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Authors:Sangyop Lee Abstract: Journal of Knot Theory and Its Ramifications, Volume 29, Issue 09, August 2020. A twisted torus knot is a torus knot with some consecutive strands twisted. More precisely, a twisted torus knot [math] is a torus knot [math] with [math] consecutive strands [math] times fully twisted. We determine which twisted torus knots [math] are a torus knot. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2020-09-05T07:00:00Z DOI: 10.1142/S0218216520500686 Issue No:Vol. 29, No. 09 (2020)
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Authors:Tomoyuki Yasuda Abstract: Journal of Knot Theory and Its Ramifications, Volume 29, Issue 09, August 2020. For any classical knot [math], we can construct a ribbon [math]-knot [math] by spinning an arc removed a small segment from [math] about [math] in [math]. A ribbon [math]-knot is an embedded [math]-sphere in [math]. If [math] has an [math]-crossing presentation, by spinning this, we can naturally construct a ribbon presentation with [math] ribbon crossings for [math]. Thus, we can define naturally a notion on ribbon [math]-knots corresponding to the crossing number on classical knots. It is called the ribbon crossing number. On classical knots, it was a long-standing conjecture that any odd crossing classical knot is not amphicheiral. In this paper, we show that for any odd integer [math] there exists an amphicheiral ribbon [math]-knot with the ribbon crossing number [math]. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2020-09-05T07:00:00Z DOI: 10.1142/S0218216520500698 Issue No:Vol. 29, No. 09 (2020)
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Authors:Noboru Ito, Yusuke Takimura Abstract: Journal of Knot Theory and Its Ramifications, Volume 29, Issue 09, August 2020. 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 Ramifications 22 (2013) 1350085]. An interesting question whether this map is also injective, which question was formulated independently by S. Kamada and Y. Nakanishi in 2013 (Question q1). This paper obtains an answer to this question. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2020-09-04T07:00:00Z DOI: 10.1142/S0218216520500613 Issue No:Vol. 29, No. 09 (2020)
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Authors:Katherine Vance Abstract: Journal of Knot Theory and Its Ramifications, Volume 29, Issue 09, August 2020. In 2003, Ozsváth and Szabó defined the concordance invariant [math] for knots in oriented 3-manifolds as part of the Heegaard Floer homology package. In 2011, Sarkar gave a combinatorial definition of [math] for knots in [math] and a combinatorial proof that [math] 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 for transverse spatial graphs in [math], extending HFK for knots. We define a [math]-filtered chain complex for balanced spatial graphs whose associated graded chain complex has homology determined by Harvey and O’Donnol’s graph Floer homology. We use this to show that there is a well-defined [math] invariant for balanced spatial graphs generalizing the [math] knot concordance invariant. In particular, this defines a [math] invariant for links in [math]. Using techniques similar to those of Sarkar, we show that our [math] invariant is an obstruction to a link being slice. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2020-09-04T07:00:00Z DOI: 10.1142/S0218216520500662 Issue No:Vol. 29, No. 09 (2020)
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Authors:Valeriy G. Bardakov, Neha Nanda, Mikhail V. Neshchadim Abstract: Journal of Knot Theory and Its Ramifications, Volume 29, Issue 09, August 2020. 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 knot groups as semi direct product and free product with amalgamation. In particular, we prove that the groups of some virtual knots are extensions of finitely generated free groups by infinite cyclic groups. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2020-08-28T07:00:00Z DOI: 10.1142/S0218216520500650 Issue No:Vol. 29, No. 09 (2020)
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Authors:Hiroshi Matsuda Abstract: Journal of Knot Theory and Its Ramifications, Volume 29, Issue 09, August 2020. Ng constructed an invariant of knots in [math], a combinatorial knot contact homology. Extending his study, we construct an invariant of surface-knots in [math] using marked graph diagrams. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2020-08-28T07:00:00Z DOI: 10.1142/S0218216520500674 Issue No:Vol. 29, No. 09 (2020)
