Authors:R. İnanç Baykur Abstract: Journal of Knot Theory and Its Ramifications, Volume 27, Issue 05, April 2018. The purpose of this note is to show that classical cobordism arguments, which go back to the pioneering works of Mandelbaum and Moishezon, provide quick and unified proofs of any knot surgered compact simply-connected 4-manifold [math] becoming diffeomorphic to [math] after a single stabilization by connected summing with [math] or [math], and almost complete decomposability of [math] for many almost completely decomposable [math], such as the elliptic surfaces. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2018-04-27T04:23:38Z DOI: 10.1142/S0218216518710013

Authors:Inasa Nakamura Abstract: Journal of Knot Theory and Its Ramifications, Volume 27, Issue 05, April 2018. A branched covering surface-knot over an oriented surface-knot [math] is a surface-knot in the form of a branched covering over [math]. A branched covering surface-knot over [math] is presented by a graph called a chart on a surface diagram of [math]. For a branched covering surface-knot, an addition of 1-handles equipped with chart loops is a simplifying operation which deforms the chart to the form of the union of free edges and 1-handles with chart loops. We investigate properties of such simplifications. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2018-04-27T04:23:37Z DOI: 10.1142/S0218216518500311

Authors:Ryoto Tange Abstract: Journal of Knot Theory and Its Ramifications, Volume 27, Issue 05, April 2018. We present a generalization of the Fox formula for twisted Alexander invariants associated to representations of knot groups over rings of [math]-integers of [math], where [math] is a finite set of finite primes of a number field [math]. As an application, we give the asymptotic growth of twisted homology groups. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2018-04-27T04:23:35Z DOI: 10.1142/S0218216518500335

Authors:Qiang E Abstract: Journal of Knot Theory and Its Ramifications, Volume 27, Issue 05, April 2018. Every surface bundle with genus [math] fiber has a canonical Heegaard splitting of genus [math]. In this paper, we discuss the topological indices of such Heegaard surfaces and prove the canonical Heegaard splitting of a surface bundle is topologically minimal if and only if it is critical, that is, its topological index is 2. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2018-04-27T04:23:33Z DOI: 10.1142/S0218216518500347

Authors:Yoshiro Yaguchi Abstract: Journal of Knot Theory and Its Ramifications, Volume 27, Issue 05, April 2018. A cord is a simple curve on a punctured plane. We introduce diagrams which represent isotopy classes of cords. Using such diagrams, we prove that there is a one-to-one correspondence from the set of the isotopy classes of cords to a set of symmetric matrices whose components are non-negative integers, and we give necessary conditions for matrices to represent the isotopy classes of cords. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2018-04-27T04:23:29Z DOI: 10.1142/S0218216518500360

Authors:Melinda Ho, Sam Nelson Abstract: Journal of Knot Theory and Its Ramifications, Volume 27, Issue 05, April 2018. We consider involutory virtual biracks with good involutions, also known as symmetric involutory virtual biracks. Any good involution on an involutory virtual birack defines an enhancement of the counting invariant. We provide examples demonstrating that the enhancement is stronger than the unenhanced counting invariant. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2018-04-27T04:23:24Z DOI: 10.1142/S0218216518500323

Authors:Natalia A. Viana Bedoya, Daciberg Lima Gonçalves, Elena A. Kudryavtseva Abstract: Journal of Knot Theory and Its Ramifications, Volume 27, Issue 05, April 2018. In this work, we study the decomposability property of branched coverings of degree [math] odd, over the projective plane, where the covering surface has Euler characteristic [math]. The latter condition is equivalent to say that the defect of the covering is greater than [math]. We show that, given a datum [math] with an even defect greater than [math], it is realizable by an indecomposable branched covering over the projective plane. The case when [math] is even is known. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2018-04-27T04:23:20Z DOI: 10.1142/S021821651850030X

Authors:Stephen J. Bigelow Abstract: Journal of Knot Theory and Its Ramifications, Volume 27, Issue 05, April 2018. In a remark in his seminal 1987 paper, Jones describes a way to define the Burau matrix of a positive braid using a metaphor of bowling a ball down a bowling alley with braided lanes. We extend this definition to allow multiple bowling balls to be bowled simultaneously. We obtain representations of the Iwahori–Hecke algebra and a cabled version of the Temperley–Lieb representation. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2018-04-27T04:23:16Z DOI: 10.1142/S0218216518500359

