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  • Kauffman skein algebras and quantum Teichmüller spaces via factorization homology
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2021-01-20
    Juliet Cooke

    We compute the factorization homology of the four-punctured sphere and punctured torus over the quantum group 𝒰q(𝔰𝔩2) explicitly as categories of equivariant modules using the framework developed by Ben-Zvi et al. We identify the algebra of 𝒰q(𝔰𝔩2)-invariants (quantum global sections) with the spherical double affine Hecke algebra of type (C1∨,C1), in the four-punctured sphere case, and with

    更新日期:2021-01-22
  • On the basic properties of GCn sets
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2021-01-20
    Hakop A. Hakopian; Navasard K. Vardanyan

    In this paper the simplest n-correct sets in the plane — GCn sets are studied. An n-correct node set 𝒳 is called GCn set if the fundamental polynomial of each node is a product of n linear factors. We say that a node uses a line if the line is a factor of the fundamental polynomial of this node. A line is called k-node line if it passes through exactly k nodes of 𝒳. At most n+1 nodes can be collinear

    更新日期:2021-01-22
  • Multivariate Alexander quandles, II. The involutory medial quandle of a link (corrected)
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-12-30
    Lorenzo Traldi

    Joyce showed that for a classical knot K, the involutory medial quandle IMQ(K) is isomorphic to the core quandle of the homology group H1(X2), where X2 is the cyclic double cover of 𝕊3, branched over K. It follows that |IMQ(K)|=|detK|. In this paper, the extension of Joyce’s result to classical links is discussed. Among other things, we show that for a classical link L of μ≥2 components, the order

    更新日期:2021-01-22
  • 3-Manifolds with nilpotent embeddings in S4
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2021-01-20
    J. A. Hillman

    We consider embeddings of 3-manifolds M in S4 such that the two complementary regions X and Y each have nilpotent fundamental group. If β=β1(M) is odd then these groups are abelian and β≤3. In general π1(X) and π1(Y) have 3-generator presentations, and β≤6. We give two examples illustrating our results.

    更新日期:2021-01-21
  • Large color R-matrix for knot complements and strange identities
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2021-01-20
    Sunghyuk Park

    The Gukov–Manolescu series, denoted by FK, is a conjectural invariant of knot complements that, in a sense, analytically continues the colored Jones polynomials. In this paper we use the large color R-matrix to study FK for some simple links. Specifically, we give a definition of FK for positive braid knots, and compute FK for various knots and links. As a corollary, we present a class of “strange

    更新日期:2021-01-21
  • Most graphs are knotted
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-12-28
    Kazuhiro Ichihara; Thomas W. Mattman

    We present four models for a random graph and show that, in each case, the probability that a graph is intrinsically knotted goes to one as the number of vertices increases. We also argue that, for n≥18, most graphs of order n are intrinsically knotted and, for n≥2m+9, most of order n are not m-apex. We observe that p(n)=1/n is the threshold for intrinsic knotting and linking in Gilbert’s model.

    更新日期:2021-01-21
  • Unknotting numbers and crossing numbers of spatial embeddings of a planar graph
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2021-01-18
    Yuta Akimoto; Kouki Taniyama

    It is known that the unknotting number u(L) of a link L is less than or equal to half the crossing number c(L) of L. We show that there are a planar graph G and its spatial embedding f such that the unknotting number u(f) of f is greater than half the crossing number c(f) of f. We study relations between unknotting number and crossing number of spatial embedding of a planar graph in general.

    更新日期:2021-01-19
  • On 2-twist-spun spherical Montesinos knots
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2021-01-18
    Yeonhee Jang; Misaki Kataoka; Rika Miyakoshi

    We give a classification of 2-twist-spun spherical Montesinos knots.

