238 results
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2. Families of curly knots.
- Author
-
Ernst, C. and Gover, S.
- Subjects
CURVATURE ,MATHEMATICS - Abstract
A spiral knot or link diagram (introduced in [C. Adams, R. Hudson, R. Morrison, W. George, L. Starkston, S. Taylor and O. Turanova, The spiral index of knots, Math. Proc. Cambridge Philos. Soc. 149(2) (2010) 297–315]) is an oriented knot or link diagram where, when traversing through the planar diagram, the curvature does not change sign. An oriented knot or link type is called curly if it admits a spiral diagram with fewer maxima than the braid index of that knot or link type. In this paper, we exhibit families of curly knots and links. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
3. The embedded homology of hypergraph pairs.
- Author
-
Ren, Shiquan, Wu, Jie, and Zhang, Mengmeng
- Subjects
HYPERGRAPHS ,MATHEMATICS - Abstract
In this paper, we generalize the embedded homology groups of hypergraphs initially given in [S. Bressan, J. Li, S. Ren and J. Wu, The embedded homology of hypergraphs and applications, Asian J. Math. 23(3) (2019) 479–500] and study the relative embedded homology groups of hypergraph pairs. We prove some long exact sequences as well as a Mayer–Vietoris sequence for the relative embedded homology groups of hypergraph pairs. Moreover, we briefly discuss the two-dimensional persistence for the relative embedded homology groups of hypergraph pairs. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
4. Arc crossing change is an unknotting operation.
- Author
-
Cericola, Christopher
- Subjects
- *
MATHEMATICS - Abstract
This paper defines a new operation through extending the idea of the 0-dimensional crossing change and Shimizu's 2-dimensional region crossing change [A. Shimizu, Region crossing change is an unknotting operation, J. Math. Soc. Jpn. 66(3) (2014) 693–708, doi:10.2969/jmsj/06630693] to a 1-dimensional version called the arc crossing change. We will also prove that the arc crossing change is an unknotting operation with the help of Gauss diagrams. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
5. Extending quasi-alternating links.
- Author
-
Chbili, Nafaa and Kaur, Kirandeep
- Subjects
POLYNOMIALS ,TOPOLOGY ,MATHEMATICS ,KNOT theory ,LOGICAL prediction ,CONSTRUCTION - Abstract
Champanerkar and Kofman [Twisting quasi-alternating links, Proc. Amer. Math. Soc.137(7) (2009) 2451–2458] introduced an interesting way to construct new examples of quasi-alternating links from existing ones. Actually, they proved that replacing a quasi-alternating crossing c in a quasi-alternating link by a rational tangle of same type yields a new quasi-alternating link. This construction has been extended to alternating algebraic tangles and applied to characterize all quasi-alternating Montesinos links. In this paper, we extend this technique to any alternating tangle of same type as c. As an application, we give new examples of quasi-alternating knots of 13 and 14 crossings. Moreover, we prove that the Jones polynomial of a quasi-alternating link that is obtained in this way has no gap if the original link has no gap in its Jones polynomial. This supports a conjecture introduced in [N. Chbili and K. Qazaqzeh, On the Jones polynomial of quasi-alternating links, Topology Appl.264 (2019) 1–11], which states that the Jones polynomial of any prime quasi-alternating link except (2 , p) -torus links has no gap. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
6. Behind maths: Federating research(ers).
- Subjects
MATHEMATICS ,COMMUNITIES ,MEMORY ,LEADERSHIP - Abstract
Patrick Dehornoy's mathematical legacy is impressive and it has been celebrated in other papers of this volume. However, I think that there is another important aspect of Patrick's work that should be stressed, which is his involvement in scientific and managerial leadership in Mathematics. This text wants therefore to be a very personal memory of Patrick Dehornoy and of the impact that he had on my perspective on the relevance and importance of commitments to our community. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
