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On twist positive braids with the 2-variable link invariant: 3-braids of width 2 J. Knot Theory Ramif. (IF 0.5) Pub Date : 2024-02-27 E. A. Elrifai, Sanaa A. Bajrim, Nouf Abdulrahman Alqahtani
The possible 2-variable link polynomial PL(v,z) for an oriented link L, which has width 2 in the variable v, is studied, where width of PL(v,z) is the minimal number of strands allowed by the index bound. It is shown that if the 2-variable link invariant PL(v,z) for an oriented link L has width k in the variable v, then it is the same as the polynomial of a closed k-braid, k=1,2. Also a complete list
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An invariant of virtual trivalent spatial graphs J. Knot Theory Ramif. (IF 0.5) Pub Date : 2024-02-27 Evan Carr, Nancy Scherich, Sherilyn Tamagawa
We create an invariant of virtual Y-oriented trivalent spatial graphs using colorings by virtual Niebrzydowski algebras. This paper generalizes the color invariants using virtual tribrackets and Niebrzydowski algebras by Nelson–Pico, and Graves-Nelson-T. We computed all tribrackets, Niebrzydowski algebras and virtual Niebrzydowski algebras of orders 3 and 4, and provide generative code for all data
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Representations of flat virtual braids which do not preserve the forbidden relations J. Knot Theory Ramif. (IF 0.5) Pub Date : 2024-02-27 Valeriy Bardakov, Bogdan Chuzhinov, Ivan Emel’yanenkov, Maxim Ivanov, Elizaveta Markhinina, Timur Nasybullov, Sergey Panov, Nina Singh, Sergey Vasyutkin, Valeriy Yakhin, Andrei Vesnin
In the paper, we construct a representation 𝜃:FVBn→Aut(F2n) of the flat virtual braid group FVBn on n strands by automorphisms of the free group F2n with 2n generators which does not preserve the forbidden relations in the flat virtual braid group. This representation gives a positive answer to the problem formulated by Bardakov in the list of unsolved problems in virtual knot theory and combinatorial
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Bounds in simple hexagonal lattice and classification of 11-stick knots J. Knot Theory Ramif. (IF 0.5) Pub Date : 2024-02-27 Yueheng Bao, Ari Benveniste, Marion Campisi, Nicholas Cazet, Ansel Goh, Jiantong Liu, Ethan Sherman
The stick number and the edge length of a knot type in the simple hexagonal lattice (sh-lattice) are the minimal numbers of sticks and edges required, respectively, to construct a knot of the given type in sh-lattice. By introducing a linear transformation between lattices, we prove that for any given knot both values in the sh-lattice are strictly less than the values in the cubic lattice. Finally
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Infinitely many arithmetic alternating links: Class number greater than one J. Knot Theory Ramif. (IF 0.5) Pub Date : 2024-01-29 M. D. Baker, A. W. Reid
We prove the existence of infinitely many alternating links in S3 whose complements are commensurable with the Bianchi orbifold ℍ3/PSL(2,O15).
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Modified symmetrized integral in G-coalgebras J. Knot Theory Ramif. (IF 0.5) Pub Date : 2024-01-24 Nathan Geer, Ngoc Phu Ha, Bertrand Patureau-Mirand
For G a commutative group, we give a purely Hopf G-coalgebra construction of G-colored 3-manifolds invariants using the notion of modified integral.
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A new condition on the Jones polynomial of a fibered positive link J. Knot Theory Ramif. (IF 0.5) Pub Date : 2024-01-06 Lizzie Buchanan
We give a new upper bound on the maximum degree of the Jones polynomial of a fibered positive link. In particular, we prove that the maximum degree of the Jones polynomial of a fibered positive knot is at most four times the minimum degree. Using this result, we can complete the classification of all knots of crossing number ≤12 as positive or not positive, by showing that the seven remaining knots
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The genus one complex quantum Chern–Simons representation of the mapping class group J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-11-09 Jørgen Ellegaard Andersen, Simone Marzioni
In this paper, we compute explicitly, following Witten’s prescription, the quantum representation of the mapping class group in genus one for complex quantum Chern–Simons theory associated to any simple and simply connected complex gauge group Gℂ. We use a generalization of the Weil-Gel’fand–Zak transform to exhibit an explicit expression for the representation.
