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Journal articles on the topic 'Diagram algebra'

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Published: 4 June 2021

Last updated: 21 February 2023

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1

COHEN, ARJEH M., DIÉ A. H. GIJSBERS, and DAVID B. WALES. "TANGLE AND BRAUER DIAGRAM ALGEBRAS OF TYPE Dn." Journal of Knot Theory and Its Ramifications 18, no. 04 (2009): 447–83. http://dx.doi.org/10.1142/s0218216509007063.

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A generalization of the Kauffman tangle algebra is given for Coxeter type D n. The tangles involve a pole of order 2. The algebra is shown to be isomorphic to the Birman–Murakami–Wenzl algebra of the same type. This result extends the isomorphism between the two algebras in the classical case, which, in our set-up, occurs when the Coxeter type is A n - 1. The proof involves a diagrammatic version of the Brauer algebra of type D n of which the generalized Temperley–Lieb algebra of type D n is a subalgebra.

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2

DUCHAMP, G. H. E., J. G. LUQUE, J. C. NOVELLI, C. TOLLU, and F. TOUMAZET. "HOPF ALGEBRAS OF DIAGRAMS." International Journal of Algebra and Computation 21, no. 06 (2011): 889–911. http://dx.doi.org/10.1142/s0218196711006418.

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We investigate several generalizations of the Hopf algebra MQSym whose constructions come from labelings of special diagrams in bijection with packed matrices. Their products come either from the Hopf algebras WSym or WQSym, respectively built on integer set partitions and set compositions. Realizations on bi-word are exhibited, and it is shown how these algebras fit into a commutative diagram. Hopf deformations and dendriform structures are also considered for some algebras in the picture.

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3

Rezaei, Akbar, Arsham Borumand Saeid, and Andrzej Walendziak. "Some results on pseudo-Q algebras." Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica 16, no. 1 (2017): 61–72. http://dx.doi.org/10.1515/aupcsm-2017-0005.

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AbstractThe notions of a dual pseudo-Q algebra and a dual pseudo-QC algebra are introduced. The properties and characterizations of them are investigated. Conditions for a dual pseudo-Q algebra to be a dual pseudo-QC algebra are given. Commutative dual pseudo-QC algebras are considered. The interrelationships between dual pseudo-Q/QC algebras and other pseudo algebras are visualized in a diagram.

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4

Matsumoto, Kengo. "C*-algebras associated with presentations of subshifts ii. ideal structure and lambda-graph subsystems." Journal of the Australian Mathematical Society 81, no. 3 (2006): 369–85. http://dx.doi.org/10.1017/s1446788700014373.

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AbstractA λ-graph system is a labeled Bratteli diagram with shift transformation. It is a generalization of finite labeled graphs and presents a subshift. InDoc. Math.7 (2002) 1–30, the author constructed aC*-algebraO£associated with a λ-graph system £ from a graph theoretic view-point. If a λ-graph system comes from a finite labeled graph, the algebra becomes a Cuntz-Krieger algebra. In this paper, we prove that there is a bijective correspondence between the lattice of all saturated hereditary subsets of £ and the lattice of all ideals of the algebraO£, under a certain condition on £ called (II). As a result, the class of theC*-algebras associated with λ-graph systems under condition (II) is closed under quotients by its ideals.

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5

Fuller, Kent R., and Koichiro Ohtake. "Strong module diagrams and frobenius diagram algebras." Communications in Algebra 17, no. 2 (1989): 259–98. http://dx.doi.org/10.1080/00927878908823727.

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6

GREEN, R. M. "GENERALIZED TEMPERLEY–LIEB ALGEBRAS AND DECORATED TANGLES." Journal of Knot Theory and Its Ramifications 07, no. 02 (1998): 155–71. http://dx.doi.org/10.1142/s0218216598000103.

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We give presentations, by means of diagrammatic generators and relations, of the analogues of the Temperley–Lieb algebras associated as Hecke algebra quotients to Coxeter graphs of type B and D. This generalizes Kauffman's diagram calculus for the Temperley–Lieb algebra.

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7

Laudone, Robert P. "The spin-Brauer diagram algebra." Journal of Algebraic Combinatorics 50, no. 2 (2018): 191–224. http://dx.doi.org/10.1007/s10801-018-0849-8.

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8

GRABOWSKI, JAN E. "ON LIE INDUCTION AND THE EXCEPTIONAL SERIES." Journal of Algebra and Its Applications 04, no. 06 (2005): 707–37. http://dx.doi.org/10.1142/s0219498805001496.

