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Diquandles and invariants for oriented dichromatic links J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210911
Sang Youl Lee, Mohd Ibrahim SheikhIn this paper, we introduce an algebraic structure called a diquandle which is a set equipped with two quandle operations satisfying the right distributive laws. We discuss various examples and some properties of diquandles and also show that a diquandle enables us to distinguish oriented dichromatic links by telling that their coloring sets are different when their arcs are colored by elements of

On arrow polynomials of checkerboard colorable virtual links J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210910
Qingying Deng, Xian’an Jin, Louis H. KauffmanIn this paper, we give two new criteria of detecting the checkerboard colorability of virtual links by using the odd writhe and the arrow polynomial of virtual links, respectively. As a result, we prove that 6 virtual knots are not checkerboard colorable, leaving only one virtual knot whose checkerboard colorability is unknown among all virtual knots up to four classical crossings.

Linking numbers, quandles and groups J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210910
Lorenzo TraldiWe introduce a quandle invariant of classical and virtual links, denoted by Qtc(L). This quandle has the property that Qtc(L)≅Qtc(L′) if and only if the components of L and L′ can be indexed in such a way that L=K1∪⋯∪Kμ, L′=K1′∪⋯∪Kμ′ and for each index i, there is a multiplier 𝜖i∈{−1,1} that connects virtual linking numbers over Ki in L to virtual linking numbers over Ki′ in L′: ℓj/i(Ki,Kj)=𝜖iℓj/i(Ki′

Tied links in various topological settings J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210908
Ioannis DiamantisTied links in S3 were introduced by Aicardi and Juyumaya as standard links in S3 equipped with some nonembedded arcs, called ties, joining some components of the link. Tied links in the Solid Torus were then naturally generalized by Flores. In this paper, we study this new class of links in other topological settings. More precisely, we study tied links in the lens spaces L(p,1), in handlebodies of

Arc presentations of Montesinos links J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210903
Hwa Jeong LeeLet L be a Montesinos link M(−p,q,r) with positive rational numbers p,q and r, each less than 1, and c(L) the minimal crossing number of L. Herein, we construct arc presentations of L with c(L), c(L)−1 and c(L)−2 arcs under some conditions for p, q and r. Furthermore, we determine the arc index of infinitely many Montesinos links.

Vertex distortion of lattice knots J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210830
Marion Campisi, Nicholas CazetThe vertex distortion of a lattice knot is the supremum of the ratio of the distance between a pair of vertices along the knot and their distance in the ℓ1norm. Inspired by Gromov, Pardon and Blair–Campisi–Taylor–Tomova, we show that results about the distortion of smooth knots hold for vertex distortion: the vertex distortion of a lattice knot is 1 only if it is the unknot, and there are minimal

A combinatorial description of the knot concordance invariant epsilon J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210708
Subhankar Dey, Hakan DoğaIn this paper, we give a combinatorial description of the concordance invariant 𝜀 defined by Hom, prove some properties of this invariant using grid homology techniques. We compute the value of 𝜀 for (p,q) torus knots and prove that 𝜀(𝔾+)=1 if 𝔾+ is a grid diagram for a positive braid. Furthermore, we show how 𝜀 behaves under (p,q)cabling of negative torus knots.

Multiflypes of rectangular diagrams of links J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210710
Ivan Dynnikov, Vera SokolovaWe 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.

Nonsemisimple TQFT of the sphere with four punctures J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210703
Jules MartelIn this work, we compute the representation of the mapping class group of the sphere with 4 punctures arising from the nonsemisimple TQFT [C. Blanchet, F. Costantino, N. Geer and B. Patureau, Nonsemisimple TQFTs, Reidemeister torsion and Kashaev’s invariants, Adv. Math. 301 (2016) 1–78]. We show that it is faithful. Lastly, we compare quantum representations of punctured spheres in general with

A relation between the crossing number and the height of a knotoid J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210726
Philipp Korablev, Vladimir TarkaevKnotoids 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