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Authors:Jonah Amundsen, Eric Anderson, Christopher William Davis Abstract: Journal of Knot Theory and Its Ramifications, Volume 29, Issue 09, August 2020. 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 [math] has vanishing pairwise linking numbers, this triple linking number gives an integer-valued invariant. When the linking numbers fail to vanish, the triple linking number is only well-defined modulo their greatest common divisor. In recent work Davis–Nagel–Orson–Powell produce a single invariant called the total triple linking number refining the triple linking number and taking values in an abelian group called the total Milnor quotient. They present examples for which this quotient is nontrivial even though each of the individual triple linking numbers take values in the trivial group, [math], and so carry no information. As a consequence, the total triple linking number carries more information than do the classical triple linking numbers. The goal of this paper is to compute this group and show that when [math] is a link of at least six components it is nontrivial. Thus, this total triple linking number carries information for every [math]-component link, even though the classical triple linking numbers often carry no information. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2020-08-17T07:00:00Z DOI: 10.1142/S0218216520500649 Issue No:Vol. 29, No. 09 (2020)
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Authors:Denis P. Ilyutko, Vassily O. Manturov Abstract: Journal of Knot Theory and Its Ramifications, Volume 29, Issue 09, August 2020. 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 [math] 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 of graphs. The main property of this invariant is that under certain conditions on the representative of the free graph-link, we can recover this representative from the value invariant on it. In addition, this invariant allows one to partially classify free graph-links. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2020-08-05T07:00:00Z DOI: 10.1142/S0218216520500637 Issue No:Vol. 29, No. 09 (2020)
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Authors:Fedor A. Dudkin, Andrey S. Mamontov Abstract: Journal of Knot Theory and Its Ramifications, Volume 29, Issue 09, August 2020. A finitely generated group [math] 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 [math] is a GBS group if and only if [math] is a torus knot group, and describe all n-knot GBS groups for [math]. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2020-07-30T07:00:00Z DOI: 10.1142/S0218216520500625 Issue No:Vol. 29, No. 09 (2020)
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Authors:Gyo Taek Jin, Hwa Jeong Lee Abstract: Journal of Knot Theory and Its Ramifications, Ahead of Print. 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 of minimal grid diagrams of the 11 crossing prime alternating knots obtained from arc presentations with 13 arcs. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2020-10-23T07:00:00Z DOI: 10.1142/S0218216520500765
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Authors:Elmas Irmak Abstract: Journal of Knot Theory and Its Ramifications, Ahead of Print. Let [math] be a compact, connected, orientable surface of genus [math] with [math] boundary components with [math], [math]. Let [math] be the nonseparating curve graph, [math] be the curve graph and [math] be the Hatcher–Thurston graph of [math]. We prove that if [math] is an edge-preserving map, then [math] is induced by a homeomorphism of [math]. We prove that if [math] is an edge-preserving map, then [math] is induced by a homeomorphism of [math]. We prove that if [math] is closed and [math] is a rectangle preserving map, then [math] is induced by a homeomorphism of [math]. We also prove that these homeomorphisms are unique up to isotopy when [math]. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2020-10-22T07:00:00Z DOI: 10.1142/S0218216520500789
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Authors:Nafaa Chbili, Kirandeep Kaur Abstract: Journal of Knot Theory and Its Ramifications, Ahead of Print. 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 [math] in a quasi-alternating link by a rational tangle of same type yields a new quasi-alternating link. This construction has been extended to alternating algebraic tangles and applied to characterize all quasi-alternating Montesinos links. In this paper, we extend this technique to any alternating tangle of same type as [math]. As an application, we give new examples of quasi-alternating knots of 13 and 14 crossings. Moreover, we prove that the Jones polynomial of a quasi-alternating link that is obtained in this way has no gap if the original link has no gap in its Jones polynomial. This supports a conjecture introduced in [N. Chbili and K. Qazaqzeh, On the Jones polynomial of quasi-alternating links, Topology Appl. 264 (2019) 1–11], which states that the Jones polynomial of any prime quasi-alternating link except [math]-torus links has no gap. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2020-10-16T07:00:00Z DOI: 10.1142/S0218216520500728