7. Ordering braids: In memory of Patrick Dehornoy.
- Subjects
SET theory ,ALGEBRA ,COMPUTER science ,MATHEMATICS ,TOPOLOGY ,BRAID group (Knot theory) - Abstract
With the untimely passing of Patrick Dehornoy in September 2019, the world of mathematics lost a brilliant scholar who made profound contributions to set theory, algebra, topology, and even computer science and cryptography. And I lost a dear friend and a strong influence in the direction of my own research in mathematics. In this paper, I will concentrate on his remarkable discovery that the braid groups are left-orderable, and its consequences, and its strong influence on my own research. I'll begin by describing how I learned of his work. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
8. Goussarov–Polyak–Viro conjecture for degree three case.
- Author
-
Ito, Noboru, Kotorii, Yuka, and Takamura, Masashi
- Subjects
LOGICAL prediction ,KNOT theory ,FINITE, The ,MATHEMATICS - Abstract
Although it is known that the dimension of the Vassiliev invariants of degree three of long virtual knots is seven, the complete list of seven distinct Gauss diagram formulas has been unknown explicitly, where only one known formula was revised without proof. In this paper, we give seven Gauss diagram formulas to present the seven invariants of the degree three (Proposition 4). We further give 2 3 Gauss diagram formulas of classical knots (Proposition 5). In particular, the Polyak–Viro Gauss diagram formula [M. Polyak and O. Viro, Gauss diagram formulas for Vassiliev invariants, Int. Math. Res. Not.1994 (1994) 445–453] is not a long virtual knot invariant; however, it is included in the list of 2 3 formulas. It has been unknown whether this formula would be available by arrow diagram calculus automatically. In consequence, as it relates to the conjecture of Goussarov-Polyak-Viro [Finite-type invariants of classical and virtual knots, Topology39 (2000) 1045–1068, Conjecture 3.C], for all the degree three finite type long virtual knot invariants, each Gauss diagram formula is represented as those of Vassiliev invariants of classical knots (Theorem 1). [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
9. Covering diagrams over surface-knot diagrams.
- Author
-
Yashiro, Tsukasa
- Subjects
KNOT theory ,GEOMETRIC surfaces ,CHARTS, diagrams, etc. ,GEOMETRIC topology ,MATHEMATICS - Abstract
A surface-knot is a closed oriented surface smoothly embedded in 4-space and a surface-knot diagram is a projected image of a surface-knot under the orthogonal projection in 3-space with crossing information. Every surface-knot diagram induces a rectangular-cell complex. In this paper, we introduce a covering diagram over a surface-knot diagram. the covering map induces a covering of the rectangular-cell complexes. As an application, a lower bound of triple point numbers for a family of surface-knots is obtained. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
10. Stick number of tangles.
- Author
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Huh, Youngsik, Lee, Jung Hoon, and Taniyama, Kouki
- Subjects
TANGLES (Knot theory) ,MATHEMATICS ,POLYMERS ,TOPOLOGY ,HOMEOMORPHISMS - Abstract
An -string tangle is a pair such that is a disjoint union of properly embedded arcs in a topological -ball . And an -string tangle is said to be trivial (or rational), if it is homeomorphic to as a pair, where is a 2-disk, is the unit interval and each is a point in the interior of . A stick tangle is a tangle each of whose arcs consists of finitely many line segments, called sticks. For an -string stick tangle its stick-order is defined to be a nonincreasing sequence of natural numbers such that, under an ordering of the arcs of the tangle, each denotes the number of sticks constituting the th arc of the tangle. And a stick-order is said to be trivial, if every stick tangle of the order is trivial. In this paper, restricting the -ball to be the standard 3-ball, we give the complete list of trivial stick-orders. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