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Lie bialgebras occur as the principal objects in the infinitesimalization of the theory of quantum groups — the semi-classical theory. Their relationship with the quantum theory has made available some new tools that we can apply to classical questions. In this paper, we study the simple complex Lie algebras using the double-bosonization construction of Majid. This construction expresses algebraically the induction process given by adding and removing nodes in Dynkin diagrams, which we call Lie induction. We first analyze the deletion of nodes, corresponding to the restriction of adjoint representations to subalgebras. This uses a natural grading associated to each node. We give explicit calculations of the module and algebra structures in the case of the deletion of a single node from the Dynkin diagram for a simple Lie (bi-)algebra. We next consider the inverse process, namely that of adding nodes, and give some necessary conditions for the simplicity of the induced algebra. Finally, we apply these to the exceptional series of simple Lie algebras, in the context of finding obstructions to the existence of finite-dimensional simple complex algebras of types E9, F5 and G3. In particular, our methods give a new point of view on why there cannot exist such an algebra of type E9.

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9

Tyler, Jason. "Every AF-algebra is Morita equivalent to a graph algebra." Bulletin of the Australian Mathematical Society 69, no. 2 (2004): 237–40. http://dx.doi.org/10.1017/s0004972700035978.

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10

Denecke, K., J. Koppitz, and R. Marszałek. "Derived Varieties and Derived Equational Theories." International Journal of Algebra and Computation 08, no. 02 (1998): 153–69. http://dx.doi.org/10.1142/s0218196798000090.

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This paper describes a derivation process for varieties and equational theories using the theory of hypersubstitutions and M-hyperidentities. A hypersubstitution σ of type τ is a map which takes each n-ary operation symbol of the type to an n-ary term of this type. If [Formula: see text] is an algebra of type τ then the algebra [Formula: see text] is called a derived algebra of [Formula: see text]. If V is a class of algebras of type τ then one can consider the variety vσ(V) generated by the class of all derived algebras from V. In the first two sections the necessary definitions are given. In Sec. 3 the properties of derived varieties and derived equational theories are described. On the set of all derived varieties of a given variety, a quasiorder is developed which gives a derivation diagram. In the final section the derivation diagram for the largest solid variety of medial semigroups is worked out.

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11

Kamada, Naoko. "Coherent double coverings of virtual link diagrams." Journal of Knot Theory and Its Ramifications 27, no. 11 (2018): 1843004. http://dx.doi.org/10.1142/s0218216518430046.

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A virtual link diagram is called normal if the associated abstract link diagram is checkerboard colorable, and a virtual link is normal if it has a normal diagram as a representative. Normal virtual links have some properties similar to classical links. In this paper, we introduce a method of converting a virtual link diagram to a normal virtual link diagram. We show that the normal virtual link diagrams obtained by this method from two equivalent virtual link diagrams are equivalent. We relate this method to some invariants of virtual links.

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12

JANSSEN, K., and J. VERCRUYSSE. "MULTIPLIER BI- AND HOPF ALGEBRAS." Journal of Algebra and Its Applications 09, no. 02 (2010): 275–303. http://dx.doi.org/10.1142/s0219498810003926.

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We propose a categorical interpretation of multiplier Hopf algebras, in analogy to usual Hopf algebras and bialgebras. Since the introduction of multiplier Hopf algebras by Van Daele [Multiplier Hopf algebras, Trans. Amer. Math. Soc.342(2) (1994) 917–932] such a categorical interpretation has been missing. We show that a multiplier Hopf algebra can be understood as a coalgebra with antipode in a certain monoidal category of algebras. We show that a (possibly nonunital, idempotent, nondegenerate, k-projective) algebra over a commutative ring k is a multiplier bialgebra if and only if the category of its algebra extensions and both the categories of its left and right modules are monoidal and fit, together with the category of k-modules, into a diagram of strict monoidal forgetful functors.

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13

Birman, Joan S., and Rolland Trapp. "BRAIDED CHORD DIAGRAMS." Journal of Knot Theory and Its Ramifications 07, no. 01 (1998): 1–22. http://dx.doi.org/10.1142/s0218216598000024.

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The notion of a braided chord diagram is introduced and studied. An equivalence relation is given which identifies all braidings of a fixed chord diagram. It is shown that finite-type invariants are stratified by braid index for knots which can be represented as closed 3-braids. Partial results are obtained about spanning sets for the algebra of chord diagrams of braid index 3.

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14

MATTHES, RALPH, and SERGEI SOLOVIEV. "Preface to the special issue: commutativity of algebraic diagrams." Mathematical Structures in Computer Science 22, no. 6 (2012): 901–3. http://dx.doi.org/10.1017/s0960129511000636.

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The problem of the commutativity of algebraic (categorical) diagrams has attracted the attention of researchers for a long time. For example, the related notion of coherence was discussed in Mac Lane's homology book Mac Lane (1963), see also his AMS presidential address Mac Lane (1976). Researchers in category theory view this problem from a specific angle, and for them it is not just a question of convenient notation, though it is worth mentioning the important role that notation plays in the development of science (take, for example, the progress made after the introduction of symbolic notation in logics or matrix notation in algebra). In 1976, Peter Freyd published the paper ‘Properties Invariant within Equivalence Types of Categories’ (Freyd 1976), where the central role is played by the notion of a ‘diagrammatic property’. We may also recall the process of ‘diagram chasing’, and its applications in topology and algebra. But before we can use diagrams (and the principal property of a diagram is its commutativity), it is vital for us to be able to check whether a diagram is commutative.