Virtual and arrow Temperley–Lieb algebras, Markov traces, and virtual link invariants J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210717
Luis Paris, Loïc RabendaLet Rf=ℤ[A±1] be the algebra of Laurent polynomials in the variable A and let Ra=ℤ[A±1,z1,z2,…] be the algebra of Laurent polynomials in the variable A and standard polynomials in the variables z1,z2,…. For n≥1 we denote by VBn the virtual braid group on n strands. We define two towers of algebras {VTLn(Rf)}n=1∞ and {ATLn(Ra)}n=1∞ in terms of diagrams. For each n≥1 we determine presentations for both

A freegroup valued invariant of free knots J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210710
Vassily Olegovich ManturovThe aim of the present paper is to construct series of invariants of free knots (flat virtual knots, virtual knots) valued in free groups (and also free products of cyclic groups).

A complete invariant for closed surfaces in the threesphere J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210818
Giovanni Bellettini, Maurizio Paolini, YiSheng WangAssociated to an embedded surface in the threesphere, we construct a diagram of fundamental groups, and prove that it is a complete invariant, whereform we deduce complete invariants of handlebody links, tunnels of handlebody links, and spatial graphs. The main ingredients in the proof of the completeness include a generalization of the Kneser conjecture for threemanifolds with boundary proved here

Arrow diagrams on spherical curves and computations J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210825
Noboru Ito, Masashi TakamuraWe give a definition of an integervalued function ∑iαixi∗ derived from arrow diagrams for the ambient isotopy classes of oriented spherical curves. Then, we introduce certain elements of the free ℤmodule generated by the arrow diagrams with at most l arrows, called relators of Type (Ǐ) ((SIǏ), (WIǏ), (SIIǏ) or (WIIǏ), respectively), and introduce another function ∑iαix̃i∗ to obtain ∑iαixi∗.

Virtual and arrow Temperley–Lieb algebras, Markov traces, and virtual link invariants J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210717
Luis Paris, Loïc RabendaLet Rf=ℤ[A±1] be the algebra of Laurent polynomials in the variable A and let Ra=ℤ[A±1,z1,z2,…] be the algebra of Laurent polynomials in the variable A and standard polynomials in the variables z1,z2,…. For n≥1 we denote by VBn the virtual braid group on n strands. We define two towers of algebras {VTLn(Rf)}n=1∞ and {ATLn(Ra)}n=1∞ in terms of diagrams. For each n≥1 we determine presentations for both

Region crossing change, bicolored diagram and Arf invariant J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210615
Kengo KawamuraWe 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.

Virtual concordance and the generalized Alexander polynomial J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210622
Hans U. Boden, Micah ChrismanWe use the BarNatan Жcorrespondence to identify the generalized Alexander polynomial of a virtual knot with the Alexander polynomial of a two component welded link. We show that the Жmap is functorial under concordance, and also that Satoh’s Tube map (from welded links to ribbon knotted tori in S4) is functorial under concordance. In addition, we extend classical results of Chen, Milnor and Hillman

The colored Jones polynomial and Kontsevich–Zagier series for double twist knots J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210701
Jeremy Lovejoy, Robert OsburnUsing a result of Takata, we prove a formula for the colored Jones polynomial of the double twist knots K(−m,−p) and K(−m,p) where m and p are positive integers. In the (−m,−p) case, this leads to new families of qhypergeometric series generalizing the Kontsevich–Zagier series. Comparing with the cyclotomic expansion of the colored Jones polynomials of K(m,p) gives a generalization of a duality at

More 1cocycles for classical knots J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210605
Thomas FiedlerLet Mreg be the topological moduli space of long knots up to regular isotopy, and for any natural number n>1 let Mnreg be the moduli space of all ncables of framed long knots which are twisted by a string link to a knot in the solid torus V3. We upgrade the Vassiliev invariant v2 of a knot to an integer valued combinatorial 1cocycle for Mnreg by a very simple formula. This 1cocycle depends on a