    更新日期:2021-01-19
  • Virtual Mosaic Knot Theory
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-12-29
    Sandy Ganzell; Allison Henrich

    Mosaic diagrams for knots were first introduced in 2008 by Lomanoco and Kauffman for the purpose of building a quantum knot system. Since then, many others have explored the structure of these knot mosaic diagrams, as they are interesting objects of study in their own right. Knot mosaics have been generalized by Garduño to virtual knots, by including an additional tile type to represent virtual crossings

    更新日期:2020-12-31
  • Polynomial invariants of virtual doodles and virtual knots
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-11-26
    Joonoh Kim

    In this study, we describe a method of making an invariant of virtual knots defined in terms of an integer labeling of the flat virtual knot diagram. We give an invariant of flat virtual knots and virtual doodles modifying the previous invariant.

    更新日期:2020-12-31
  • Interior polynomial for signed bipartite graphs and the HOMFLY polynomial
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-12-08
    Keiju Kato

    The interior polynomial is a Tutte-type invariant of bipartite graphs, and a part of the HOMFLY polynomial of a special alternating link coincides with the interior polynomial of the Seifert graph of the link. We extend the interior polynomial to signed bipartite graphs, and we show that, in the planar case, it is equal to the maximal z-degree part of the HOMFLY polynomial of a naturally associated

    更新日期:2020-12-31
  • Genera of knots in the complex projective plane
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-11-23
    Jake Pichelmeyer

    Our goal is to systematically compute the ℂP2-genus of as many prime knots up to 8-crossings as possible. We obtain upper bounds on the ℂP2-genus via coherent band surgery. We obtain lower bounds by obstructing homological degrees of potential slice discs. The obstructions are pulled from a variety of sources in low-dimensional topology and adapted to ℂP2. There are 27 prime knots and distinct mirrors

    更新日期:2020-12-31
  • A note on representations of welded braid groups
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-11-23
    Paolo Bellingeri; Arthur Soulié

    In this paper, we adapt the procedure of the Long-Moody procedure to construct linear representations of welded braid groups. We exhibit the natural setting in this context and compute the first examples of representations we obtain thanks to this method. We take this way also the opportunity to review the few known linear representations of welded braid groups.

    更新日期:2020-12-31
  • Properties of Casson–Gordon’s rectangle condition
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-11-18
    Bo-Hyun Kwon; Jung Hoon Lee

    For a Heegaard splitting of a 3-manifold, Casson–Gordon’s rectangle condition, simply rectangle condition, is a condition on its Heegaard diagram that guarantees the strong irreducibility of the splitting; it requires nine types of rectangles for every combination of two pairs of pants from opposite sides. The rectangle condition is also applied to bridge decompositions of knots. We give examples of

    更新日期:2020-12-31
  • Quandle Module Quivers
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-11-23
    Karma Istanbouli; Sam Nelson

    We enhance the quandle coloring quiver invariant of oriented knots and links with quandle modules. This results in a two-variable polynomial invariant which specializes to the previous quandle module polynomial invariant as well as to the quandle counting invariant. We provide example computations to show that the enhancement is proper in the sense that it distinguishes knots and links with the same

    更新日期:2020-12-31
  • Hexagonal mosaic links generated by saturation
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-11-20
    J. Bush; P. Commins; T. Gomez; J. McLoud-Mann

    Square mosaic knots have many applications in algebra, such as modeling quantum states. In this paper, we extend mosaic knot theory to a theory of hexagonal mosaic links, which are links embedded in a plane tiling of regular hexagons. We investigate hexagonal mosaic links created from particular patches of hextiles with a high number of crossings, which we describe as saturated diagrams. Considering

    更新日期:2020-12-31
  • Non-left-orderable surgeries on 1-bridge braids
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-12-16
    Shiyu Liang

    Boyer, Gordon and Watson have conjectured that an irreducible rational homology 3-sphere is an L-space if and only if its fundamental group is not left-orderable. Since Dehn surgeries on knots in S3 can produce large families of L-spaces, it is natural to examine the conjecture on these 3-manifolds. Greene, Lewallen and Vafaee have proved that all 1-bridge braids are L-space knots. In this paper, we

    更新日期:2020-12-31
  • Shake slice and shake concordant links
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-12-10
    Anthony M. Bosman