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The H(n)-move is an unknotting operation for virtual and welded links J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-08-21 Danish Ali, Zhiqing Yang, Abid Hussain, Mohd Ibrahim Sheikh
An unknotting operation is a local move such that any knot diagram can be transformed into a diagram of the trivial knot by a finite sequence of these operations plus some Reidemeister moves. It is known that for all n≥2 the H(n)-move is an unknotting operation for classical knots and links. In this paper, we extend the classical unknotting operation H(n)-move to virtual knots and links. Virtualization
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A polynomial invariant of Kauffman type for knotoids J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-08-17 Yasuyuki Miyazawa
We construct a three-variable polynomial invariant for unoriented kd(1)-linkoid diagrams including knotoid ones as a certain weighted sum of polynomials on oriented kd(1)-linkoid diagrams associated with a given kd(1)-linkoid diagram.
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A polynomial invariant of Kauffman type for knotoids II J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-08-17 Yasuyuki Miyazawa
In this paper, we construct a polynomial invariant of Kauffman type for kd(1)-linkoids and compute the polynomials for knotoids with up to three crossings. As a consequence, it is shown that the polynomial is different from the previous one in [Y. Miyazawa, A polynomial invariant of Kauffman type for knotoids, preprint].
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On the swap move of letters in words and the forbidden moves in virtual diagrams J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-08-17 Tomonori Fukunaga
In the theory of topology of words, the move to swap two adjacent letters in words is a simple operator that makes an S-homotopy class of a nanoword into the S-homotopy class of the empty word. In this paper, we study the relation between the swap move of letters and the forbidden moves of virtual diagrams, that is, local moves of virtual diagrams which unknot any virtual knot.
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A lower bound on the average genus of a 2-bridge knot J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-08-12 Moshe Cohen
Experimental data from Dunfield et al. using random grid diagrams suggest that the genus of a knot grows linearly with respect to the crossing number. Using billiard table diagrams of Chebyshev knots developed by Koseleff and Pecker and a random model of 2-bridge knots via these diagrams developed by the author with Krishnan and then with Even-Zohar and Krishnan, we introduce a further-truncated model
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An alternate approach to Habiro and Lê’s universal invariant of homology three-spheres J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-08-03 Stephen F. Sawin
An alternative construction of the invariant of homology three spheres valued in a completion of an integral polynomial ring associated to each quantized complex simple Lie algebra by Habiro and Lê [Unified quantum invariants for integral homology spheres associated with simple Lie algebras, Geom. Topol.20 (2016) 2687–2835, doi:10.2140/gt.2016.20.2687, arXiv:1503:03549v1] is given.
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A polyhedral approach to the arithmetic and geometry of hyperbolic link complements J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-08-04 Ruth Kellerhals
We provide an elementary polyhedral approach to study and deduce results about the arithmeticity and commensurability of an infinite family of hyperbolic link complements Mn for n≥3. The manifold Mn is the complement of 𝕊3 by the (2n)-link chain 𝒟2n and has 2n cusps. We show that Mn is closely related to a hyperbolic Coxeter orbifold that is commensurable to an orbifold with a single cusp. Vinberg’s
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On torsion in linearized Legendrian contact homology J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-08-03 Roman Golovko
In this short note, we discuss certain examples of Legendrian submanifolds, whose linearized Legendrian contact (co)homology groups over integers have non-vanishing algebraic torsion. More precisely, for a given arbitrary finitely generated abelian group G and a positive integer n≥3, n≠4, we construct examples of Legendrian submanifolds of the standard contact vector space ℝ2n+1, whose n−1th linearized
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The coefficients of the Jones polynomial J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-08-01 V. A. Manathunga
It has been known that the coefficients of the series expansion of the Jones polynomial evaluated at ex are rational-valued Vassiliev invariants. In this paper, we calculate minimal multiplying factor, λ, needed for these rational-valued invariants to become integer-valued Vassiliev invariants. By doing that, we obtain a set of integer-valued Vassiliev invariants.
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The coefficients of the Jones polynomial J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-08-01 V. A. Manathunga
It has been known that the coefficients of the series expansion of the Jones polynomial evaluated at ex are rational-valued Vassiliev invariants. In this paper, we calculate minimal multiplying factor, λ, needed for these rational-valued invariants to become integer-valued Vassiliev invariants. By doing that, we obtain a set of integer-valued Vassiliev invariants.