11. The special rank of virtual knot groups.
- Author
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Mira-Albanés, Jhon Jader, Rodríguez-Nieto, José Gregorio, and Salazar-Díaz, Olga Patricia
- Subjects
GROUPOIDS ,NUMBER theory ,ORBIFOLDS ,KNOT theory ,MATHEMATICS - Abstract
In this paper we introduce the special rank for virtual knots and some properties of this number are studied. Although we do not know if it can be considered as a nontrivial extension of the meridional rank given by [H. U. Boden and A. I. Gaudreau, Bridge number for virtual and welded knots, J. Knot Theory Ramifications24 (2015), Article ID: 1550008] and by [M. Boileau and B. Zimmermann, The π -orbifold group of a link, Math. Z.200 (1989) 187–208], we prove that classical knots with special rank 2 are 2 -bridge knots. Therefore, a modified version of the so called Cappell and Shaneson conjecture could be considered. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
12. JONES POLYNOMIALS OF LONG VIRTUAL KNOTS.
- Author
-
ITO, NOBORU
- Subjects
POLYNOMIALS ,KNOT theory ,HOMOLOGY theory ,MATHEMATICAL mappings ,CHARTS, diagrams, etc. ,MATHEMATICS - Abstract
This paper defines versions of the Jones polynomial and Khovanov homology by using several maps from the set of Gauss diagrams to its variant. Through calculation of some examples, this paper also shows that these versions behave differently from the original ones. [ABSTRACT FROM AUTHOR]
- Published
- 2013
- Full Text
- View/download PDF
13. HOMOTOPY CLASSIFICATION OF NANOPHRASES IN TURAEV'S THEORY OF WORDS.
- Author
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FUKUNAGA, TOMONORI
- Subjects
HOMOTOPY groups ,CLASSIFICATION ,INVARIANTS (Mathematics) ,GROUP theory ,MATHEMATICS - Abstract
The purpose of this paper is to give the homotopy classification of nanophrases of length 2 with 4 letters. To do it we construct some new invariants of nanophrases γ and T. The invariant γ defined in this paper is an extension of the invariant γ for nanowords introduced by Turaev. The invariant T is a new invariant of nanophrases. As a corollary of these results, we give the classification of two-component pointed, ordered, oriented curves on surfaces with minimum crossing number less than or equal to 2. [ABSTRACT FROM AUTHOR]
- Published
- 2009
- Full Text
- View/download PDF
14. QUATERNION ALGEBRAS AND INVARIANTS OF VIRTUAL KNOTS AND LINKS I:: THE ELLIPTIC CASE.
- Author
-
FENN, ROGER
- Subjects
UNIVERSAL algebra ,POLYNOMIALS ,MATRICES (Mathematics) ,KNOT theory ,MATHEMATICS - Abstract
In this paper, we show how generalized quaternions including some 2 × 2 matrices, can be used to find solutions of the equation \[ [B,(A - 1)(A,B)] = 0. \] These solutions can then be used to find polynomial invariants of virtual knots and links. The remaining 2 × 2 matrices will be considered in a later paper. [ABSTRACT FROM AUTHOR]
- Published
- 2008
- Full Text
- View/download PDF
15. QUATERNION ALGEBRAS AND INVARIANTS OF VIRTUAL KNOTS AND LINKS II:: THE HYPERBOLIC CASE.
- Author
-
BUDDEN, STEPHEN and FENN, ROGER
- Subjects
ALGEBRA ,HYPERBOLIC groups ,KNOT theory ,LOW-dimensional topology ,MATHEMATICS - Abstract
Let A, B be invertible, non-commuting elements of a ring R. Suppose that A - 1 is also invertible and that the equation \[ [B,(A - 1)(A,B)] = 0 \] called the fundamental equation is satisfied. Then an invariant R-module is defined for any diagram of a (virtual) knot or link. Solutions in the classic quaternion case have been found by Bartholomew, Budden and Fenn. Solutions in the generalized quaternion case have been found by Fenn in an earlier paper. These latter solutions are only partial in the case of 2 × 2 matrices and the aim of this paper is to provide solutions to the missing cases. [ABSTRACT FROM AUTHOR]
- Published
- 2008
- Full Text
- View/download PDF
16. LUNE-FREE KNOT GRAPHS.
- Author
-
ELIAHOU, SHALOM, HARARY, FRANK, and KAUFFMAN, LOUIS H.