    We can construct a 4-manifold by attaching 2-handles to a 4-ball with framing r along the components of a link in the boundary of the 4-ball. We define a link as r-shake slice if there exists embedded spheres that represent the generators of the second homology of the 4-manifold. This naturally extends r-shake slice, a generalization of slice that has previously only been studied for knots, to links

    更新日期:2020-12-31
  • The special rank of virtual knot groups
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-12-12
    Jhon Jader Mira-Albanés; José Gregorio Rodríguez-Nieto; Olga Patricia Salazar-Díaz

    In this paper we introduce the special rank for virtual knots and some properties of this number are studied. Although we do not know if it can be considered as a nontrivial extension of the meridional rank given by [H. U. Boden and A. I. Gaudreau, Bridge number for virtual and welded knots, J. Knot Theory Ramifications24 (2015), Article ID: 1550008] and by [M. Boileau and B. Zimmermann, The π-orbifold

    更新日期:2020-12-31
  • 𝔤𝔩n-webs, categorification and Khovanov–Rozansky homologies
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-11-05
    Daniel Tubbenhauer

    In this paper, we define an explicit basis for the 𝔤𝔩n-web algebra Hn(k) (the 𝔤𝔩n generalization of Khovanov’s arc algebra) using categorified q-skew Howe duality. Our construction is a 𝔤𝔩n-web version of Hu–Mathas’ graded cellular basis and has two major applications: it gives rise to an explicit isomorphism between a certain idempotent truncation of a thick calculus cyclotomic KLR algebra and

    更新日期:2020-12-30
  • Edge-preserving maps of the nonseparating curve graphs, curve graphs and rectangle preserving maps of the Hatcher–Thurston graphs
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-10-22
    Elmas Irmak

    Let R be a compact, connected, orientable surface of genus g with n boundary components with g≥2, n≥0. Let 𝒩(R) be the nonseparating curve graph, 𝒞(R) be the curve graph and ℋ𝒯(R) be the Hatcher–Thurston graph of R. We prove that if λ:𝒩(R)→𝒩(R) is an edge-preserving map, then λ is induced by a homeomorphism of R. We prove that if 𝜃:𝒞(R)→𝒞(R) is an edge-preserving map, then 𝜃 is induced by

    更新日期:2020-12-30
  • Generating links that are both quasi-alternating and almost alternating
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-12-28
    Hamid Abchir; Mohammed Sabak

    We construct an infinite family of links which are both almost alternating and quasi-alternating from a given either almost alternating diagram representing a quasi-alternating link, or connected and reduced alternating tangle diagram. To do that we use what we call a dealternator extension which consists in replacing the dealternator by a rational tangle extending it. We note that all non-alternating

    更新日期:2020-12-29
  • The sixth Russian-Chinese conference on knot theory and related topics
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-12-04
    Nikolay Abrosimov; Zhiyun Cheng; Louis H. Kauffman; Andrei Vesnin; Zhiqing Yang

    This is a preface to the special issue on Proceedings of the Sixth Russian-Chinese Conference on Knot Theory and Related Topics which was held in Novosibirsk in June 2019.

    更新日期:2020-12-23
  • Studying complex manifolds by using groups Gnk and Γnk
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-12-16
    Vassily Olegovich Manturov; Zheyan Wan

    In this paper, we study several complex manifolds by using the following idea. First, we construct a certain moduli space and study the fundamental group of this space. This fundamental group is naturally mapped to the groups Gnk and Γnk. This is the step towards “complexification” of the Gnk and Γnk approach first developed in [V. O. Manturov, D. A. Fedoseev, S. Kim and I. M. Nikonov, On groups Gnk

    更新日期:2020-12-18
  • From chord parity to chord index
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-12-16
    Zhiyun Cheng; Denis A. Fedoseev; Hongzhu Gao; Vassily O. Manturov; Mengjian Xu

    We give a brief survey of virtual knot invariants derived from chord parity or chord index. These invariants have grown into an area in its own right due to rapid developing in the last decade. Several similar invariants of flat virtual knots and free knots are also discussed.

