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Authors:Teruo Nagase, Akiko Shima Abstract: Journal of Knot Theory and Its Ramifications, Ahead of Print. Charts are oriented labeled graphs in a disk. Any simple surface braid (2-dimensional braid) can be described by using a chart. Also, a chart represents an oriented closed surface embedded in 4-space. In this paper, we investigate embedded surfaces in 4-space by using charts. Let [math] be a chart, and we denote by [math] the union of all the edges of label [math]. A chart [math] is of type [math] if there exists a label [math] such that [math], [math] where [math] is the number of white vertices in [math]. In this paper, we prove that there is no minimal chart of type [math]. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2022-05-12T07:00:00Z DOI: 10.1142/S0218216522500171

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Authors:Min Hoon Kim, Changhee Lee, Minkyoung Song Abstract: Journal of Knot Theory and Its Ramifications, Ahead of Print. Greene-Jabuka and Lecuona confirmed the slice-ribbon conjecture for 3-stranded pretzel knots except for an infinite family [math], where [math] is an odd integer greater than [math]. Lecuona and Miller showed that [math] are not slice unless [math]. In this note, we show that four-fifths of the remaining knots in the family are not slice. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2022-05-12T07:00:00Z DOI: 10.1142/S0218216522500183

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Authors:Anshul Guha Abstract: Journal of Knot Theory and Its Ramifications, Ahead of Print. An [math]-crossing projection of a link [math] is a projection of [math] onto a plane such that [math] points on [math] are superimposed on top of each other at every crossing. We prove that for all [math] and all links [math], the inequality c2k+1(L) ≥ 2g(L) + r(L) − 1 k2 holds, where [math], [math] and [math] are the [math]-crossing number, [math]-genus, and the number of components of [math] respectively. This result is used to prove a new bound on the odd crossing numbers of torus knots and generalizes a result of Jablonowski (see [M. Jabłonowski, Triple-crossing number, the genus of a knot or link and torus knots, Topology Appl. 285 (2020) 107389]). We also prove a new upper bound on the [math]-crossing numbers of the 2-torus knots and links. Furthermore, we improve the lower bounds on the [math]-crossing numbers of [math] knots with [math]-crossing number at most [math]. Finally, we improve the lower bounds on the [math]-crossing numbers of [math] knots with [math]-crossing number at most [math]. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2022-05-06T07:00:00Z DOI: 10.1142/S0218216522500080

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Authors:Sam Nelson, Yuqi Zhao Abstract: Journal of Knot Theory and Its Ramifications, Ahead of Print. In this paper, we generalize unoriented handlebody-links to the twisted virtual case, obtaining Reidemeister moves for handlebody-links in ambient spaces of the form [math] for [math] a compact closed 2-manifold up to stable equivalence. We introduce a related algebraic structure known as twisted virtual bikeigebras whose axioms are motivated by the twisted virtual handlebody-link Reidemeister moves. We use twisted virtual bikeigebras to define [math]-colorability for twisted virtual handlebody-links and define an integer-valued invariant [math] of twisted virtual handlebody-links. We provide example computations of the new invariants and use them to distinguish some twisted virtual handlebody-links. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2022-05-06T07:00:00Z DOI: 10.1142/S0218216522500109

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Authors:Khaydar Nurligareev, Ivan Reshetnikov Abstract: Journal of Knot Theory and Its Ramifications, Ahead of Print. A finite subset [math] is basic, if for any function [math] there exists a collection of functions [math] such that for each element [math], we have [math]. For certain finite sets, we prove a criterion for a set to be basic, and we show that it cannot be extended to the general case. In addition, we interpret the above criterion in terms of doubly-weighted graphs and give an estimation for the number of elements in certain basic and non-basic subsets. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2022-05-06T07:00:00Z DOI: 10.1142/S0218216522500110

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Authors:Shiquan Ren, Chengyuan Wu, Jie Wu Abstract: Journal of Knot Theory and Its Ramifications, Ahead of Print. Let [math] be a simplicial complex. In this paper, we study random sub-hypergraphs and random sub-complexes of [math]. By considering the minimal complex that a sub-hypergraph can be embedded in and the maximal complex that can be embedded in a sub-hypergraph, we define some maps on the space of probability functions on sub-hypergraphs of [math]. We study the compositions of these maps as well as their actions on the space of probability functions. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2022-05-06T07:00:00Z DOI: 10.1142/S0218216522500158