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Torus knots and links in V1 that are invariant under the ℤm × ℤl-actions on V1 that form the quotient orbifold (A0,k) J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-08-01 Aruna Bandara
We demonstrate how the genus one handlebody orbifold (A0,k) is obtained as a quotient of orientation preserving ℤm×ℤl-action on the solid torus V1=𝕊1×D2, where m, k, and l are positive integers, m divides k and k divides l. We determine the torus knots and links in V1 that are invariant under the corresponding action. In that case, the image of T(p,q) in (A0,k) is always a torus knot or link on the
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Relationship of the Hennings and Witten–Reshetikhan–Turaev invariants for higher rank quantum groups J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-07-29 Winston Cheong, Alexander Doser, McKinley Gray, Stephen F. Sawin
The Hennings invariant for the small quantum group associated to an arbitrary simple Lie algebra at a root of unity is shown to agree with the Witten–Reshetikhin–Turaev (WRT) three-manifold invariant arising from Chern–Simons field theory for the same Lie algebra and the same root of unity on all integer homology three-spheres, at roots of unity where both are defined. This partially generalizes the
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Quantum invariants of links and 3-manifolds with boundary defined via virtual links J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-07-19 Louis H. Kauffman, Eiji Ogasa
We introduce new topological quantum invariants, surface link quantum invariants, of compact oriented 3-manifolds with boundary where the boundary is a disjoint union of two identical surfaces. The invariants are constructed by using virtual knots and links. These invariants are new, nontrivial, and calculable examples of quantum invariants of 3-manifolds with non-vacuous boundary. Our new invariants
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A new framework for the Jones polynomial of fluid knots J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-07-06 Renzo L. Ricca, Xin Liu
Here we illustrate how Jones’ polynomials are derived from the kinetic helicity of vortical flows, and how they can be used to measure the topological complexity of fluid knots by numerical values. Relying on this new findings, we show how to use our adapted Jones polynomial in a new framework by introducing a knot polynomial space whose discrete points are the adapted Jones polynomials of fluid knots
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Personal recollection of Vaughan F. R. Jones (1952–2020) J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-07-05 Józef H. Przytycki
Vaughan Jones was my mentor and friend. I provide here a few personal recollections including the “knotted antennas” project which brought me to Berkeley.
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Regular isotopy classes of link diagrams from Thompson’s groups J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-07-05 Rushil Raghavan, Dennis Sweeney
In 2014, Vaughan Jones developed a method to produce links from elements of Thompson’s group F, and showed that all links arise this way. He also introduced a subgroup F→ of F and a method to produce oriented links from elements of this subgroup. In 2018, Valeriano Aiello showed that all oriented links arise from this construction. We classify exactly those regular isotopy classes of links that arise
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An improvement of the lower bound for the minimum number of link colorings by quandles J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-07-05 H. Abchir, S. Lamsifer
We improve the lower bound for the minimum number of colors for linear Alexander quandle colorings of a knot given in Theorem 1.2 of [L. H. Kauffman and P. Lopes, Colorings beyond Fox: The other linear Alexander quandles, Linear Algebra Appl. 548 (2018) 221–258]. We express this lower bound in terms of the degree k of the reduced Alexander polynomial of the knot. We show that it is exactly k+1 for
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Jones rational coincidences J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-07-04 Ruth Lawrence, Ori Rosenstein
We investigate coincidences of the (one-variable) Jones polynomial amongst rational knots, what we call “Jones rational coincidences”. We provide moves on the continued fraction expansion of the associated rational which we prove do not change the Jones polynomial and conjecture (based on experimental evidence from all rational knots with determinant <900) that these moves are sufficient to generate
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Algebra, topology and the discoveries of Vaughan Jones J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-06-30 Joan S. Birman
In this paper, the discovery of the Jones polynomial will be discussed, emphasizing the way in which it illustrated the remarkable unity between distinct parts of mathematics, each with its own language, but initially without a dictionary.
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Framed Thompson groups J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-06-30 Aristides Kontogeorgis, Sofia Lambropoulou
We introduce the notion of the framed Thompson group, which can be seen as a categorification of the ordinary Thompson group, and we show how framed links can be obtained from elements of the framed Thompson group.
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HOMFLY-PT polynomials of open links J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-06-30 Kenneth C. Millett, Eleni Panagiotou
We numerically estimate the superposition of the HOMFLY-PT polynomial of an open two-component link, define its spread, and describe how this quantity may be employed to quantify the degree of entanglement of confined two component open links.