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15

Ehrig, Michael, and Catharina Stroppel. "2-row Springer Fibres and Khovanov Diagram Algebras for Type D." Canadian Journal of Mathematics 68, no. 6 (2016): 1285–333. http://dx.doi.org/10.4153/cjm-2015-051-4.

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AbstractWe study in detail two row Springer fibres of even orthogonal type from an algebraic as well as a topological point of view. We show that the irreducible components and their pairwise intersections are iterated ℙ1-bundles. Using results of Kumar and Procesi we compute the cohomology ring with its action of the Weyl group. The main tool is a type D diagram calculus labelling the irreducible components in a convenient way that relates to a diagrammatical algebra describing the category of perverse sheaves on isotropic Grassmannians based on work of Braden. The diagram calculus generalizes Khovanov's arc algebra to the type D setting and should be seen as setting the framework for generalizing well-known connections of these algebras in type A to other types.

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16

Kawauchi, Akio. "Knotting probability of an arc diagram." Journal of Knot Theory and Its Ramifications 29, no. 10 (2020): 2042004. http://dx.doi.org/10.1142/s0218216520420043.

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The knotting probability of an arc diagram is defined as the quadruplet of four kinds of finner knotting probabilities which are invariant under a reasonable deformation containing an isomorphism on an arc diagram. In a separated paper, it is shown that every oriented spatial arc admits four kinds of unique arc diagrams up to isomorphisms determined from the spatial arc and the projection, so that the knotting probability of a spatial arc is defined. The definition of the knotting probability of an arc diagram uses the fact that every arc diagram induces a unique chord diagram representing a ribbon 2-knot. Then the knotting probability of an arc diagram is set to measure how many nontrivial ribbon genus 2 surface-knots occur from the chord diagram induced from the arc diagram. The conditions for an arc diagram with the knotting probability 0 and for an arc diagram with the knotting probability 1 are given together with some other properties and some examples.

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17

BALDRIDGE, SCOTT, and ADAM M. LOWRANCE. "CUBE DIAGRAMS AND 3-DIMENSIONAL REIDEMEISTER-LIKE MOVES FOR KNOTS." Journal of Knot Theory and Its Ramifications 21, no. 05 (2012): 1250033. http://dx.doi.org/10.1142/s0218216511009832.

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In this paper we introduce a representation of knots and links called a cube diagram. We show that a property of a cube diagram is a link invariant if and only if the property is invariant under five cube diagram moves. A knot homology is constructed from cube diagrams and shown to be equivalent to knot Floer homology.

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18

MATSUMOTO, KENGO. "FACTOR MAPS OF LAMBDA-GRAPH SYSTEMS AND INCLUSIONS OF C*-ALGEBRAS." International Journal of Mathematics 15, no. 04 (2004): 313–39. http://dx.doi.org/10.1142/s0129167x04002351.

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A λ-graph system is a labeled Bratteli diagram with shift transformation. It is a generalization of finite labeled graphs and presents a subshift. In [Doc. Math. 7 (2002), 1–30], the author introduced a C*-algebra [Formula: see text] associated with a λ-graph system [Formula: see text] as a generalization of the Cuntz–Krieger algebras. In this paper, we study a functorial property between factor maps of λ-graph systems and inclusions of the associated C*-algebras with gauge actions. We prove that if there exists a surjective left-covering λ-graph system homomorphism [Formula: see text], there exists a unital embedding of the C*-algebra [Formula: see text] into the C*-algebra [Formula: see text] compatible to its gauge actions. We also show that a sequence of left-covering graph homomorphisms of finite labeled graphs gives rise to a λ-graph system such that the associated C*-algebra is an inductive limit of the Cuntz–Krieger algebras for the finite labeled graphs.

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19

Shimizu, Ayaka. "Prime alternating knots of minimal warping degree two." Journal of Knot Theory and Its Ramifications 29, no. 08 (2020): 2050060. http://dx.doi.org/10.1142/s0218216520500601.

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The warping degree of an oriented knot diagram is the minimal number of crossing changes which are required to obtain a monotone diagram from the diagram. The minimal warping degree of a knot is the minimal value of the warping degree for all oriented minimal diagrams of the knot. In this paper, all prime alternating knots with minimal warping degree two are determined.

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20

Kawauchi, Akio, and Ayaka Shimizu. "On the orientations of monotone knot diagrams." Journal of Knot Theory and Its Ramifications 26, no. 10 (2017): 1750053. http://dx.doi.org/10.1142/s0218216517500535.