    更新日期:2020-12-18
  • Persistent homology for hypergraphs and computational tools — A survey for users
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-12-16
    Shiquan Ren

    In this paper, we give the users an introduction about the framework of persistent homology methods for hypergraphs. We list the steps for standard computations of the persistent homology and discuss about the algorithms. We also give some potential mathematical tools for efficient computing.

    更新日期:2020-12-18
  • Normal forms in braid groups with respect to some Gröbner–Shirshov basis
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-12-11
    Xing Chen; Xuezhi Zhao

    In 2008, Bokut obtained a Gröbner–Shirshov basis S of the braid group Bn+1 in the Artin–Garside generators and showed that S-irreducible words of the Bn+1 coincided with the Garside normal forms of words. Using this basis S, we obtain the concrete expression of the S-irreducible words, i.e. normal forms, of the Bn+1, and hence give a new understanding of the word problem and Garside normal forms of

    更新日期:2020-12-13
  • New criteria and constructions of brunnian links
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-12-11
    Sheng Bai; Weibiao Wang

    We present two practical and widely applicable methods, including some criteria and a general procedure, for detecting Brunnian property of a link, if each component is known to be unknot. The methods are based on observation and handwork. They are used successfully for all Brunnian links known so far. Typical examples and extensive experiments illustrate their efficiency. As an application, infinite

    更新日期:2020-12-13
  • On generalized configuration space and its homotopy groups
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-12-10
    Jun Wang; Xuezhi Zhao

    Let M be a subset of vector space or projective space. The authors define generalized configuration space of M which is formed by n-tuples of elements of M, where any k elements of each n-tuple are linearly independent. The generalized configuration space gives a generalization of Fadell’s classical configuration space, and Stiefel manifold. Denote generalized configuration space of M by Wk,n(M). For

    更新日期:2020-12-11
  • Cohomological invariants of representations of 3-manifold groups
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-12-10
    Haimiao Chen

    Suppose Γ is a discrete group, and α∈Z3(BΓ;A), with A an abelian group. Given a representation ρ:π1(M)→Γ, with M a closed 3-manifold, put F(M,ρ)=〈(Bρ)∗[α],[M]〉, where Bρ:M→BΓ is a continuous map inducing ρ which is unique up to homotopy, and 〈−,−〉:H3(M;A)×H3(M;ℤ)→A is the pairing. We extend the definition of F(M,ρ) to manifolds with corners, and establish a gluing law. Based on these, we present a

    更新日期:2020-12-11
  • Hyperbolic tunnel-number-one knots with Seifert-fibered Dehn surgeries
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-11-03
    Sungmo Kang

    Suppose α and R are disjoint simple closed curves in the boundary of a genus two handlebody H such that H[R] (i.e. a 2-handle addition along R) embeds in S3 as the exterior of a hyperbolic knot k (thus, k is a tunnel-number-one knot), and α is Seifert in H (i.e. a 2-handle addition H[α] is a Seifert-fibered space) and not the meridian of H[R]. Then for a slope γ of k represented by α, γ-Dehn surgery

    更新日期:2020-11-03
  • On fertility of knot shadows
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-11-02
    Ryo Hanaki

    A knot K is a parent of a knot H if there exists a minimal crossing diagram D of K such that a subset of the crossings of D can be changed to produce a diagram of H. A knot K with crossing number n is fertile if for any prime knot H with crossing number less than n, K is a parent of H. It is known that only 01,31,41,52,62,63,76 are fertile for knots up to 10 crossings. However it is unknown whether

    更新日期:2020-11-03
  • Morphisms between categorified spin networks
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-10-28
    Matthew Hogancamp

    We introduce a graphical calculus for computing morphism spaces between the categorified spin networks of Cooper and Krushkal. The calculus, phrased in terms of planar compositions of categorified Jones–Wenzl projectors and their duals, is then used to study the module structure of spin networks over the colored unknots.