Authors:Margarita Toro, Mauricio Rivera Abstract: Journal of Knot Theory and Its Ramifications, Volume 27, Issue 04, April 2018. We introduce the Schubert form for a [math]-bridge link diagram, as a generalization of the Schubert normal form of a [math]-bridge link. It consists of a set of six positive integers, written as [math], with some conditions and it is based on the concept of [math]-butterfly. Using the Schubert normal form of a [math]-bridge link diagram, we give two presentations of the 3-bridge link group. These presentations are given by concrete formulas that depend on the integers [math]. The construction is a generalization of the form the link group presentation of the [math]-bridge link [math] depends on the integers [math] and [math]. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2018-04-24T09:57:53Z DOI: 10.1142/S0218216518500293

Authors:Román Aranda, Seungwon Kim, Maggy Tomova Abstract: Journal of Knot Theory and Its Ramifications, Volume 27, Issue 04, April 2018. We study the essential surfaces in the exterior of a cable knot to compute the representativity and waist of most cable knots. Our computation answers Ozawa’s question [5] about the relationship between the representativity and the waist of a knot in the negative. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2018-04-24T09:57:47Z DOI: 10.1142/S0218216518500256

Authors:Byung Hee An, Hwa Jeong Lee Abstract: Journal of Knot Theory and Its Ramifications, Volume 27, Issue 04, April 2018. In this paper, we define the set of singular grid diagrams [math] which provides a unified description for singular links, singular Legendrian links, singular transverse links, and singular braids. We also classify the complete set of all equivalence relations on [math] which induce the bijection onto each singular object. This is an extension of the known result of Ng–Thurston [Grid diagrams, braids, and contact geometry, in Proc. Gökova Geometry-Topology Conf. 2008, Gökova Geometry/Topology Conference (GGT), Gökova, 2009, pp. 120–136] for nonsingular links and braids. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2018-04-24T09:57:38Z DOI: 10.1142/S0218216518500232

Authors:Hideo Takioka Abstract: Journal of Knot Theory and Its Ramifications, Volume 27, Issue 04, April 2018. For coprime integers [math] and [math], the [math]-cable [math]-polynomial of a knot is the [math]-polynomial of the [math]-cable knot of the knot, where the [math]-polynomial is the common zeroth coefficient polynomial of the HOMFLYPT and Kauffman polynomials. Since it is known that the [math]-polynomial is computable in polynomial time, the [math]-cable [math]-polynomial is also computable in polynomial time. In this paper, we show that the [math]-cable [math]-polynomial completely classifies the unoriented knots up to ten crossings including the chirality information. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2018-04-24T09:57:34Z DOI: 10.1142/S0218216518500281

Authors:Sangbum Cho, Yuya Koda Abstract: Journal of Knot Theory and Its Ramifications, Volume 27, Issue 04, April 2018. We give a purely combinatorial proof of a result of Kobayashi and Saeki that every genus one 1-bridge position of a nontrivial 2-bridge knot is a stabilization. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2018-04-24T09:57:23Z DOI: 10.1142/S021821651850027X

Authors:Joonoh Kim, Mihaw Shin Abstract: Journal of Knot Theory and Its Ramifications, Volume 27, Issue 04, April 2018. In this paper, we describe a method of making an invariant of virtual knots that is defined in terms of an integer labeling of the flat virtual knot diagram. We give an invariant of flat virtual knots using the invariant above. Moreover, we derive a relation of two invariants. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2018-04-24T09:57:08Z DOI: 10.1142/S0218216518500244