- Subjects
CHARTS, diagrams, etc. ,GRAPHIC methods ,ALGEBRA ,MATHEMATICS - Abstract
This paper is an exploration of simple four-regular graphs in the plane (i.e. loop-free and with no more than one edge between any two nodes). Such graphs are fundamental to the theory of knots and links in three dimensional space, and their planar diagrams. We dedicate this paper to Frank Harary (1921–2005), whose fascination with graphs of knots inspired this work, and with whom we had the pleasure of developing this paper. [ABSTRACT FROM AUTHOR]
- Published
- 2008
- Full Text
- View/download PDF
17. The minimal coloring number of any non-splittable -colorable link is four.
- Author
-
Zhang, Meiqiao, Jin, Xian'an, and Deng, Qingying
- Subjects
MATHEMATICS ,CHARTS, diagrams, etc. ,POLYNOMIALS ,REIDEMEISTER moves ,KNOT theory - Abstract
Ichihara and Matsudo introduced the notions of -colorable links and the minimal coloring number for -colorable links, which is one of invariants for links. They proved that the lower bound of minimal coloring number of a non-splittable -colorable link is 4. In this paper, we show the minimal coloring number of any non-splittable -colorable link is exactly 4. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
18. Knot fertility and lineage.
- Author
-
Cantarella, Jason, Henrich, Allison, Magness, Elsa, O'Keefe, Oliver, Perez, Kayla, Rawdon, Eric, and Zimmer, Briana
- Subjects
CHARTS, diagrams, etc. ,MATHEMATICS ,INTEGERS ,POLYNOMIALS ,KNOT theory - Abstract
In this paper, we introduce a new type of relation between knots called the descendant relation. One knot is a descendant of another knot if can be obtained from a minimal crossing diagram of by some number of crossing changes. We explore properties of the descendant relation and study how certain knots are related, paying particular attention to those knots, called fertile knots, that have a large number of descendants. Furthermore, we provide computational data related to various notions of knot fertility and propose several open questions for future exploration. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
19. The -Gordian complex of knots.
- Author
-
Zhang, Kai, Yang, Zhiqing, and Lei, Fengchun
- Subjects
COMPLEX numbers ,INTEGERS ,SET theory ,MATHEMATICS ,MATHEMATICS theorems - Abstract
In this paper, the -Gordian complex of knots is defined. The authors show that for any knot and any positive integer , there is a finite set of knots of size containing , such that the -Gordian distance between any two knots in this set is 1 for all . [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
20. THE PROBLEM OF DETECTING THE SATELLITE STRUCTURE OF A LINK BY MONOTONIC SIMPLIFICATION.
- Author
-
KAZANTSEV, ALEXANDR
- Subjects
KNOT theory ,MONOTONIC functions ,GRAPH theory ,TORUS ,EMBEDDINGS (Mathematics) ,RECTANGLES ,MATHEMATICS - Abstract
In a recent work "Arc-presentation of links: Monotonic simplification", Dynnikov shows that each rectangular diagram of the unknot, composite link, or split link can be monotonically simplified into a trivial, composite, or split diagram, respectively. The following natural question arises: Is it always possible to simplify monotonically a rectangular diagram of a satellite knot or link into one where the satellite structure is seen? Here we give a negative answer to that question both for knot and link cases. An example of a torus embedding that cannot be obtained from ordinary "thin" torus by methods of paper [2] is also constructed. [ABSTRACT FROM AUTHOR]
- Published
- 2011
- Full Text
- View/download PDF
21. VIRTUAL HOMOTOPY.
- Author
-
DYE, H. A. and KAUFFMAN, LOUIS H.