    更新日期:2020-10-30
  • Generalized torsion elements and hyperbolic links
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-10-28
    Masakazu Teragaito

    In a group, a generalized torsion element is a non-identity element whose some non-empty finite product of its conjugates yields the identity. Such an element is an obstruction for a group to be bi-orderable. We show that the Weeks manifold, the figure-eight sister manifold, and the complement of Whitehead sister link admit generalized torsion elements in their fundamental groups. In particular, the

    更新日期:2020-10-30
  • Minimal grid diagrams of 11 crossing prime alternating knots
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-10-23
    Gyo Taek Jin; Hwa Jeong Lee

    The arc index of a knot is the minimal number of arcs in all arc presentations of the knot. An arc presentation of a knot can be shown in the form of a grid diagram which is a closed plane curve consisting of finitely many horizontal line segments and the same number of vertical line segments. The arc index of an alternating knot is its minimal crossing number plus two. In this paper, we give a list

    更新日期:2020-10-30
  • Extending quasi-alternating links
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-10-16
    Nafaa Chbili; Kirandeep Kaur

    Champanerkar and Kofman [Twisting quasi-alternating links, Proc. Amer. Math. Soc.137(7) (2009) 2451–2458] introduced an interesting way to construct new examples of quasi-alternating links from existing ones. Actually, they proved that replacing a quasi-alternating crossing c in a quasi-alternating link by a rational tangle of same type yields a new quasi-alternating link. This construction has been

    更新日期:2020-10-17
  • Labeled singular knots
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-10-13
    Khaled Bataineh

    We introduce labeled singular knots and equivalently labeled 4-valent rigid vertex spatial graphs. Labeled singular knots are singular knots with labeled singularities. These knots are considered subject to isotopies preserving the labelings. We provide a topological invariant schema similar to that of Henrich and Kauffman in [A. Henrich and L. H. Kauffman, Tangle insertion invariants for pseudoknots

    更新日期:2020-10-15
  • Knot diagrams on a punctured sphere as a model of string figures
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-10-13
    Masafumi Arai; Kouki Taniyama

    A string figure is topologically a trivial knot lying on an imaginary plane orthogonal to the fingers with some crossings. The fingers prevent cancellation of these crossings. As a mathematical model of string figure, we consider a knot diagram on the xy-plane in xyz-space missing some straight lines parallel to the z-axis. These straight lines correspond to fingers. We study minimal number of crossings

    更新日期:2020-10-15
  • A note on the orderability of Dehn fillings of the manifold v2503
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-10-06
    Konstantinos Varvarezos

    We show that Dehn filling on the manifold v2503 results in a non-orderable space for all rational slopes in the interval (−∞,−1). This is consistent with the L-space conjecture, which predicts that all fillings will result in a non-orderable space for this manifold.

    更新日期:2020-10-07
  • Positive knots and weak (1, 3) homotopy
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-09-04
    Noboru Ito; Yusuke Takimura

    It is known that there exists a surjective map from the set of weak (1, 3) homotopy classes of knot projections to the set of positive knots [N. Ito and Y. Takimura, (1, 2) and weak (1, 3) homotopies on knot projections, J. Knot Theory Ramifications22 (2013) 1350085]. An interesting question whether this map is also injective, which question was formulated independently by S. Kamada and Y. Nakanishi

    更新日期:2020-09-23
  • On knot groups acting on trees
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-07-30
    Fedor A. Dudkin; Andrey S. Mamontov

    A finitely generated group G acting on a tree with infinite cyclic edge and vertex stabilizers is called a generalized Baumslag–Solitar group (GBS group). We prove that a one-knot group G is a GBS group if and only if G is a torus knot group, and describe all n-knot GBS groups for n≥3.