Authors:Hoang-An Nguyen, Anh T. Tran Abstract: Journal of Knot Theory and Its Ramifications, Volume 27, Issue 04, April 2018. The adjoint twisted Alexander polynomial has been computed for twist knots [A. Tran, Twisted Alexander polynomials with the adjoint action for some classes of knots, J. Knot Theory Ramifications 23(10) (2014) 1450051], genus one two-bridge knots [A. Tran, Adjoint twisted Alexander polynomials of genus one two-bridge knots, J. Knot Theory Ramifications 25(10) (2016) 1650065] and the Whitehead link [J. Dubois and Y. Yamaguchi, Twisted Alexander invariant and nonabelian Reidemeister torsion for hyperbolic three dimensional manifolds with cusps, Preprint (2009), arXiv:0906.1500]. In this paper, we compute the adjoint twisted Alexander polynomial and nonabelian Reidemeister torsion of twisted Whitehead links. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2018-04-24T09:56:54Z DOI: 10.1142/S0218216518500268

Authors:Jerzy Kocik Abstract: Journal of Knot Theory and Its Ramifications, Ahead of Print. An alternative framework underlying connection between tensor [math]-calculus and spin networks is suggested. New sign convention for the inner product in the dual spinor space leads to a simpler and direct set of initial rules for the diagrammatic recoupling methods. Yet, it preserves the standard chromatic graph evaluations. In contrast with the standard formulation, the background space is that of symmetric tensor spaces, which seems to be in accordance with the representation theory of [math]. An example of Apollonian disk packing is shown to be a source of spin networks. The graph labeling is extended to non-integer values, resulting in the complex values of chromatic evaluations. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2018-04-23T09:43:16Z DOI: 10.1142/S0218216518410031

Authors:Zhiyun Cheng, Sujoy Mukherjee, Józef H. Przytycki, Xiao Wang, Seung Yeop Yang Abstract: Journal of Knot Theory and Its Ramifications, Ahead of Print. We classify rooted trees which have strictly unimodal plucking polynomials ([math]-polynomials). We also give criteria for a trapezoidal shape of a plucking polynomial. We generalize results of Pak and Panova on strict unimodality of [math]-binomial coefficients. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2018-04-19T08:25:23Z DOI: 10.1142/S0218216518410092

Authors:Igor Szczyrba Abstract: Journal of Knot Theory and Its Ramifications, Ahead of Print. We study the asymptotic behavior of integer sequences related to knots that are generated by linear recurrences. We determine which of the 85 such sequences cataloged in Online Encyclopedia of Integer Sequences have ratios of their consecutive terms converging to a limit. We show that all but one of the ratio limits can be expressed by means of the [math]-anacci constants with [math] equal to 1 or 2. Finally, we demonstrate how the [math]-anacci constants are linked to affine geometry. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2018-04-19T08:25:23Z DOI: 10.1142/S0218216518410110

Authors:Józef H. Przytycki, Xiao Wang Abstract: Journal of Knot Theory and Its Ramifications, Ahead of Print. In 2004, Carter, Elhamdadi and Saito defined a homology theory for set-theoretic Yang–Baxter operators (we will call it the “algebraic” version in this paper). In 2012, Przytycki defined another homology theory for pre-Yang–Baxter operators which has a nice graphic visualization (we will call it the “graphic” version in this paper). We show that they are equivalent. The “graphic” homology is also defined for pre-Yang–Baxter operators, and we give some examples of its one-term and two-term homologies. In the two-term case, we have found torsion in homology of Yang–Baxter operator that yields the Jones polynomial. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2018-04-19T08:25:23Z DOI: 10.1142/S0218216518410134

Authors:L. Sbitneva Abstract: Journal of Knot Theory and Its Ramifications, Ahead of Print. The original approach of Lie to the theory of transformation groups acting on smooth manifolds, on the basis of differential equations, being applied to smooth loops, has permitted the development of the infinitesimal theory of smooth loops generalizing the Lie group theory. A loop with the law of associativity verified for its binary operation is a group. It has been shown that the system of differential equations characterizing a smooth loop with the right Bol identity and the integrability conditions lead to the binary-ternary algebra as a proper infinitesimal object, which turns out to be the Bol algebra (i.e. a Lie triple system with an additional bilinear skew-symmetric operation). There exist the analogous considerations for Moufang loops. We will consider the differential equations of smooth loops, generalizing smooth left Bol loops, with the identities that are the characteristic identities for the algebraic description of some relativistic space-time models. Further examinations of the integrability conditions for the differential equations allow us to introduce the proper infinitesimal object for some subclass of loops under consideration. The geometry of corresponding homogeneous spaces can be described in terms of tensors of curvature and torsion. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2018-04-19T08:25:22Z DOI: 10.1142/S0218216518410043