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Set-theoretic Yang–Baxter (co)homology theory of involutive non-degenerate solutions J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-06-30 Józef H. Przytycki, Petr Vojtěchovský, Seung Yeop Yang
W. Rump showed that there exists a one-to-one correspondence between involutive right non-degenerate solutions of the Yang–Baxter equation and cycle sets. J. S. Carter, M. Elhamdadi, and M. Saito, meanwhile, introduced a homology theory of set-theoretic solutions of the Yang–Baxter equation in order to define cocycle invariants of classical knots. In this paper, we introduce the normalized homology
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An introduction to Thompson knot theory and to Jones subgroups J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-06-29 Valeriano Aiello
We review a constructions of knots from elements of the Thompson groups due to Vaughan Jones, which comes in two flavors: oriented and unoriented.
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Virtual multicrossings and petal diagrams for virtual knots and links J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-06-27 Colin Adams, Chaim Even-Zohar, Jonah Greenberg, Reuben Kaufman, David Lee, Darin Li, Dustin Ping, Theodore Sandstrom, Xiwen Wang
Multicrossings, which have previously been defined for classical knots and links, are extended to virtual knots and links. In particular, petal diagrams are shown to exist for all virtual knots.
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Knot invariants for rail knotoids J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-06-27 Dimitrios Kodokostas, Sofia Lambropoulou
To each rail knotoid we associate two unoriented knots along with their oriented counterparts, thus deriving invariants for rail knotoids based on these associations. We then translate them to invariants of rail isotopy for rail arcs.
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Tri-plane diagrams for simple surfaces in S4 J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-06-27 Wolfgang Allred, Manuel Aragón, Zack Dooley, Alexander Goldman, Yucong Lei, Isaiah Martinez, Nicholas Meyer, Devon Peters, Scott Warrander, Ana Wright, Alexander Zupan
Meier and Zupan proved that an orientable surface 𝒦 in S4 admits a tri-plane diagram with zero crossings if and only if 𝒦 is unknotted, so that the crossing number of 𝒦 is zero. We determine the minimal crossing numbers of nonorientable unknotted surfaces in S4, proving that c(𝒫n,m)=max{1,|n−m|}, where 𝒫n,m denotes the connected sum of n unknotted projective planes with normal Euler number +2
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B-type Catalan states of lattice crossing J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-06-26 Mieczyslaw K. Dabkowski, Mihaja Rakotomalala
We show which crossingless connections between 2(m+n) outer boundary points of an annulus can be realized as Kauffman states of the B-type lattice crossing LB(m,n). Furthermore, we give a closed-form formula for the number of realizable B-type Catalan states, and we find coefficients of those obtained as Kauffman states of LB(m,1) and LB(m,2).
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Combinatorial knot theory and the Jones polynomial J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-06-26 Louis H. Kauffman
This paper is an introduction to combinatorial knot theory via state summation models for the Jones polynomial and its generalizations. It is also a story about the developments that ensued in relation to the discovery of the Jones polynomial and a remembrance of Vaughan Jones and his mathematics.
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Tangles in affine Hecke algebras J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-06-26 Hugh Morton
Framed oriented n-tangle diagrams in the annulus, subject to the Homfly skein relations, are used to produce an algebra isomorphic to the affine Hecke algebras Ḣn of type A. The use of closed curves and braids gives neat pictures for central elements in the algebras.
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Over then under tangles J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-06-23 Dror Bar-Natan, Zsuzsanna Dancso, Roland van der Veen
Over-then-Under (OU) tangles are oriented tangles whose strands travel through all of their over crossings before any under crossings. In this paper, we discuss the idea of gliding: an algorithm by which tangle diagrams could be brought to OU form. By analyzing cases in which the algorithm converges, we obtain a braid classification result, which we also extend to virtual braids, and provide a Mathematica
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Link-homotopy classes of 4-component links, claspers and the Habegger–Lin algorithm J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-06-26 Yuka Kotorii, Atsuhiko Mizusawa
Two links are link-homotopic if they are transformed into each other by a sequence of self-crossing changes and ambient isotopies. The link-homotopy classes of links with up to three components were classified by the Milnor homotopy invariants. Levine investigated the indeterminacy of Milnor homotopy invariants and gave a classification of the link-homotopy classes of 4-component links. In this paper