    更新日期:2020-09-23
  • An invariant of graph-links valued in graphs and an almost classification of graph-links
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-08-05
    Denis P. Ilyutko; Vassily O. Manturov

    In [V. O. Manturov, An almost classification of free knots, Dokl. Math.88(2) (2013) 556–558.] the second author constructed an invariant which in some sense generalizes the quantum sl(3) link invariant of Kuperberg to the case of free links. In this paper, we generalize this construction to free graph-links. As a result, we obtain an invariant of free graph-links with values in linear combinations

    更新日期:2020-09-23
  • On the indeterminacy of Milnor’s triple linking number
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-08-17
    Jonah Amundsen; Eric Anderson; Christopher William Davis

    In the 1950s Milnor defined a family of higher-order invariants generalizing the linking number. Even the first of these new invariants, the triple linking number, has received fruitful study since its inception. In the case that a link L has vanishing pairwise linking numbers, this triple linking number gives an integer-valued invariant. When the linking numbers fail to vanish, the triple linking

    更新日期:2020-09-23
  • On the lower central series of some virtual knot groups
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-08-28
    Valeriy G. Bardakov; Neha Nanda; Mikhail V. Neshchadim

    We study groups of some virtual knots with small number of crossings and prove that there is a virtual knot with long lower central series which, in particular, implies that there is a virtual knot with residually nilpotent group. This gives a possibility to construct invariants of virtual knots using quotients by terms of the lower central series of knot groups. Also, we study decomposition of virtual

    更新日期:2020-09-23
  • Tau invariants for balanced spatial graphs
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-09-04
    Katherine Vance

    In 2003, Ozsváth and Szabó defined the concordance invariant τ for knots in oriented 3-manifolds as part of the Heegaard Floer homology package. In 2011, Sarkar gave a combinatorial definition of τ for knots in S3 and a combinatorial proof that τ gives a lower bound for the slice genus of a knot. Recently, Harvey and O’Donnol defined a relatively bigraded combinatorial Heegaard Floer homology theory

    更新日期:2020-09-23
  • 2-Knot homology and Yoshikawa move
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-08-28
    Hiroshi Matsuda

    Ng constructed an invariant of knots in ℝ3, a combinatorial knot contact homology. Extending his study, we construct an invariant of surface-knots in ℝ4 using marked graph diagrams.

    更新日期:2020-09-23
  • Twisted torus knots T(p,q,p − kq,−1) which are torus knots
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-09-05
    Sangyop Lee

    A twisted torus knot is a torus knot with some consecutive strands twisted. More precisely, a twisted torus knot T(p,q,r,s) is a torus knot T(p,q) with r consecutive strands s times fully twisted. We determine which twisted torus knots T(p,q,p−kq,−1) are a torus knot.

    更新日期:2020-09-23
  • Amphicheirality of ribbon 2-knots
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-09-05
    Tomoyuki Yasuda

    For any classical knot k1, we can construct a ribbon 2-knot spun(k1) by spinning an arc removed a small segment from k1 about R2 in R4. A ribbon 2-knot is an embedded 2-sphere in R4. If k1 has an n-crossing presentation, by spinning this, we can naturally construct a ribbon presentation with n ribbon crossings for spun(k1). Thus, we can define naturally a notion on ribbon 2-knots corresponding to the

    更新日期:2020-09-23
  • Positive twist knots and the uniform thickness property
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-08-08
    Whitney George; Mark Myers

    We extend results of Positive Twist Knots and Thickenings [W. George, Positive twist knots and thickenings, J. Knot Theory Ramifications22 (2013) 1350046] to show that positive twist knots 𝒦m, for m≥3, satisfy the Uniform Thickness Property.

    更新日期:2020-09-22
  • Black holes and topological surgery
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-09-14
    Stathis Antoniou; Louis H. Kauffman; Sofia Lambropoulou

    We directly connect topological changes that can occur in mathematical three-space via surgery, with black hole formation, the formation of wormholes and new generalizations of these phenomena. This work widens the bridge between topology and natural sciences and creates a new platform for exploring geometrical physics.

    更新日期:2020-09-14
  • Torsion calculations in Khovanov cohomology
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-06-24
    Dirk Schütz

    We obtain information on torsion in Khovanov cohomology by performing calculations directly over ℤ/pkℤ for p prime and k≥2. In particular, we get that the torus knots T(9,10) and T(9,11) contain torsion of order 9 and 27 in their Khovanov cohomology.