On classification of genus g knots which admit a (1,1)decomposition J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210705
Mario EudaveMuñoz, Fabiola ManjarrezGutiérrez, Enrique RamírezLosadaGiven an oriented minimal genus Seifert surface F′ for a (1,1)knot K it is possible to surger F′ along annuli to obtain a simple minimal Seifert surfaceF. Such a surface can be put in a very nice position with respect to the (1,1)position of the knot K. Using this kind of surfaces we give a description of a (1,1)knot of genus g as a vertical banding of (1,1)knots of genus smaller than g. In addition

On the third Ohtsuki invariant for the Brieskorn–Hamm manifolds J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210609
Yasuyoshi TsutsumiWe calculate the Ohtsuki invariants λi(M)(i=0,1,2,3) of every Brieskorn–Hamm manifold M which is a rational homology 3sphere. We denote the order of H1(M;ℤ) by H. By the result, we show that the third Ohtsuki invariant λ3(M) of the Brieskorn–Hamm manifolds M with H=1 is negative, and the third Ohtsuki invariant λ3(M) of most Brieskorn–Hamm manifolds M with H≥2 is positive.

A diagrammatic approach for determining the braid index of alternating links J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210701
Yuanan Diao, Claus Ernst, Gabor Hetyei, Pengyu LiuThis paper concerns the braid index of an alternating link. It is well known that the braid index of any link equals the minimum number of Seifert circles among all link diagrams representing it. For an alternating link represented by a reduced alternating diagram D, it is known that s(D), the number of Seifert circles in D, equals the braid index b(D) of D if D contains no lone crossings, where a

Quotients of the Gordian and H(2)Gordian graphs J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210705
Christopher Flippen, Allison H. Moore, Essak SeddiqThe Gordian graph and H(2)Gordian graphs of knots are abstract graphs whose vertex sets represent isotopy classes of unoriented knots, and whose edge sets record whether pairs of knots are related by crossing changes or H(2)moves, respectively. We investigate quotients of these graphs under equivalence relations defined by several knot invariants including the determinant, the span of the Jones polynomial

Integral leftorderable surgeries on genus one fibered knots J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210522
Kazuhiro Ichihara, Yasuharu NakaeFollowing the classification of genus one fibered knots in lens spaces by Baker, we determine hyperbolic genus one fibered knots in lens spaces on whose all integral Dehn surgeries yield closed 3manifolds with leftorderable fundamental groups.

Thesaurus racks: Categorizing rack objects J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210505
Tobias GrøsfjeldWe define and explore rack objects internal to categories with products. In demonstration, we classify the groupracks, and use homotopy to prove both existence and exclusion theorems for pathconnected topological racks.

The braid group and its presentation J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210505
Bronislaw WajnrybIn this paper, we recall the geometric definition of the braid group by Emil Artin and we give a complete, elementary geometric/topological proof of the standard presentation of the braid group on n strings.

Crosscap number and epimorphisms of twobridge knot groups J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210518
Jim Hoste, Patrick D. Shanahan, Cornelia A. Van CottWe consider the relationship between the crosscap number γ of knots and a partial order on the set of all prime knots, which is defined as follows. For two knots K and J, we say K≥J if there exists an epimorphism f:π1(S3−K)→π1(S3−J). We prove that if K and J are 2bridge knots and K>J, then γ(K)≥3γ(J)−4. We also classify all pairs (K,J) for which the inequality is sharp. A similar result relating the

Polynomial invariants, knot homologies, and higher twist numbers of weaving knots W(3,n) J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210531
Rama Mishra, Ross StaffeldtWe investigate several conjectures in geometric topology by assembling computer data obtained by studying weaving knots, a doubly infinite family W(p,n) of examples of hyperbolic knots. In particular, we compute some important polynomial knot invariants, as well as knot homologies, for the subclass W(3,n) of this family. We use these knot invariants to conclude that all knots W(3,n) are fibered knots

A lower bound on critical points of the electric potential of a knot J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210510
Max LiptonConsider a knot K in S3 with charge uniformly distributed on it. From the standpoint of both physics and knot theory, it is natural to try to understand the critical points of the potential and their behavior. We show the number of critical points of the potential is at least 2t(K)+2, where t(K) is the tunnel number, defined as the smallest number of arcs one must add to K such that its complement