- Subjects
HOMOTOPY theory ,INVARIANTS (Mathematics) ,KNOT theory ,LOW-dimensional topology ,MATHEMATICS - Abstract
Two welded (respectively virtual) link diagrams are homotopic if one may be transformed into the other by a sequence of extended Reidemeister moves, classical Reidemeister moves, and self crossing changes. In this paper, we extend Milnor's μ and $\bar{\mu}$ invariants to welded and virtual links. We conclude this paper with several examples, and compute the μ invariants using the Magnus expansion and Polyak's skein relation for the μ invariants. [ABSTRACT FROM AUTHOR]
- Published
- 2010
- Full Text
- View/download PDF
22. INTRODUCTION TO GRAPH-LINK THEORY.
- Author
-
ILYUTKO, DENIS PETROVICH and MANTUROV, VASSILY OLEGOVICH
- Subjects
KNOT theory ,SET theory ,POLYNOMIALS ,MUTATIONS (Algebra) ,MATHEMATICAL analysis ,MATHEMATICS - Abstract
The present paper is an introduction to a combinatorial theory arising as a natural generalization of classical and virtual knot theory. There is a way to encode links by a class of "realizable" graphs. When passing to generic graphs with the same equivalence relations we get "graph-links". On one hand graph-links generalize the notion of virtual link, on the other hand they do not detect link mutations. We define the Jones polynomial for graph-links and prove its invariance. We also prove some a generalization of the Kauffman–Murasugi–Thistlethwaite theorem on "minimal diagrams" for graph-links. [ABSTRACT FROM AUTHOR]
- Published
- 2009
- Full Text
- View/download PDF
23. MINIMAL DEGREE SEQUENCE FOR TORUS KNOTS OF TYPE (p,2p-1).
- Author
-
MADETI, PRABHAKAR and MISHRA, RAMA
- Subjects
KNOT theory ,LOW-dimensional topology ,MATHEMATICS ,ALGEBRAIC topology ,MANIFOLDS (Mathematics) - Abstract
In this paper, we explore the issue of minimizing the degree sequence for torus knots. We find the minimal degree sequence for torus knots of type (p, 2p-1) for any integer p ≥2. We use some results from algebraic geometry to prove our main result. [ABSTRACT FROM AUTHOR]
- Published
- 2006
- Full Text
- View/download PDF
24. SOME RESULTS ABOUT THE KAUFFMAN BRACKET SKEIN MODULE OF THE TWIST KNOT EXTERIOR.
- Author
-
GELCA, RĂZVAN and NAGASATO, FUMIKAZU
- Subjects
KNOT theory ,LOW-dimensional topology ,KNOT polynomials ,POLYNOMIALS ,MATHEMATICS - Abstract
In this paper, we list in explicit form the factoring relations of the Kauffman bracket skein module (KBSM for short) of a twist knot exterior. This is done using curves decorated by characters of irreducible SL(2, ℂ)-representations. In the process, we exhibit a relation which holds in the KBSM of the knot exterior, called the minimal relation. In the final section we prove that, when specializing the variable of the Kauffman bracket at t = -1, the minimal relation becomes the defining polynomial of the SL(2, ℂ)-character variety of the twist knot. [ABSTRACT FROM AUTHOR]
- Published
- 2006
- Full Text
- View/download PDF
25. VIRTUAL BRAIDS AND THE L-MOVE.
- Author
-
KAUFFMAN, LOUIS H. and LAMBROPOULOU, SOFIA
- Subjects
BRAID theory ,MARKOV processes ,KNOT theory ,LOW-dimensional topology ,ALGEBRA ,MATHEMATICS - Abstract
In this paper we prove a Markov theorem for virtual braids and for analogs of this structure including flat virtual braids and welded braids. The virtual braid group is the natural companion to the category of virtual knots, just as the Artin braid group is the natural companion to classical knots and links. In this paper we follow L-move methods to prove the Virtual Markov theorems. One benefit of this approach is a fully local algebraic formulation of the theorems in each category. [ABSTRACT FROM AUTHOR]