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Authors:Zipei Zhuang Abstract: Journal of Knot Theory and Its Ramifications, Ahead of Print. For a connected cobordism [math] between two knots [math] in [math], we establish an inequality involving the number of local maxima, the genus of [math], and the torsion orders of [math], where [math] denotes Lee’s perturbation of Khovanov homology. This shows that the torsion order gives a lower bound for the band-unlinking number. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2022-04-30T07:00:00Z DOI: 10.1142/S0218216522500122

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Authors:M. Y. Avetisyan, R. L. Mkrtchyan Abstract: Journal of Knot Theory and Its Ramifications, Ahead of Print. In the study of finite (Vassiliev’s) knot invariants, Vogel introduced the so-called universal parameters, belonging to the projective plane, in particular parametrizing the simple Lie algebras by Vogel’s table. Subsequently, a number of quantities, such as some universal knot invariants and (quantum) dimensions of simple Lie algebras, have been represented in terms of these parameters, i.e., in the universal form. We prove that at the points from Vogel’s table all known universal quantum dimension formulae are linearly resolvable, i.e., yield finite answers even if these points are singular, provided one restricts them to the appropriate lines. We show that the same phenomenon takes place for another three distinguished points in Vogel’s plane — [math] and [math]. We also examine the same formulae on linear resolvability at the remaining 48 distinguished points in Vogel’s plane, which correspond to the so-called [math]-objects. Among them, three points were found to be regular for all known quantum dimension formulae. Two of them happen to be sharing a remarkable similarity with the simple Lie algebras, namely, the universal formulae yield integer-valued outputs (dimensions) at the indicated points in the classical limit. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2022-04-30T07:00:00Z DOI: 10.1142/S0218216522500146

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Authors:Youfa Han, Boxin Zhou Abstract: Journal of Knot Theory and Its Ramifications, Ahead of Print. This paper mainly studies the minimum number of colorings for all non-trivially 19-colored diagrams of any 19-colorable knot K. By using some special Reidemeister move, we successfully eliminated 13 colors from 19 colors. It can be seen that for any 19-colorable knot K, at least six colors are enough to color K, that is, the minimum number of 19-colorable knot is six. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2022-04-27T07:00:00Z DOI: 10.1142/S0218216522500134

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Authors:Igor Nikonov Abstract: Journal of Knot Theory and Its Ramifications, Ahead of Print. A parity is a labeling of the crossings of knot diagrams which is compatible with Reidemeister moves. We define the notion of parity for based matrices — algebraic objects are introduced by Turaev in his research on virtual strings. We present the reduced stable parity on based matrix which gives a new example of a parity of virtual knots. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2022-04-22T07:00:00Z DOI: 10.1142/S0218216522500079

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Authors:Blake Mellor Abstract: Journal of Knot Theory and Its Ramifications, Ahead of Print. To better understand the fundamental quandle of a knot or link, it can be useful to look at finite quotients of the quandle. One such quotient is the [math]-quandle (or, when [math], the involutory quandle). Hoste and Shanahan [J. Hoste and P. D. Shanahan, Links with finite n-quandles, Algebraic Geom. Topol. 17 (2017) 2807–2823.] gave a complete list of the links which have finite [math]-quandles; it remained to give explicit descriptions of these quandles. This has been done for several cases in [A. Crans, J. Hoste, B. Mellor and P. D. Shanahan, Finite n-qundles of torus and two-bridge links, J. Knot Theory Ramifications 28 (2019) 1950028; J. Hoste and P. D. Shanahan, Involutory quandles of (2, 2, r)-Montesinos links, J. Knot Theory Ramifications 26 (2017)]; in this work, we continue this project and explicitly describe the Cayley graphs for the finite involutory quandles of two-bridge links with an axis. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2022-04-22T07:00:00Z DOI: 10.1142/S0218216522500092

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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