Shrinking braids and left distributive monoid J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210531
Linjun LiWe consider a natural generalization of braids which we call shrinking braids. We state the relations of shrinking braids and use them to define algebraically the monoid R. We endow a subset of R with a left distributive monoid structure and use it to extend the Dehornoy order on B∞ to an order on R. By using this order, we prove that R is isomorphic to the monoid which is generated (geometrically)

Ribbonlength and crossing number for folded ribbon knots J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210605
Elizabeth DenneWe study Kauffman’s model of folded ribbon knots: knots made of a thin strip of paper folded flat in the plane. The ribbonlength is the length to width ratio of such a folded ribbon knot. We show for any knot or link type that there exist constants c1,c2>0 such that the ribbonlength is bounded above by c1Cr(K)2, and also by c2Cr(K)3/2. We use a different method for each bound. The constant c1 is quite

Brunnian braids over the 2sphere and Artin combed form J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210412
Fedor Duzhin, Sher En Jessica LohFinding homotopy group of spheres is an old open problem in topology. Berrick et al. derive in [A. J. Berrick, F. Cohen, Y. L. Wong and J. Wu, Configurations, braids, and homotopy groups, J. Amer. Math. Soc. 19 (2006)] an exact sequence that relates Brunnian braids to homotopy groups of spheres. We give an interpretation of this exact sequence based on the combed form for braids over the sphere developed

Stick numbers of Montesinos knots J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210320
Hwa Jeong Lee, Sungjong No, Seungsang OhNegami found an upper bound on the stick number s(K) of a nontrivial knot K in terms of the minimal crossing number c(K): s(K)≤2c(K). Huh and Oh found an improved upper bound: s(K)≤32(c(K)+1). Huh, No and Oh proved that s(K)≤c(K)+2 for a 2bridge knot or link K with at least six crossings. As a sequel to this study, we present an upper bound on the stick number of Montesinos knots and links. Let K

The reduced Dijkgraaf–Witten invariant of twist knots in the Bloch group of a finite field J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210412
Hiroaki KaruoLet M be a closed oriented 3manifold and let G be a discrete group. We consider a representation ρ:π1(M)→G. For a 3cocycle α, the Dijkgraaf–Witten invariant is given by (ρ∗α)[M], where ρ∗:H3(G)→H3(M) is the map induced by ρ, and [M] denotes the fundamental class of M. Note that (ρ∗α)[M]=α(ρ∗[M]), where ρ∗:H3(M)→H3(G) is the map induced by ρ, we consider an equivalent invariant ρ∗[M]∈H3(G), and we

Palettes of Dehn colorings for spatial graphs and the classification of vertex conditions J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210320
Kanako Oshiro, Natsumi OyamaguchiIn this paper, we study Dehn colorings of spatial graph diagrams, and classify the vertex conditions, equivalently the palettes. We give some example of spatial graphs which can be distinguished by the number of Dehn colorings with selecting an appropriate palette. Furthermore, we also discuss the generalized version of palettes, which is defined for knottheoretic ternaryquasigroups and region colorings

Twisted torus knots T(mn + m + 1,mn + 1,mn + m + 2,−1) and T(n + 1,n,2n − 1,−1) are torus knots J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210331
Sangyop LeeA twisted torus knot T(p,q,r,s) is a torus knot T(p,q) with r adjacent strands twisted fully s times. In this paper, we determine the braid index of the knot T(p,q,r,s) when the parameters p,q,r satisfy 1

On alternating closed braids J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210407
María de los Angeles GuevaraHernández, Akio KawauchiWe introduce a numerical invariant called the braid alternation number that measures how far a link is from being an alternating closed braid. This invariant resembles the alternation number, which was previously introduced by the second author. However, these invariants are not equal, even for alternating links. We study the relation of this invariant with others and calculate this invariant for some

Verification of the Jones unknot conjecture up to 24 crossings J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210429
Robert E. Tuzun, Adam S. SikoraExtending upon our previous work, we verify the Jones Unknot Conjecture for all knots up to 24 crossings. We describe the method of our approach and analyze the growth of the computational complexity of its different components.