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A note on Rivin’s interpretation of McShane’s identity as identities for closed geodesics with one self-intersection J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-06-26 Da Lei, Ying Zhang
Rivin interpreted McShane’s identity as an identity for closed geodesics with one self-intersection on a one-cusped hyperbolic torus (cf. [I. Rivin, Geodesics with one self-intersection, and other stories, Adv. Math. 231 (2012) 2391–2412, Theorem 3.2]). In this note we point out that only those geodesics of non-hyperelliptic type are included in the interpreted identity, while those of hyperelliptic
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Generalizing the relation between the Kauffman bracket and Jones polynomial J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-06-21 Uwe Kaiser
We generalize Kauffman’s famous formula defining the Jones polynomial of an oriented link in 3-space from his bracket and the writhe of an oriented diagram [L. Kauffman, State models and the Jones polynomial, Topology26(3) (1987) 395–407]. Our generalization is an epimorphism between skein modules of tangles in compact connected oriented 3-manifolds with markings in the boundary. Besides the usual
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The colored Jones polynomial of a cable of the figure-eight knot J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-06-15 Hitoshi Murakami, Anh T. Tran
We study the asymptotic behavior of the N-dimensional colored Jones polynomial of a cable of the figure-eight knot, evaluated at exp(ξ/N) for a real number ξ. We show that if ξ is sufficiently large, the colored Jones polynomial grows exponentially when N goes to the infinity. Moreover the growth rate is related to the Chern–Simons invariant of the knot exterior associated with an SL(2;ℝ) representation
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Biquandle bracket quivers J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-06-15 Pia Cosma Falkenburg, Sam Nelson
Biquandle brackets define invariants of classical and virtual knots and links using skein invariants of biquandle-colored knots and links. Biquandle coloring quivers categorify the biquandle counting invariant in the sense of defining quiver-valued enhancements which decategorify to the counting invariant. In this paper, we unite the two ideas to define biquandle bracket quivers, providing new categorifications
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On invariants of surfaces in the 3-sphere J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-06-16 Hiroaki Kurihara
In this paper we study isotopy classes of closed connected orientable surfaces in the standard 3-sphere. Such a surface splits the 3-sphere into two compact connected submanifolds, and by using their Heegaard splittings, we obtain a 2-component handlebody-link. In this paper, we first show that the equivalence class of such a 2-component handlebody-link up to attaching trivial 1-handles can recover
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Petal number of torus knots of type (r,r + 2) J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-06-12 Hwa Jeong Lee, Gyo Taek Jin
Let r be an odd integer, r≥3. Then the petal number of the torus knot of type (r,r+2) is equal to 2r+3.
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Remembering Vaughan Jones J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-06-12 Ruth Lawrence
Personal reminiscences of meetings with Vaughan Jones.
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Linking number of monotonic cycles in random book embeddings of complete graphs J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-06-13 Yasmin Aguillon, Eric Burkholder, Xingyu Cheng, Spencer Eddins, Emma Harrell, Kenji Kozai, Elijah Leake, Pedro Morales
A book embedding of a complete graph is a spatial embedding whose planar projection has the vertices located along a circle, consecutive vertices are connected by arcs of the circle, and the projections of the remaining “interior” edges in the graph are straight line segments between the points on the circle representing the appropriate vertices. A random embedding of a complete graph can be generated
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Linking in tree-manifolds J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-06-13 Xueqi Wang
Let T be a tree with vertices 2m-dimensional oriented vector bundles over S2m and M(T) be the boundary of the oriented manifold obtained by plumbing according to T. In this paper, we calculate linking numbers between the fiber spheres, provided M(T) is a rational homology sphere, and thereby provides a correction and completion of a work of tom Dieck.
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Distinguishing some genus one knots using finite quotients J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-05-31 Tamunonye Cheetham-West
We give a criterion for distinguishing a prime knot K in S3 from every other knot in S3 using the finite quotients of π1(S3∖K). Using recent work of Baldwin–Sivek, we apply this criterion to the hyperbolic knots 52, 15n43522, and the three-strand pretzel knots P(−3,3,2n+1) for every integer n.