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An oriented monotone knot diagram is a knot diagram such that one meets each crossing as an over-crossing first as one travels the diagram with the orientation by starting at a point on the diagram. In this paper, unoriented knot projections which are monotone with an orientation and any over/under information are characterized. Also, monotone diagrams which are monotone with exactly one orientation and unique basepoint are characterized. As an application, a necessary condition for a knot projection with reductivity four is given.

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21

LEE, H. C. "ON SEIFERT CIRCLES AND FUNCTORS FOR TANGLES." International Journal of Modern Physics A 07, supp01b (1992): 581–610. http://dx.doi.org/10.1142/s0217751x9200394x.

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The properties of the Seifert circles in an oriented tangle diagram are exploited to prove a theorem that asserts that every (n,n)-tangle diagram is isotopic to a partially closed braid, and a second one that facilitates the assignment of wrong-way edges, one on each Seifert circle, in a tangle diagram. These result are used to identify the structure of an abstract algebra on which a functor for the isotopy of general tangles may be constructed. Any finite dimensional irreducible representation of a quasitriangular Hopf algebra is a realization of this algebra.

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22

Clikeman, Miles, Rachel Morris, and Heather M. Russell. "Ineffective sets and the region crossing change operation." Journal of Knot Theory and Its Ramifications 29, no. 03 (2020): 2050010. http://dx.doi.org/10.1142/s0218216520500108.

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Region crossing change (RCC) is an operation on link diagrams in which all crossings incident to a selected region are changed. Two diagrams are called RCC-equivalent if one can be transformed to the other via a sequence of RCCs. RCC is an unknotting operation but not an unlinking operation. A set of regions of a diagram is called ineffective if RCCs at every region in that set have no net effect on the crossings of the diagram. The main result of this paper is a construction of the complete collection of ineffective sets for any link diagram. This involves a combination of linear algebraic and diagrammatic techniques including a generalization of checkerboard shading called tricoloring. Using this construction of ineffective sets, we provide sharp upper bounds on the maximum number of RCCs needed to transform between RCC-equivalent knot diagrams and reduced 2- and 3-component link diagrams with fixed underlying projections.

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23

LIU, DONGLUN, STEVEN MACKEY, NEIL R. NICHOLSON, TYLER SCHROEDER, and KYLE THOMAS. "AVERAGE BRIDGE NUMBER OF SHADOW RESOLUTIONS." Journal of Knot Theory and Its Ramifications 22, no. 10 (2013): 1350064. http://dx.doi.org/10.1142/s0218216513500648.

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A Gauss code for a virtual knot diagram is a sequence of crossing labels, each repeated twice and assigned a + or - symbol to identify over and undercrossings. Eliminate these symbols and what remains is a Gauss code for the shadow of the diagram, one type of virtual pseudodiagram. While it is now impossible to determine which particular virtual diagram the shadow resulted from, we can consider the collection of all diagrams, called resolutions of the shadow, that would yield such a code. We compute the average virtual bridge number over all these diagrams and show that for a shadow with n classical precrossings, the average virtual bridge number is [Formula: see text].

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24

PRZYTYCKI, JÓZEF H., and KOUKI TANIYAMA. "ALMOST POSITIVE LINKS HAVE NEGATIVE SIGNATURE." Journal of Knot Theory and Its Ramifications 19, no. 02 (2010): 187–289. http://dx.doi.org/10.1142/s0218216510007838.

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We analyze properties of links which have diagrams with a small number of negative crossings. We show that if a nontrivial link has a diagram with all crossings positive except possibly one, then the signature of the link is negative. If a link diagram has two negative crossings, we show that the signature of the link is nonpositive with the exception of the left-handed Hopf link (possibly, with extra trivial components). We also characterize those links which have signature zero and diagrams with two negative crossings. In particular, we show that if a nontrivial knot has a diagram with two negative crossings then the signature of the knot is negative, unless the knot is a twist knot with negative clasp. We completely determine all trivial link diagrams with two or fewer negative crossings. For a knot diagram with three negative crossings, the signature of the knot is nonpositive except for the left-handed trefoil knot. These results generalize those of Rudolph, Cochran, Gompf, Traczyk and Przytycki, solve [27, Conjecture 5], and give a partial answer to [3, Problem 2.8] about knots dominating the trefoil knot or the trivial knot. We also describe all unknotting number one positive knots.

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25

Matsumoto, Kengo. "Construction and pure infiniteness of $C^*$-algebras associated with lambda-graph systems." MATHEMATICA SCANDINAVICA 97, no. 1 (2005): 73. http://dx.doi.org/10.7146/math.scand.a-14964.