Group presentations for links in thickened surfaces J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210423
Daniel S. Silver, Susan G. WilliamsUsing a combinatorial argument, we prove the wellknown result that the Wirtinger and Dehn presentations of a link in 3space describe isomorphic groups. The result is not true for links ℓ in a thickened surface S×[0,1]. Their precise relationship, as given in [R. E. Byrd, On the geometry of virtual knots, M.S. Thesis, Boise State University (2012)], is established here by an elementary argument. When

Codimension two spinal open book embeddings of 3manifolds J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210429
Suhas Pandit, A. SelvakumarIn this paper, we show that every spinal open book decomposition of a closed oriented 3manifold M spinal open book embeds into a certain spinal open book decomposition of #kS2×̃S3, the connected sum of k copies of the twisted S3bundle over S2, where k depends on the spinal open book decomposition of M. We also discuss spinal open book embeddings of a huge class of spinal open books of closed oriented

Unknotting operations on knots and links J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210313
Ayaka ShimizuBy considering unknotting operations, we obtain ways of measuring how knotted a knot is. Unknotting phenomena can be seen not only in knot theory, but also in various settings such as DNA knots, mind knots and so on ([C. C. Adams, The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots (American Mathematical Society, Providence, RI, 2004); A. Kawauchi, K. Kishimoto and A. Shimizu

On twobridge knots and a conjecture of Hirasawa–Murasugi J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210308
Wenzhao ChenFox conjectured the Alexander polynomial of an alternating knot is trapezoidal, i.e. the absolute values of the coefficients first increase, then stabilize and finally decrease in a symmetric way. Recently, Hirasawa and Murasugi further conjectured a relation between the number of the stable coefficients in the Alexander polynomial and the signature invariant. In this paper we prove the Hirasawa–Murasugi

Welded extensions and ribbon restrictions of diagrammatical moves J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210320
Boris ColombariIn this paper, we consider local moves on classical and welded diagrams of string links, and the notion of welded extension of a classical move. Such extensions being nonunique in general, the idea is to find a topological criterion which could isolate one extension from the others. To that end, we turn to the relation between welded string links and knotted surfaces in ℝ4, and the ribbon subclass

Superbridge and bridge indices for knots J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210331
Colin Adams, Nikhil Agarwal, Rachel Allen, Tirasan Khandhawit, Alex Simons, Rebecca Winarski, Mary WoottersWe improve the upper bound on superbridge index sb[K] in terms of bridge index b[K] from sb[K]≤5b[K]−3 to sb[K]≤3b[K]−1.

Flat plumbing basket and contact structure J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210313
Tetsuya Ito, Keiji TagamiA flat plumbing basket is a Seifert surface consisting of a disk and bands contained in distinct pages of the disk open book decomposition of the 3sphere. In this paper, we examine close connections between flat plumbing baskets and the contact structure supported by the open book. As an application we give lower bounds for the flat plumbing basket numbers and determine the flat plumbing basket numbers

Quandle coloring quivers of links using dihedral quandles J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210320
Yuta TaniguchiCho and Nelson introduced the notion of a quandle coloring quiver, which is a quivervalued link invariant. This invariant is in general a stronger link invariant than the quandle coloring number. In this paper, we study quandle coloring quiver using dihedral quandle. We show that when we use a dihedral quandle of prime order, the quandle coloring quivers are equivalent to the quandle coloring numbers

Span of the Jones polynomials of certain vadequate virtual links J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210228
Minori Okamura, Keiichi SakaiIt is known that the Kauffman–Murasugi–Thislethwaite type inequality becomes an equality for any (possibly virtual) adequate link diagram. We refine this condition. As an application we obtain a criterion for virtual link diagram with exactly one virtual crossing to represent a properly virtual link.