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Fundamental heap for framed links and ribbon cocycle invariants J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-05-30 Masahico Saito, Emanuele Zappala
A heap is a set with a certain ternary operation that is self-distributive and exemplified by a group with the operation (x,y,z)↦xy−1z. We introduce and investigate framed link invariants using heaps. In analogy with the knot group, we define the fundamental heap of framed links using group presentations. The fundamental heap is determined for some classes of links such as certain families of torus
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Classification of doubly periodic untwisted (p,q)-weaves by their crossing number and matrices J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-05-29 Mizuki Fukuda, Motoko Kotani, Sonia Mahmoudi
A weave is the lift to the thickened Euclidean plane of a particular type of quadrivalent planar connected graph with an over or under crossing information to each vertex and such that the lifted components are non-intersecting simple open curves. In this paper, we introduce a formal topological definition of weaves as three-dimensional entangled structures and characterize the equivalence classes
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Bilinear enhancements of quandle invariants J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-05-25 Will Gilroy, Sam Nelson
We enhance the quandle counting invariants of oriented classical and virtual links using a construction similar to quandle modules but inspired by symplectic quandle operations rather than Alexander quandle operations. Given a finite quandle X and a vector space V over a field, sets of bilinear forms on V indexed by pairs of elements of X satisfying certain conditions yield new enhanced multiset- and
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On the nonorientable 4-genus of double twist knots J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-05-23 Jim Hoste, Patrick D. Shanahan, Cornelia A. Van Cott
We investigate the nonorientable 4-genus γ4 of a special family of 2-bridge knots, the double twist knots C(m,n). Because the nonorientable 4-genus is bounded by the nonorientable 3-genus, it is known that γ4(C(m,n))≤3. By using explicit constructions to obtain upper bounds on γ4 and known obstructions derived from Donaldson’s diagonalization theorem to obtain lower bounds on γ4, we produce infinite
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On the triple point number of surface-links in Yoshikawa’s table J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-05-09 Nicholas Cazet
Yoshikawa made a table of knotted surfaces in ℝ4 with ch-index 10 or less. This remarkable table is the first to enumerate knotted surfaces analogous to the classical prime knot table. A broken sheet diagram of a surface-link is a generic projection of the surface in ℝ3 with crossing information along its singular set. The minimal number of triple points among all broken sheet diagrams representing
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On the generalized virtual Goeritz matrix for virtual knots J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-05-09 Kyeonghui Lee, Sera Kim
Im et al. [Signature, nullity and determinant of checkerboard colorable virtual links, J. Knot Theory Ramifications 19(8) (2010) 1093–1114] introduced how to define Goeritz matrices for checkerboard colorable virtual links. In this paper, we extend this for the Goeritz matrices of virtual knots. And we consider its signature and determinant and show they are invariants for virtual knots.
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Intersection formulas for parities on virtual knots J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-04-28 Igor Nikonov
We prove that parities on virtual knots come from invariant 1-cycles on the arcs of knot diagrams. In turn, the invariant cycles are determined by quasi-indices on the crossings of the diagrams. The found connection between the parities and the (quasi)-indices allows one to define a new series of parities on virtual knots.
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The Gordian complexes of knots given by 4-move J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-04-27 Danish Ali, Zhiqing Yang, Abid Hussain, Muqadar Ali
The Gordian complex of knots is a simplicial complex whose vertices consist of all knot types in 𝕊3. Local moves play an important role in defining knot invariants. There are many local moves known as unknotting operations for knots. In this paper, we discuss the 4-move operation. We show that for any knot K0 and for any given natural number n, there exists a family of knots {K0,K1,…,Kn} such that
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Arithmetic Chern–Simons theory with real places J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-04-19 Jungin Lee, Jeehoon Park
The goal of this paper is two-fold: we generalize the arithmetic Chern–Simons theory over totally imaginary number fields studied in [H.-J. Chung, D. Kim, M. Kim, J. Park and H. Yoo, Arithmetic Chern–Simons theory II, preprint (2016), arXiv:1609.03012v3; M. Kim, Arithmetic Chern–Simons theory I, preprint (2015) arXiv:1510.05818v4] to arbitrary number fields (with real places) and provide new examples
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Bi-Legendrian rack colorings of Legendrian knots J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-04-19 Naoki Kimura
We introduce a bi-Legendrian rack and show that a bi-Legendrian rack coloring number is an invariant of Legendrian knots. We prove that bi-Legendrian rack coloring numbers can distinguish all Legendrian unknots with the same Thurston–Bennequin number. We also consider pairs of Legendrian knots which cannot be distinguished by bi-Legendrian rack coloring numbers.
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The stick number of rail arcs J. Knot Theory Ramif. (IF 0.5) Pub Date : 2023-04-19 Nicholas Cazet
Consider two parallel lines ℓ1 and ℓ2 in ℝ3. A rail arc is an embedding of an arc in ℝ3 such that one endpoint is on ℓ1, the other is on ℓ2, and its interior is disjoint from ℓ1∪ℓ2. Rail arcs are considered up to rail isotopies, ambient isotopies of ℝ3 with each self-homeomorphism mapping ℓ1 and ℓ2 onto themselves. When the manifolds and maps are taken in the piecewise linear category, these rail arcs