- Published
- 2006
- Full Text
- View/download PDF
26. HAMILTONIAN CYCLES AND ROPE LENGTHS OF CONWAY ALGEBRAIC KNOTS.
- Author
-
Diao, Yuanan and Ernst, Claus
- Subjects
KNOT theory ,HAMILTONIAN graph theory ,HAMILTONIAN operator ,ARITHMETIC ,COMPLEX numbers ,MATHEMATICS - Abstract
For a knot or link K, let L(K) denote the rope length of K and let Cr(K) denote the crossing number of K. An important problem in geometric knot theory concerns the bound on L(K) in terms of Cr(K). It is well-known that there exist positive constants c
1 , c2 such that for any knot or link K, c1 · (Cr(K))3/4 ≤ L(K) ≤ c2 · (Cr(K))3/2 . It is also known that for any real number p such that 3/4 ≤ p ≤ 1, there exists a family of knots {Kn } with the property that Cr(Kn ) → ∞ (as n → ∞) such that L(Kn ) = O(Cr(Kn )p ). However, it is still an open question whether there exists a family of knots {Kn } with the property that Cr(Kn ) → ∞ (as n → ∞) such that L(Kn ) = O(Cr(Kn )p ) for some 1 < p ≤ 3/2. In this paper, we show that there are many families of prime alternating Conway algebraic knots {Kn } with the property that Cr(Kn ) → ∞ (as n → ∞) such that L(Kn ) can grow no faster than linearly with respect to Cr(Kn ). [ABSTRACT FROM AUTHOR]- Published
- 2006
- Full Text
- View/download PDF
27. A NOTE ON STRONGLY n-TRIVIAL LINKS.
- Author
-
Torisu, Ichiro
- Subjects
MATHEMATICAL models ,KNOT theory ,LOW-dimensional topology ,GEOMETRIC modeling ,HYPERBOLIC geometry ,MATHEMATICS - Abstract
In this paper, we extend a Howards–Luecke's theorem on strongly n-trivial knots to link case. We show that a link is strongly n-trivial for all n if and only if it is a trivial link. [ABSTRACT FROM AUTHOR]
- Published
- 2005
- Full Text
- View/download PDF
28. THE BRAID GROUP $B_{n,m}(\mathbb{S}^{2})$ AND A GENERALISATION OF THE FADELL–NEUWIRTH SHORT EXACT SEQUENCE.
- Author
-
Gonçalves, Daciberg Lima and Guaschi, John
- Subjects
BRAID theory ,KNOT theory ,GROUP presentations (Mathematics) ,GROUP theory ,CONFIGURATION space ,MATHEMATICS - Abstract
Let m,n ∈ ℕ. We define $B_{n,m}(\mathbb{S}^{2})$ to be the set of (n+m)-braids of the sphere whose associated permutation lies in the subgroup S
n × Sm of the symmetric group Sn+m on n+m letters. In a previous paper [13], we showed that if n ≥ 3, then there exists the following generalisation of the Fadell–Neuwirth short exact sequence: \[ 1\to B_m(\mathbb{S}^{2}\setminus\{x_1,\ldots,x_n\})\to B_{n,m}(\mathbb{S}^{2})\stackrel{p_{\ast}}{\longrightarrow} B_n(\mathbb{S}^{2})\to 1, \] where ${p_{\ast}}{:}\, B_{n,m}(\mathbb{S}^{2}) \to B_n(\mathbb{S}^{2})$ is the group homomorphism (defined for all n ∈ ℕ) given geometrically by forgetting the last m strings. In this paper we study the splitting of this short exact sequence, as well as the existence of a cross-section for the fibration $p{:}\, D_{n,m}(\mathbb{S}^{2}) \to D_n(\mathbb{S}^{2})$ of the quotients of the corresponding configuration spaces. Our main results are as follows: if n = 1 (respectively, n = 2) then the homomorphism p* and the fibration p admit (respectively, do not admit) a section. If n = 3, then p* and p admit a section if and only if m ≡ 0,2 (mod 3). If n ≥ 4, we show that if p* and p admit a section then m ≡ ε1 (n - 1)(n - 2) - ε2 n(n - 2) (mod n(n - 1)(n - 2)), where ε1 ,ε2 ∈ {0,1}. Finally, we show that $B_n(\mathbb{S}^{2})$ is generated by two of its torsion elements. [ABSTRACT FROM AUTHOR]- Published