Quandle coloring quivers of surfacelinks J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210218
Jieon Kim, Sam Nelson, Minju SeoQuandle coloring quivers are directed graphvalued invariants of oriented knots and links, defined using a choice of finite quandle X and set S⊂Hom(X,X) of endomorphisms. From a quandle coloring quiver, a polynomial knot invariant known as the indegree quiver polynomial is defined. We consider quandle coloring quiver invariants for oriented surfacelinks, represented by marked graph diagrams. We provide

Polynomial invariants of singular knots and links J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210225
Jose Ceniceros, Indu R. Churchill, Mohamed ElhamdadiWe generalize the notion of the quandle polynomial to the case of singquandles. We show that the singquandle polynomial is an invariant of finite singquandles. We also construct a singular link invariant from the singquandle polynomial and show that this new singular link invariant generalizes the singquandle counting invariant. In particular, using the new polynomial invariant, we can distinguish

Writhelike invariants of alternating links J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210218
Yuanan Diao, Van PhamIt is known that the writhe calculated from any reduced alternating link diagram of the same (alternating) link has the same value. That is, it is a link invariant if we restrict ourselves to reduced alternating link diagrams. This is due to the fact that reduced alternating link diagrams of the same link are obtainable from each other via flypes and flypes do not change writhe. In this paper, we introduce

Heegaard distance of the link complements in S3 J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210223
Xifeng JinWe show that, for any integers, g≥3 and n≥2, there exists a link in S3 such that its complement has a genus g Heegaard splitting with distance n.

Tied links in the solid torus J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210228
Marcelo FloresWe introduce the concept of tied links in the solid torus, which generalizes naturally the concept of tied links in S3 previously introduced by Aicardi and Juyumaya. We also define an invariant of these tied links by using skein relations, and we then recover this invariant by using Jones’ method over the btalgebra of type B and the Markov trace defined on this.

Multivariate Alexander quandles, II. The involutory medial quandle of a link (corrected) J. Knot Theory Ramif. (IF 0.379) Pub Date : 20201230
Lorenzo TraldiJoyce 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

On 2twistspun spherical Montesinos knots J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210118
Yeonhee Jang, Misaki Kataoka, Rika MiyakoshiWe give a classification of 2twistspun spherical Montesinos knots.

Studying complex manifolds by using groups Gnk and Γnk J. Knot Theory Ramif. (IF 0.379) Pub Date : 20201216
Vassily Olegovich Manturov, Zheyan WanIn 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

From chord parity to chord index J. Knot Theory Ramif. (IF 0.379) Pub Date : 20201216
Zhiyun Cheng, Denis A. Fedoseev, Hongzhu Gao, Vassily O. Manturov, Mengjian XuWe 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.

Representations of Gn3 and related groups J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210125
Denis A. Fedoseev, S. Kim, Vassily O. ManturovIn this paper, we study representations of Gn3like groups. The group Gn3 itself appeared in works of the third named author on nonReidemeister knot (and braid) theory. This group is closely related to dynamical systems of points and their invariants. Representations of Gn3like groups are useful both for the study of those groups themselves, and constructing invariants of knots and braids based on

Persistent homology for hypergraphs and computational tools — A survey for users J. Knot Theory Ramif. (IF 0.379) Pub Date : 20201216
Shiquan RenIn 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.

The Disk Complex and Topologically Minimal Surfaces in the 3Sphere J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210125
Marion Campisi, Luis TorresWe show that the disk complex of a closed, connected surface of genus $g>1$, properly embedded in the 3sphere is homotopy equivalent to a wedge of $(2g2)$dimensional spheres. This implies that genus $g>1$ Heegaard surfaces of the 3sphere are topologically minimal with index $2g1$.

3Manifolds with Nilpotent Embeddings in S4 J. Knot Theory Ramif. (IF 0.379) Pub Date : 20210120
J. A. HillmanWe consider embeddings of 3manifolds $M$ in $S^4$ such that the two complementary regions $X$ and $Y$ each have nilpotent fundamental group. If $\beta=\beta_1(M)$ is odd then these groups are abelian and $\beta\leq3$. In general, $\pi_1(X)$ and $\pi_1(Y)$ have 3generator presentations, and $\beta\leq6$. We determine all such nilpotent groups which are torsionfree and have Hirsch length $\leq5$.