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A $\lambda$-graph system is a labeled Bratteli diagram with shift transformation. It is a generalization of finite labeled graphs and presents a subshift. In [16] the author has introduced a $C^*$-algebra $\mathcal{O}_{\mathfrak{L}}$ associated with a $\lambda$-graph system $\mathfrak{L}$ by using groupoid method as a generalization of the Cuntz-Krieger algebras. In this paper, we concretely construct the $C^*$-algebra $\mathcal{O}_{\mathfrak{L}}$ by using both creation operators and projections on a sub Fock Hilbert space associated with $\mathfrak{L}$. We also introduce a new irreducible condition on $\mathfrak{L}$ under which the $C^*$-algebra $\mathcal{O}_{\mathfrak{L}}$ becomes simple and purely infinite.

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26

Zhu, Jun. "On Kauffman Brackets." Journal of Knot Theory and Its Ramifications 06, no. 01 (1997): 125–48. http://dx.doi.org/10.1142/s021821659700011x.

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An antichain is associated to each link diagram so that the highest degree of the Kauffman bracket can be determined. As an application, we show that the span of the Kauffman bracket is less than or equal to 4(n - m) for dealternator connected m-alternating diagrams and the upper bound is best possible. This completely solves a conjecture of [1]. Finally, we show that a semi-alternating diagram may be not a minimal diagram which disproves a conjecture of K. Murasugi [10].

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27

SCHELLHORN, WILLIAM J. "FILAMENTATIONS FOR VIRTUAL LINKS." Journal of Knot Theory and Its Ramifications 15, no. 03 (2006): 327–38. http://dx.doi.org/10.1142/s0218216506004464.

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In 2002, Hrencecin and Kauffman defined a filamentation invariant on oriented chord diagrams that may determine whether the corresponding flat virtual knot diagrams are non-trivial. A virtual knot diagram is non-classical if its related flat virtual knot diagram is non-trivial. Hence filamentations can be used to detect non-classical virtual knots. We extend these filamentation techniques to virtual links with more than one component. We also give examples of virtual links that they can detect as non-classical.

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28

Okazaki, Kenta. "An elementary proof of a non-triviality of the E8subfactor planar algebra." International Journal of Mathematics 26, no. 05 (2015): 1550037. http://dx.doi.org/10.1142/s0129167x15500378.

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In this paper, we show in a combinatorial way that the 0-box space of the E8subfactor planar algebra is 1-dimensional. In the proof, we improve on Bigelow's relations for the E8subfactor planar algebra and give an efficient algorithm to reduce any planar diagram to the empty diagram.

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29

SHIMIZU, AYAKA. "THE WARPING DEGREE OF A KNOT DIAGRAM." Journal of Knot Theory and Its Ramifications 19, no. 07 (2010): 849–57. http://dx.doi.org/10.1142/s0218216510008194.

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For an oriented knot diagram D, the warping degree d(D) is the smallest number of crossing changes which are needed to obtain the monotone diagram from D in the usual way. We show that d(D) + d(-D) + 1 is less than or equal to the crossing number of D. Moreover, the equality holds if and only if D is an alternating diagram. For a knot K, we also estimate the minimum of d(D) + d(-D) for all diagrams D of K with c(D) = c(K).

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30

Adams, Colin, Jim Hoste, and Martin Palmer. "Triple-crossing number and moves on triple-crossing link diagrams." Journal of Knot Theory and Its Ramifications 28, no. 11 (2019): 1940001. http://dx.doi.org/10.1142/s0218216519400017.

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Every link in the 3-sphere has a projection to the plane where the only singularities are pairwise transverse triple points. The associated diagram, with height information at each triple point, is a triple-crossing diagram of the link. We give a set of diagrammatic moves on triple-crossing diagrams analogous to the Reidemeister moves on ordinary diagrams. The existence of [Formula: see text]-crossing diagrams for every [Formula: see text] greater than one allows the definition of the [Formula: see text]-crossing number. We prove that for any nontrivial, nonsplit link, other than the Hopf link, its triple-crossing number is strictly greater than its quintuple-crossing number.

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31

Lobb, Andrew. "Computable bounds for Rasmussen’s concordance invariant." Compositio Mathematica 147, no. 2 (2010): 661–68. http://dx.doi.org/10.1112/s0010437x10005117.

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AbstractGiven a diagram D of a knot K, we give easily computable bounds for Rasmussen’s concordance invariant s(K). The bounds are not independent of the diagram D chosen, but we show that for diagrams satisfying a given condition the bounds are tight. As a corollary we improve on previously known Bennequin-type bounds on the slice genus.

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32

HARDIMAN, LEONARD, and ALASTAIR KING. "DECOMPOSING THE TUBE CATEGORY." Glasgow Mathematical Journal 62, no. 2 (2019): 441–58. http://dx.doi.org/10.1017/s001708951900020x.