- 2005
29. HIGHER DEGREE INVARIANTS OF ALMOST GENERIC PLANE IMMERSED CURVES.
- Author
-
MOHANTY, SARA
- Subjects
CURVES ,MATHEMATICS ,CONIC sections ,ENUMERATIVE geometry ,CALCULUS ,ANALYTIC geometry ,GEOMETRY - Abstract
In earlier works (by Arnold and the author) the approach of singularity theory was used to construct invariants of degree 1 of plane immersed curves. This provides a finer classification of these immersions. In this paper, we construct higher degree invariants by "integrating" some of the basic invariants found earlier of almost generic immersed curves. [ABSTRACT FROM AUTHOR]
- Published
- 2005
- Full Text
- View/download PDF
30. TWIST MOVES AND VASSILIEV INVARIANTS.
- Author
-
JEONG, MYEONG-JU, KIM, EUN-JIN, and PARK, CHAN-YOUNG
- Subjects
KNOT polynomials ,POLYNOMIALS ,KNOT theory ,INVARIANT imbedding ,INVARIANTS (Mathematics) ,MATHEMATICS - Abstract
The transforms of two oriented parallel strands to a k-half twist of two strands are called t
k -move and &tmacr;k -move respectively depending on the orientations of the two strands. In this paper we give criterions to detect whether a knot K can be transformed to a knot K' by t2k -moves and t2k -moves respectively and if so, we give some results on how many moves are needed in these transformations respectively, by using some Vassiliev invariants. Moreover we give a relation between the Δ-move and the t2k -move by considering the coefficient of z2 in the Conway polynomial of a knot, which is a Vassiliev invariant of degree 2. [ABSTRACT FROM AUTHOR]- Published
- 2004
- Full Text
- View/download PDF
31. LEGENDRIAN SURGERY IS NOT CATEGORY-PRESERVING FOR TIGHT CONTACT STRUCTURES.
- Author
-
Jin-hong Kim
- Subjects
SURGERY ,CATEGORIES (Mathematics) ,TIGHT junctions ,MANIFOLDS (Mathematics) ,MATHEMATICS ,VECTOR spaces - Abstract
The aim of this paper is to show that the Seifert fibered space Σ(-½, ⅓, ⅓) over S
2 does not admit any tight contact structures. As a consequence, we can conclude that Legendrian surgery is not category-preserving for tight contact structures on closed 3-manifolds. [ABSTRACT FROM AUTHOR]- Published
- 2004
- Full Text
- View/download PDF
32. A NOTE ON THE INDEPENDENCE OF REIDEMEISTER MOVES.
- Author
-
CHENG, ZHIYUN and GAO, HONGZHU
- Subjects
REIDEMEISTER moves ,MATHEMATICS ,SET theory ,GRAPHIC methods ,NUMERICAL analysis ,MATHEMATICAL analysis - Abstract
In this paper, from the viewpoint of quandle construction we prove that for any link type L and any diagram of it, there exists another diagram of L such that these two diagrams are Ω
1 -dependent, Ω2 -dependent and Ω3 -dependent. [ABSTRACT FROM AUTHOR]- Published
- 2012
- Full Text
- View/download PDF
33. HOMOLOGICALLY PERIPHERAL HYPERBOLIC LINKS WHICH ARE NOT PARTIALLY PERIPHERAL.
- Author
-
IKEDA, TORU
- Subjects
MANIFOLDS (Mathematics) ,HYPERBOLIC spaces ,MATHEMATICS ,HOMOLOGY theory ,MATHEMATICAL analysis ,MATHEMATICAL proofs ,SPHERES - Abstract
Partially peripheral 3-manifolds and homologically peripheral 3-manifolds are defined by generalizing the notion of totally peripheral 3-manifolds. The aim of this paper is to provide a method for constructing hyperbolic links in 3-manifolds whose exteriors are not partially peripheral but homologically peripheral, and to prove that the connected sum with a homology 3-sphere enables every closed connected orientable 3-manifold to contain infinitely many such hyperbolic links. [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
