES II." Journal of Knot Theory and Its Ramifications 12, no. 08 (December 2003): 1041–92. http://dx.doi.org/10.1142/s0218216503002949.

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The number of colorings of a knot diagram by a given quandle is a knot invariant. This follows naturally from the fact that the knot quandle is a knot invariant, and is observed again in the context of the CJKLS invariant and also in the context of the equivalence relation on colored diagrams introduced by the second author. Here we investigate the effectiveness of this invariant on prime knots of up to and including ten crossings, by computing the number of colorings using a list of ten quandles. This distinguishes these knots in all but less than 3% of the cases.

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Clark, W. Edwin, and Masahico Saito. "Longitudinal mapping knot invariant for SU(2)." Journal of Knot Theory and Its Ramifications 27, no. 11 (October 2018): 1843014. http://dx.doi.org/10.1142/s0218216518430149.

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The knot coloring polynomial defined by Eisermann for a finite pointed group is generalized to an infinite pointed group as the longitudinal mapping invariant of a knot. In turn, this can be thought of as a generalization of the quandle 2-cocycle invariant for finite quandles. If the group is a topological group, then this invariant can be thought of as a topological generalization of the 2-cocycle invariant. The longitudinal mapping invariant is based on a meridian–longitude pair in the knot group. We also give an interpretation of the invariant in terms of quandle colorings of a 1-tangle for generalized Alexander quandles without use of a meridian–longitude pair in the knot group. The invariant values are concretely evaluated for the torus knots [Formula: see text], their mirror images, and the figure eight knot for the group SU(2).

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40

Horvat, E., and A. S. Crans. "From biquandle structures to Hom-biquandles." Journal of Knot Theory and Its Ramifications 29, no. 02 (February 2020): 2040008. http://dx.doi.org/10.1142/s0218216520400088.

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We investigate the relationship between the quandle and biquandle coloring invariant and obtain an enhancement of the quandle and biquandle coloring invariants using biquandle structures. We also continue the study of biquandle homomorphisms into a medial biquandle begun in [Hom quandles, J. Knot Theory Ramifications 23(2) (2014)], finding biquandle analogs of results therein. We describe the biquandle structure of the Hom-biquandle, and consider the relationship between the Hom-quandle and Hom-biquandle.

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41

Traldi, Lorenzo. "Multivariate Alexander quandles, III. Sublinks." Journal of Knot Theory and Its Ramifications 28, no. 14 (December 2019): 1950090. http://dx.doi.org/10.1142/s0218216519500901.

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If [Formula: see text] is a classical link then the multivariate Alexander quandle, [Formula: see text], is a substructure of the multivariate Alexander module, [Formula: see text]. In the first paper of this series, we showed that if two links [Formula: see text] and [Formula: see text] have [Formula: see text], then after an appropriate re-indexing of the components of [Formula: see text] and [Formula: see text], there will be a module isomorphism [Formula: see text] of a particular type, which we call a “Crowell equivalence.” In this paper, we show that [Formula: see text] (up to quandle isomorphism) is a strictly stronger link invariant than [Formula: see text] (up to re-indexing and Crowell equivalence). This result follows from the fact that [Formula: see text] determines the [Formula: see text] quandles of all the sublinks of [Formula: see text], up to quandle isomorphisms.

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42

Taniguchi, Yuta. "Quandle coloring quivers of links using dihedral quandles." Journal of Knot Theory and Its Ramifications 30, no. 02 (February 2021): 2150011. http://dx.doi.org/10.1142/s0218216521500115.

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Cho and Nelson introduced the notion of a quandle coloring quiver, which is a quiver-valued link invariant. This invariant is in general a stronger link invariant than the quandle coloring number. In this paper, we study quandle coloring quiver using dihedral quandle. We show that when we use a dihedral quandle of prime order, the quandle coloring quivers are equivalent to the quandle coloring numbers. We also discuss shadow versions of their invariants.

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Kim, Jieon, Sam Nelson, and Minju Seo. "Quandle coloring quivers of surface-links." Journal of Knot Theory and Its Ramifications 30, no. 01 (January 2021): 2150002. http://dx.doi.org/10.1142/s0218216521500024.

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Quandle coloring quivers are directed graph-valued invariants of oriented knots and links, defined using a choice of finite quandle [Formula: see text] and set [Formula: see text] of endomorphisms. From a quandle coloring quiver, a polynomial knot invariant known as the in-degree quiver polynomial is defined. We consider quandle coloring quiver invariants for oriented surface-links, represented by marked graph diagrams. We provide example computations for all oriented surface-links with ch-index up to 10 for choices of quandles and endomorphisms.