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AbstractThe tube category of a modular tensor category is a variant of the tube algebra, first introduced by Ocneanu. As a category, it can be decomposed in two different, but related, senses. Firstly, via the Yoneda embedding, the Hom spaces decompose into summands factoring through irreducible functors, in a manner analogous to decomposing an algebra as a sum of matrix algebras. We describe these summands. Secondly, under the Yoneda embedding, each object decomposes into irreducibles, which correspond to primitive idempotents in the category itself. We identify these idempotents. We make extensive use of diagram calculus in the description and proof of these decompositions.

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33

PARVATHI, M., and B. SIVAKUMAR. "THE KLEIN-4 DIAGRAM ALGEBRAS." Journal of Algebra and Its Applications 07, no. 02 (2008): 231–62. http://dx.doi.org/10.1142/s0219498808002795.

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In this paper we study a new class of diagram algebras, the Klein-4 diagram algebras denoted by Rk(n). These algebras are the centralizer algebras of the group Gn := (ℤ2 × ℤ2)≀Sn acting on V⊗k, where V is the signed permutation module for Gn These algebras have been realized as subalgebras of the extended G-vertex colored partition algebras introduced by Parvathi and Kennedy in [7]. In this paper we give a combinatorial rule for the decomposition of the tensor powers of the signed permutation representation of Gn by explicitly constructing the basis for the irreducible modules. In the process we also give the basis for the irreducible modules appearing in the decomposition of V⊗k in [5]. We then use this rule to describe the Bratteli diagram of Klein-4 diagram algebras.

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34

Korablev, Philipp, and Vladimir Tarkaev. "A relation between the crossing number and the height of a knotoid." Journal of Knot Theory and Its Ramifications 30, no. 06 (2021): 2150040. http://dx.doi.org/10.1142/s0218216521500401.

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Knotoids are open ended knot diagrams regarded up to Reidemeister moves and isotopies. The notion is introduced by Turaev in 2012. Two most important numeric characteristics of a knotoid are the crossing number and the height. The latter is the least number of intersections between a diagram and an arc connecting its endpoints, where the minimum is taken over all representative diagrams and all such arcs which are disjoint from crossings. In the paper, we answer the question: are there any relations between the crossing number and the height of a knotoid. We prove that the crossing number of a knotoid is greater than or equal to twice the height of the knotoid. Combining the inequality with known lower bounds of the height we obtain a lower bounds of the crossing number of a knotoid via the extended bracket polynomial, the affine index polynomial and the arrow polynomial of the knotoid. As an application of our result we prove an upper bound for the length of a bridge in a minimal diagram of a classical knot: the number of crossings in a minimal diagram of a knot is greater than or equal to three times the length of a longest bridge in the diagram.

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35

Miyazawa, Haruko A., Kodai Wada, and Akira Yasuhara. "Linking invariants of even virtual links." Journal of Knot Theory and Its Ramifications 26, no. 12 (2017): 1750072. http://dx.doi.org/10.1142/s0218216517500729.

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A virtual link diagram is even if the virtual crossings divide each component into an even number of arcs. The set of even virtual link diagrams is closed under classical and virtual Reidemeister moves, and it contains the set of classical link diagrams. For an even virtual link diagram, we define a certain linking invariant which is similar to the linking number. In contrast to the usual linking number, our linking invariant is not preserved under the forbidden moves. In particular, for two fused isotopic even virtual link diagrams, the difference between the linking invariants of them gives a lower bound of the minimal number of forbidden moves needed to deform one into the other. Moreover, we give an example which shows that the lower bound is best possible.

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36

Wang, Junchang, and Steffen Koenig. "Cyclotomic Extensions of Diagram Algebras." Communications in Algebra 36, no. 5 (2008): 1739–57. http://dx.doi.org/10.1080/00927870801940350.

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37

WADA, MASAAKI. "CODING LINK DIAGRAMS." Journal of Knot Theory and Its Ramifications 02, no. 02 (1993): 233–37. http://dx.doi.org/10.1142/s0218216593000143.

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38

ISIHARA, PAUL, and ANDREA KOENIGSBERG. "A NOTE ON THE UTILITY OF KAUFFMAN'S STATE SUMMATION FORMULA FOR THE BRACKET POLYNOMIAL." Journal of Knot Theory and Its Ramifications 11, no. 01 (2002): 13–79. http://dx.doi.org/10.1142/s0218216502001470.

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Kauffman's state summation formula is especially useful for computing the bracket polynomials of projection diagrams which are related by smoothings or crossing changes. This facilitates the writing of a symbolic algebra program which computes the normalized bracket polynomials and frequencies of knots and links whose projection diagrams result from a given knot's oriented projection diagram either by crossing changes or by orientation preserving smoothings called natural smoothings. These frequencies provide insight into the unknotting game (and similar resultant games) whose object is to specify crossing changes or natural smoothings that will transform a given projection diagram of a knot into a projection diagram representing an unknot (or some other specified knot or link). The practical utility of the state summation formula is greatly enhanced by means of diagrams for closed tangle sums. These diagrams offer a special cost-reducing method to obtain crucial information needed to compute the state summation formula. This special method also gives insight into why the bracket is unchanged by mutation and contributes a strategy to the enigmatic search for a non-trivial knot with Jones polynomial equal to one.