Authors:Piotr Stachura Abstract: Journal of Knot Theory and Its Ramifications, Ahead of Print. The algebraic part of an approach to groupoids started by Zakrzewski is presented together with some new results on formal properties of morphisms. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2018-04-19T08:25:22Z DOI: 10.1142/S0218216518410109

Authors:Louis H. Kauffman Abstract: Journal of Knot Theory and Its Ramifications, Ahead of Print. This paper shows how, in principle, simplicial methods, including the well-known Dold–Kan construction can be applied to convert link homology theories into homotopy theories. The paper studies particularly the case of Khovanov homology and shows how simplicial structures are implicit in the construction of the Khovanov complex from a link diagram and how the homology of the Khovanov category, with coefficients in an appropriate Frobenius algebra, is related to Khovanov homology. This Khovanov category leads to simplicial groups satisfying the Kan condition that are relevant to a homotopy theory for Khovanov homology. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2018-04-19T08:25:21Z DOI: 10.1142/S021821651841002X

Authors:Trang Ha, Valentina Harizanov Abstract: Journal of Knot Theory and Its Ramifications, Ahead of Print. We investigate algebraic and computability-theoretic properties of orderable magmas. A magma is an algebraic structure with a single binary operation. A right order on a magma is a linear ordering of its domain, which is right-invariant with respect to the magma operation. We use tools of computability theory to investigate Turing complexity of orders on computable orderable magmas. A magma is computable if it is finite, or if its domain can be identified with the set of natural numbers and the magma operation is computable. Interesting orderable magmas that are not even associative come from knot theory. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2018-04-19T08:25:20Z DOI: 10.1142/S0218216518410018

Authors:Atsushi Takemura Abstract: Journal of Knot Theory and Its Ramifications, Ahead of Print. We show that for any positive integers [math] and [math], the Alexander polynomial of the [math]-Turk’s head link is divisible by that of the [math]-Turk’s head link. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2018-04-10T07:47:22Z DOI: 10.1142/S0218216518500402

Authors:Aaron Heap, Douglas Knowles Abstract: Journal of Knot Theory and Its Ramifications, Ahead of Print. In this paper, we introduce the concept of a space-efficient knot mosaic. That is, we seek to determine how to create knot mosaics using the least number of non-blank tiles necessary to depict the knot. This least number is called the tile number of the knot. We determine strict bounds for the tile number of a knot in terms of the mosaic number of the knot. We also determine the tile number of several knots and provide space-efficient knot mosaics for each of them. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2018-04-10T07:47:22Z DOI: 10.1142/S0218216518500414

Authors:Efstratia Kalfagianni, Christine Ruey Shan Lee Abstract: Journal of Knot Theory and Its Ramifications, Ahead of Print. We describe a normal surface algorithm that decides whether a knot, with known degree of the colored Jones polynomial, satisfies the Strong Slope Conjecture. We also discuss possible simplifications of our algorithm and state related open questions. We establish a relation between the Jones period of a knot and the number of sheets of the surfaces that satisfy the Strong Slope Conjecture (Jones surfaces). We also present numerical and experimental evidence supporting a stronger such relation which we state as an open question. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2018-03-28T07:39:10Z DOI: 10.1142/S0218216518500396

Authors:Thomas Fleming, Joel Foisy Abstract: Journal of Knot Theory and Its Ramifications, Ahead of Print. We consider intrinsic linking and knotting in the context of directed graphs. We construct an example of a directed graph that contains a consistently oriented knotted cycle in every embedding. We also construct examples of intrinsically 3-linked and 4-linked directed graphs. We introduce two operations, consistent edge contraction and H-cyclic subcontraction, as special cases of minors for digraphs, and show that the property of having a linkless embedding is closed under these operations. We analyze the relationship between the number of distinct knots and links in an undirected graph [math] and its corresponding symmetric digraph [math]. Finally, we note that the maximum number of edges for a graph that is not intrinsically linked is [math] in the undirected case, but [math] for directed graphs. Citation: Journal of Knot Theory and Its Ramifications PubDate: 2018-03-19T08:23:19Z DOI: 10.1142/S0218216518500372