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Authors:Khaled Bataineh Abstract: Journal of Knot Theory and Its Ramifications, Ahead of Print. 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, singular knots, and rigid vertex spatial graphs, Contemp. Math. 689 (2017) 1–10] by inserting rational tangles at the labeled singularities to extend usual knot invariants to our class of singular knots. We show that we can use invariants of labeled singular knots to serve usual singular knots. Labeled framed pseudoknots are also introduced and discussed. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2020-10-13T07:00:00Z DOI: 10.1142/S0218216520500704
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Authors:Masafumi Arai, Kouki Taniyama Abstract: Journal of Knot Theory and Its Ramifications, Ahead of Print. 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 [math]-plane in [math]-space missing some straight lines parallel to the [math]-axis. These straight lines correspond to fingers. We study minimal number of crossings of these knot diagrams under Reidemeister moves missing these lines. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2020-10-13T07:00:00Z DOI: 10.1142/S0218216520500716
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Authors:Konstantinos Varvarezos Abstract: Journal of Knot Theory and Its Ramifications, Ahead of Print. We show that Dehn filling on the manifold [math] results in a non-orderable space for all rational slopes in the interval [math]. This is consistent with the L-space conjecture, which predicts that all fillings will result in a non-orderable space for this manifold. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2020-10-06T07:00:00Z DOI: 10.1142/S0218216520710029
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Authors:Stathis Antoniou, Louis H. Kauffman, Sofia Lambropoulou Abstract: Journal of Knot Theory and Its Ramifications, Ahead of Print. 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. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2020-09-14T07:00:00Z DOI: 10.1142/S0218216520420109
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Authors:Fengling Li, Dongxu Wang, Liang Liang, Fengchun Lei Abstract: Journal of Knot Theory and Its Ramifications, Ahead of Print. In the paper, we give an equivalent description of the lens space [math] with [math] prime in terms of any corresponding Heegaard diagrams as follows: Let [math] be a closed orientable 3-manifold with [math] and [math] a Heegaard splitting of genus [math] for [math] with an associated Heegaard diagram [math]. Assume [math] is a prime integer. Then [math] is homeomorphic to the lens space [math] if and only if there exists an embedding [math] such that [math] bounds a complete system of surfaces for [math]. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2020-08-21T07:00:00Z DOI: 10.1142/S0218216520420055
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Authors:Amrendra Gill, Madeti Prabhakar, Andrei Vesnin Abstract: Journal of Knot Theory and Its Ramifications, Ahead of Print. Gordian complex of knots was defined by Hirasawa and Uchida as the simplicial complex whose vertices are knot isotopy classes in [math]. Later Horiuchi and Ohyama defined Gordian complex of virtual knots using [math]-move and forbidden moves. In this paper, we discuss Gordian complex of knots by region crossing change and Gordian complex of virtual knots by arc shift move. Arc shift move is a local move in the virtual knot diagram which results in reversing orientation locally between two consecutive crossings. We show the existence of an arbitrarily high-dimensional simplex in both the Gordian complexes, i.e. by region crossing change and by the arc shift move. For any given knot (respectively, virtual knot) diagram we construct an infinite family of knots (respectively, virtual knots) such that any two distinct members of the family have distance one by region crossing change (respectively, arc shift move). We show that the constructed virtual knots have the same affine index polynomial. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2020-08-21T07:00:00Z DOI: 10.1142/S0218216520420080
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Authors:Tushar K. Naik, Neha Nanda, Mahender Singh Abstract: Journal of Knot Theory and Its Ramifications, Ahead of Print. The twin group [math] is a right angled Coxeter group generated by [math] involutions and having only far commutativity relations. These groups can be thought of as planar analogues of Artin braid groups. In this paper, we study some properties of twin groups whose analogues are well known for Artin braid groups. We give an algorithm for two twins to be equivalent under individual Markov moves. Further, we show that twin groups [math] have [math]-property and are not co-Hopfian for [math]. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2020-08-19T07:00:00Z DOI: 10.1142/S0218216520420067