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Tuzun, Robert E., and Adam S. Sikora. "Verification of the Jones unknot conjecture up to 22 crossings." Journal of Knot Theory and Its Ramifications 27, no. 03 (2018): 1840009. http://dx.doi.org/10.1142/s0218216518400096.

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We proved by computer enumeration that the Jones polynomial distinguishes the unknot for knots up to 22 crossings. Following an approach of Yamada, we generated knot diagrams by inserting algebraic tangles into Conway polyhedra, computed their Jones polynomials by a divide-and-conquer method, and tested those with trivial Jones polynomials for unknottedness with the computer program SnapPy. We employed numerous novel strategies for reducing the computation time per knot diagram and the number of knot diagrams to be considered. That made computations up to 21 crossings possible on a single processor desktop computer. We explain these strategies in this paper. We also provide total numbers of algebraic tangles up to 18 crossings and of Conway polyhedra up to 22 vertices. We encountered new unknot diagrams with no crossing-reducing pass moves in our search. We report one such diagram in this paper.

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Kawamura, Kengo. "Region crossing change, bicolored diagram and Arf invariant." Journal of Knot Theory and Its Ramifications 30, no. 05 (2021): 2150029. http://dx.doi.org/10.1142/s0218216521500292.

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We introduce the notion of bicolored diagrams which are closely related to the region crossing changes. Moreover, we refine Cheng’s results on the region crossing changes and propose a certain way to calculate the Arf invariant of a proper link using a bicolored diagram.

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41

Adler, M. "An algebra for data flow diagram process decomposition." IEEE Transactions on Software Engineering 14, no. 2 (1988): 169–83. http://dx.doi.org/10.1109/32.4636.

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42

Ebrahimi-Fard, Kurusch, and Dirk Kreimer. "The Hopf algebra approach to Feynman diagram calculations." Journal of Physics A: Mathematical and General 38, no. 50 (2005): R385—R407. http://dx.doi.org/10.1088/0305-4470/38/50/r01.

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43

DIAO, YUANAN, CLAUS ERNST, and ANDRZEJ STASIAK. "A PARTIAL ORDERING OF KNOTS AND LINKS THROUGH DIAGRAMMATIC UNKNOTTING." Journal of Knot Theory and Its Ramifications 18, no. 04 (2009): 505–22. http://dx.doi.org/10.1142/s0218216509007026.

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In this paper we define a partial ordering of knots and links using a special property derived from their minimal diagrams. A link [Formula: see text] is called a predecessor of a link [Formula: see text] if [Formula: see text] and a diagram of [Formula: see text] can be obtained from a minimal diagram D of [Formula: see text] by a single crossing change. In such a case, we say that [Formula: see text]. We investigate the sets of links that can be obtained by single crossing changes over all minimal diagrams of a given link. We show that these sets are specific for different links and permit partial ordering of all links. Some interesting results are presented and many questions are raised.

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44

Aung, Pye Phyo. "Gorenstein dimensions over some rings of the form R ⊕ C." Journal of Algebra and Its Applications 15, no. 03 (2016): 1650043. http://dx.doi.org/10.1142/s0219498816500432.

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Given a semidualizing module [Formula: see text] over a commutative Noetherian ring, Holm and Jørgensen [Semi-dualizing modules and related Gorenstein homological dimensions, J. Pure Appl. Algebra 205(2) (2006) 423–445] investigate some connections between [Formula: see text]-Gorenstein dimensions of an [Formula: see text]-complex and Gorenstein dimensions of the same complex viewed as a complex over the “trivial extension” [Formula: see text]. We generalize some of their results to a certain type of retract diagram. We also investigate some examples of such retract diagrams, namely D’Anna and Fontana’s amalgamated duplication [An amalgamated duplication of a ring along an ideal: The basic properties, J. Algebra Appl. 6(3) (2007) 443–459] and Enescu’s pseudocanonical cover [A finiteness condition on local cohomology in positive characteristic, J. Pure Appl. Algebra 216(1) (2012) 115–118].

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GONZÁLEZ-MENESES, JUAN, and PEDRO M. G. MANCHÓN. "A GEOMETRIC CHARACTERIZATION OF THE UPPER BOUND FOR THE SPAN OF THE JONES POLYNOMIAL." Journal of Knot Theory and Its Ramifications 20, no. 07 (2011): 1059–71. http://dx.doi.org/10.1142/s0218216511009005.