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Zablow, Joel. "Ping-ponging between relations in groups and relations in quandles." Journal of Knot Theory and Its Ramifications 24, no. 07 (June 2015): 1550034. http://dx.doi.org/10.1142/s0218216515500340.

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We give elementary proofs of certain relations in the mapping class group of a closed surface of genus 2, MCG (F2, 0). We generalize portions of these to relations in quandles with certain types of elements, associated to a relatively broad class of groups (including mapping class groups), and derive further similar quandle relations. We show these quandle relations correspond to 2-cycles in the homology of racks of tuples of quandle elements, and thence to families of commutation relations back in the groups. The recurrence of some of these phenomena within higher level structures is also explored, as are multiple types of modifications yielding different relations. The constructions are quite malleable in this respect.

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45

Traldi, Lorenzo. "Multivariate Alexander colorings." Journal of Knot Theory and Its Ramifications 27, no. 14 (December 2018): 1850076. http://dx.doi.org/10.1142/s0218216518500761.

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We extend the notion of link colorings with values in an Alexander quandle to link colorings with values in a module [Formula: see text] over the Laurent polynomial ring [Formula: see text]. If [Formula: see text] is a diagram of a link [Formula: see text] with [Formula: see text] components, then the colorings of [Formula: see text] with values in [Formula: see text] form a [Formula: see text]-module [Formula: see text]. Extending a result of Inoue [Knot quandles and infinite cyclic covering spaces, Kodai Math. J. 33 (2010) 116–122], we show that [Formula: see text] is isomorphic to the module of [Formula: see text]-linear maps from the Alexander module of [Formula: see text] to [Formula: see text]. In particular, suppose [Formula: see text] is a field and [Formula: see text] is a homomorphism of rings with unity. Then [Formula: see text] defines a [Formula: see text]-module structure on [Formula: see text], which we denote [Formula: see text]. We show that the dimension of [Formula: see text] as a vector space over [Formula: see text] is determined by the images under [Formula: see text] of the elementary ideals of [Formula: see text]. This result applies in the special case of Fox tricolorings, which correspond to [Formula: see text] and [Formula: see text]. Examples show that even in this special case, the higher Alexander polynomials do not suffice to determine [Formula: see text]; this observation corrects erroneous statements of Inoue [Quandle homomorphisms of knot quandles to Alexander quandles, J. Knot Theory Ramifications 10 (2001) 813–821; op. cit.].

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46

IWAKIRI, MASAHIDE. "CALCULATION OF DIHEDRAL QUANDLE COCYCLE INVARIANTS OF TWIST SPUN 2-BRIDGE KNOTS." Journal of Knot Theory and Its Ramifications 14, no. 02 (March 2005): 217–29. http://dx.doi.org/10.1142/s0218216505003798.

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Carter, Jelsovsky, Kamada, Langford and Saito introduced the quandle cocycle invariants of 2-knots, and calculated the cocycle invariant of a 2-twist-spun trefoil knot associated with a 3-cocycle of the dihedral quandle of order 3. Asami and Satoh calculated the cocycle invariants of twist-spun torus knots τrT(m,n) associated with 3-cocycles of some dihedral quandles. They used tangle diagrams of the torus knots. In this paper, we calculate the cocycle invariants of twist-spun 2-bridge knots τrS(α,β) by a similar method.

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47

Nosaka, Takefumi. "On quandle homology groups of Alexander quandles of prime order." Transactions of the American Mathematical Society 365, no. 7 (January 30, 2013): 3413–36. http://dx.doi.org/10.1090/s0002-9947-2013-05754-6.

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48

Kamada, Seiichi. "Quandles and symmetric quandles for higher dimensional knots." Banach Center Publications 103 (2014): 145–58. http://dx.doi.org/10.4064/bc103-0-6.

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49

Lebed, Victoria, and Arnaud Mortier. "Abelian quandles and quandles with abelian structure group." Journal of Pure and Applied Algebra 225, no. 1 (January 2021): 106474. http://dx.doi.org/10.1016/j.jpaa.2020.106474.

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50

Bardakov, Valeriy G., Mahender Singh, and Manpreet Singh. "Free quandles and knot quandles are residually finite." Proceedings of the American Mathematical Society 147, no. 8 (April 8, 2019): 3621–33. http://dx.doi.org/10.1090/proc/14488.

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