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Authors:Valeriy G. Bardakov, Jie Wu Abstract: Journal of Knot Theory and Its Ramifications, Ahead of Print. This paper is dedicated to cabling on virtual braids. This construction gives a new generating set for the virtual pure braid group [math]. We define simplicial group [math] and its simplicial subgroup [math] which is generated by [math]. Consequently, we describe [math] as HNN-extension and find presentation of [math] and [math]. As an application to classical braids, we find a new presentation of the Artin pure braid group [math] in terms of the cabled generators. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2020-08-17T07:00:00Z DOI: 10.1142/S021821652042002X
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Authors:Zhiyun Cheng, Hongzhu Gao, Mengjian Xu Abstract: Journal of Knot Theory and Its Ramifications, Ahead of Print. In this paper, we discuss how to define a chord index via smoothing a real crossing point of a virtual knot diagram. Several polynomial invariants of virtual knots and links can be recovered from this general construction. We also explain how to extend this construction from virtual knots to flat virtual knots. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2020-08-08T07:00:00Z DOI: 10.1142/S0218216520420031
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Authors:Valeriy G. Bardakov, Tatyana A. Kozlovskaya Abstract: Journal of Knot Theory and Its Ramifications, Ahead of Print. In this paper, we study the singular pure braid group [math] for [math]. We find generators, defining relations and the algebraical structure of these groups. In particular, we prove that [math] is a semi-direct product [math], where [math] is an HNN-extension with base group [math] and cyclic associated subgroups. We prove that the center [math] of [math] is a direct factor in [math]. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2020-08-05T07:00:00Z DOI: 10.1142/S0218216520420018
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Authors:Akio Kawauchi Abstract: Journal of Knot Theory and Its Ramifications, Ahead of Print. 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. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2020-08-05T07:00:00Z DOI: 10.1142/S0218216520420043
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Authors:Wonjun Chang, Byung Chun Kim, Yongjin Song Abstract: Journal of Knot Theory and Its Ramifications, Ahead of Print. The [math]-fold ([math]) branched coverings on a disk give an infinite family of nongeometric embeddings of braid groups into mapping class groups. We, in this paper, give new explicit expressions of these braid group representations into automorphism groups of free groups in terms of the actions on the generators of free groups. We also give a systematic way of constructing and expressing these braid group representations in terms of a new gadget, called covering groupoid. We prove that each generator [math] of braid group inside mapping class group induced by [math]-fold covering is the product of [math] Dehn twists on the surface. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2020-08-05T07:00:00Z DOI: 10.1142/S0218216520420079
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Authors:Yulai Wu, Ximin Liu Abstract: Journal of Knot Theory and Its Ramifications, Ahead of Print. In this paper, we study the minimal symplectic elliptic surfaces [math] with homologically trivial symplectic finite group actions, and get a rigidity theorem under some restriction. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2020-08-05T07:00:00Z DOI: 10.1142/S0218216520420092
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Authors:Masaya Kameyama, Satoshi Nawata Abstract: Journal of Knot Theory and Its Ramifications, Ahead of Print. We formulate large [math] duality of [math] refined Chern–Simons theory with a torus knot/link in [math]. By studying refined BPS states in M-theory, we provide the explicit form of low-energy effective actions of Type IIA string theory with D4-branes on the [math]-background. This form enables us to relate refined Chern–Simons invariants of a torus knot/link in [math] to refined BPS invariants in the resolved conifold. Assuming that the extra [math] global symmetry acts on BPS states trivially, the duality predicts graded dimensions of cohomology groups of moduli spaces of M2–M5 bound states associated to a torus knot/link in the resolved conifold. Thus, this formulation can be also interpreted as a positivity conjecture of refined Chern–Simons invariants of torus knots/links. We also discuss about an extension to non-torus knots. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2020-07-08T07:00:00Z DOI: 10.1142/S0218216520410011
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