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Let D be a link diagram with n crossings, sA and sB be its extreme states and |sAD| (respectively, |sBD|) be the number of simple closed curves that appear when smoothing D according to sA (respectively, sB). We give a general formula for the sum |sAD| + |sBD| for a k-almost alternating diagram D, for any k, characterizing this sum as the number of faces in an appropriate triangulation of an appropriate surface with boundary. When D is dealternator connected, the triangulation is especially simple, yielding |sAD| + |sBD| = n + 2 - 2k. This gives a simple geometric proof of the upper bound of the span of the Jones polynomial for dealternator connected diagrams, a result first obtained by Zhu [On Kauffman brackets, J. Knot Theory Ramifications6(1) (1997) 125–148.]. Another upper bound of the span of the Jones polynomial for dealternator connected and dealternator reduced diagrams, discovered historically first by Adams et al. [Almost alternating links, Topology Appl.46(2) (1992) 151–165.], is obtained as a corollary. As a new application, we prove that the Turaev genus is equal to the number k of dealternator crossings for any dealternator connected diagram.

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Ilyutko, Denis P., and Vassily O. Manturov. "Picture-valued parity-biquandle bracket." Journal of Knot Theory and Its Ramifications 29, no. 02 (2020): 2040004. http://dx.doi.org/10.1142/s0218216520400040.

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In V. O. Manturov, On free knots, preprint (2009), arXiv:math.GT/0901.2214], the second named author constructed the bracket invariant [Formula: see text] of virtual knots valued in pictures (linear combinations of virtual knot diagrams with some crossing information omitted), such that for many diagrams [Formula: see text], the following formula holds: [Formula: see text], where [Formula: see text] is the underlying graph of the diagram, i.e. the value of the invariant on a diagram equals the diagram itself with some crossing information omitted. This phenomenon allows one to reduce many questions about virtual knots to questions about their diagrams. In [S. Nelson, M. E. Orrison and V. Rivera, Quantum enhancements and biquandle brackets, preprint (2015), arXiv:math.GT/1508.06573], the authors discovered the following phenomenon: having a biquandle coloring of a certain knot, one can enhance various state-sum invariants (say, Kauffman bracket) by using various coefficients depending on colors. Taking into account that the parity can be treated in terms of biquandles, we bring together the two ideas from these papers and construct the picture-valued parity-biquandle bracket for classical and virtual knots. This is an invariant of virtual knots valued in pictures. Both the parity bracket and Nelson–Orrison–Rivera invariants are partial cases of this invariant, hence this invariant enjoys many properties of various kinds. Recently, the authors together with E. Horvat and S. Kim have found that the picture-valued phenomenon works in the classical case.

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47

Dean, Andrew J. "Classification of AF Flows." Canadian Mathematical Bulletin 46, no. 2 (2003): 164–77. http://dx.doi.org/10.4153/cmb-2003-018-0.

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AbstractAn AF flow is a one-parameter automorphism group of an AF C*-algebra A such that there exists an increasing sequence of invariant finite dimensional sub-C*-algebras whose union is dense in A. In this paper, a classification of C*-dynamical systems of this form up to equivariant isomorphism is presented. Two pictures of the actions are given, one in terms of a modified Bratteli diagram/pathspace construction, and one in terms of a modified K0 functor.

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HAYASHI, CHUICHIRO. "A LOWER BOUND FOR THE NUMBER OF REIDEMEISTER MOVES FOR UNKNOTTING." Journal of Knot Theory and Its Ramifications 15, no. 03 (2006): 313–25. http://dx.doi.org/10.1142/s0218216506004488.

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How many Reidemeister moves do we need for unknotting a given diagram of the trivial knot? The absolute value of the writhe gives a lower bound of the number of Reidemeister I moves. That of a complexity of knot diagram "cowrithe" works for Reidemeister II, III moves. In Appendix A, we give an example of an infinite sequence of diagrams Dn of the trivial knot with an O(n) number of crossings such that the author expects the number of Reidemeister moves needed for unknotting it to be O(n2). However, writhe and cowrithe do not prove this.

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BAE, YONGJU, and HUGH R. MORTON. "THE SPREAD AND EXTREME TERMS OF JONES POLYNOMIALS." Journal of Knot Theory and Its Ramifications 12, no. 03 (2003): 359–73. http://dx.doi.org/10.1142/s0218216503002512.

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We adapt Thistlethwaite's alternating tangle decomposition of a knot diagram to identify the potential extreme terms in its bracket polynomial, and give a simple combinatorial calculation for their coefficients, based on the intersection graph of certain chord diagrams.

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50

Dynnikov, Ivan, and Vera Sokolova. "Multiflypes of rectangular diagrams of links." Journal of Knot Theory and Its Ramifications 30, no. 06 (2021): 2150038. http://dx.doi.org/10.1142/s0218216521500383.

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We introduce a new very large family of transformations of rectangular diagrams of links that preserve the isotopy class of the link. We provide an example when two diagrams of the same complexity are related by such a transformation and are not obtained from one another by any sequence of “simpler” moves not increasing the complexity of the diagram along the way.

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