    更新日期:2020-09-02
  • F-polynomials of tabulated virtual knots
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-06-24
    Maxim Ivanov; Andrei Vesnin

    A sequence of F-polynomials {FKn(t,ℓ)}n=1∞ of virtual knots K was defined by Kaur et al. in 2018. These polynomials have been expressed in terms of index value of crossing and n-writhe of K. By the construction, F-polynomials are generalizations of Kauffman’s Affine Index Polynomial, and are invariants of virtual knot K. We present values of F-polynomials of oriented virtual knots having at most four

    更新日期:2020-09-02
  • On Bennequin-type inequalities for links in tight contact 3-manifolds
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-07-20
    Alberto Cavallo

    We prove that a version of the Thurston–Bennequin inequality holds for Legendrian and transverse links in a rational homology contact 3-sphere (M,ξ), whenever ξ is tight. More specifically, we show that the self-linking number of a transverse link T in (M,ξ), such that the boundary of its tubular neighborhood consists of incompressible tori, is bounded by the Thurston norm ∥T∥T of T. A similar inequality

    更新日期:2020-09-02
  • Satellite knots and trivializing bands
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-07-25
    Lorena Armas-Sanabria; Mario Eudave-Muñoz

    We show an infinite family of satellite knots that can be unknotted by a single band move, but such that there is no band unknotting the knots which is disjoint from the satellite torus.

    更新日期:2020-09-02
  • Links in surfaces and Laplacian modules
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-08-05
    Daniel S. Silver; Susan G. Williams

    Laplacian matrices of signed graphs in surfaces S are used to define module and polynomial invariants of ℤ/2-homologically trivial links in S×[0,1]. Information about virtual genus is obtained.

    更新日期:2020-09-02
  • Minimal coloring numbers on minimal diagrams of torus links
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-07-25
    Kazuhiro Ichihara; Katsumi Ishikawa; Eri Matsudo

    We determine the minimal number of colors for nontrivial ℤ-colorings on the standard minimal diagrams of ℤ-colorable torus links. Also included is a complete classification of such ℤ-colorings, which are shown by using rack colorings on link diagrams.

    更新日期:2020-09-02
  • Prime alternating knots of minimal warping degree two
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-07-08
    Ayaka Shimizu

    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.

    更新日期:2020-09-02
  • DNA origami and unknotted A-trails in torus graphs
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-07-20
    Ada Morse; William Adkisson; Jessica Greene; David Perry; Brenna Smith; Jo Ellis-Monaghan; Greta Pangborn

    Motivated by problem of determining the unknotted routes for the scaffolding strand in DNA origami self-assembly, we examine the existence and knottedness of A-trails in graphs embedded on the torus. We show that any A-trail in a checkerboard-colorable torus graph is unknotted and characterizes the existence of A-trails in checkerboard-colorable torus graphs in terms of pairs of quasitrees in associated

    更新日期:2020-09-01
  • Refined Chern–Simons theory in genus two
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-07-15
    S. Arthamonov; Sh. Shakirov

    Reshetikhin–Turaev (a.k.a. Chern–Simons) TQFT is a functor that associates vector spaces to two-dimensional genus g surfaces and linear operators to automorphisms of surfaces. The purpose of this paper is to demonstrate that there exists a Macdonald q,t-deformation — refinement — of these operators that preserves the defining relations of the mapping class groups beyond genus 1. For this, we explicitly

    更新日期:2020-09-01
  • A multiple group rack and oriented spatial surfaces
    J. Knot Theory Ramif. (IF 0.426) Pub Date : 2020-07-15
    Atsushi Ishii; Shosaku Matsuzaki; Tomo Murao

    A spatial surface is a compact surface embedded in the 3-sphere. In this paper, we provide several typical examples of spatial surfaces and construct a coloring invariant to distinguish them. The coloring is defined by using a multiple group rack, which is a rack version of a multiple conjugation quandle.

    更新日期:2020-09-01
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