- View/download PDF
34. ON COMPUTING KAUFFMAN BRACKET POLYNOMIAL OF MONTESINOS LINKS.
- Author
-
JIN, XIAN'AN and ZHANG, FUJI
- Subjects
POLYNOMIALS ,MATHEMATICS ,ALGEBRA ,MATHEMATICAL analysis ,RINGS of integers - Abstract
It is well known that Jones polynomial (hence, Kauffman bracket polynomial) of links is, in general, hard to compute. By now, Jones polynomials or Kauffman bracket polynomials of many link families have been computed, see [4, 7–11]. In recent years, the computer algebra (Maple) techniques were used to calculate link polynomials for various link families, see [7, 12–14]. In this paper, we try to design a maple program to calculate the explicit expression of the Kauffman bracket polynomial of Montesinos links. We first introduce a family of "ring of tangles" links, which includes Montesinos links as a special subfamily. Then, we provide a closed-form formula of Kauffman bracket polynomial for a "ring of tangles" link in terms of Kauffman bracket polynomials of the numerators and denominators of the tangles building the link. Finally, using this formula and known results on rational links, the Maple program is designed. [ABSTRACT FROM AUTHOR]
- Published
- 2010
- Full Text
- View/download PDF
35. ON THE JONES POLYNOMIAL OF ADEQUATE VIRTUAL LINKS.
- Author
-
BAE, YONGJU, LEE, HYE SOOK, and PARK, CHAN-YOUNG
- Subjects
KNOT theory ,POLYNOMIALS ,NUMBER theory ,LOW-dimensional topology ,MATHEMATICS - Abstract
In this paper, we prove that an adequate virtual link diagram of an adequate virtual link has minimal real crossing number. [ABSTRACT FROM AUTHOR]
- Published
- 2010
- Full Text
- View/download PDF
36. REGULAR PROJECTIONS OF GRAPHS WITH AT MOST THREE DOUBLE POINTS.
- Author
-
HUH, YOUNGSIK and NIKKUNI, RYO
- Subjects
GRAPHIC methods ,IMMERSIONS (Mathematics) ,MANIFOLDS (Mathematics) ,MATHEMATICAL mappings ,KNOT theory ,MATHEMATICS - Abstract
A generic immersion of a planar graph into the 2-space is said to be knotted if there does not exist a trivial embedding of the graph into the 3-space obtained by lifting the immersion with respect to the natural projection from the 3-space to the 2-space. In this paper, we show that if a generic immersion of a planar graph is knotted then the number of double points of the immersion is more than or equal to three. To prove this, we also show that an embedding of a graph obtained from a generic immersion of the graph (does not need to be planar) with at most three double points is totally free if it contains neither a Hopf link nor a trefoil knot. [ABSTRACT FROM AUTHOR]
- Published
- 2010
- Full Text
- View/download PDF
37. A NOTE ON THE CROSSING NUMBER AND THE BRAID INDEX FOR VIRTUAL LINKS.
- Author
-
TAKEDA, YASUSHI
- Subjects
BRAID theory ,KNOT theory ,LOW-dimensional topology ,MATHEMATICAL inequalities ,MATHEMATICS - Abstract
It is well known that any virtual link is described as the closure of a virtual braid. Therefore, we can define the virtual braid index. Ohyama proved an inequality for the crossing number and the braid index of a classical link. In this paper, we prove an analogous inequality for the (total) crossing number and the braid index of a virtual link. [ABSTRACT FROM AUTHOR]
- Published
- 2010
- Full Text
- View/download PDF
38. DEHN SURGERIES ON 2-BRIDGE LINKS WHICH YIELD REDUCIBLE 3-MANIFOLDS.
- Author
-
GODA, HIROSHI, HAYASHI, CHUICHIRO, and SONG, HYUN-JONG
- Subjects
MANIFOLDS (Mathematics) ,TORUS ,KNOT theory ,SURGERY (Topology) ,MATHEMATICS - Abstract
We completely determine which Dehn surgeries on 2-bridge links yield reducible 3-manifolds. Further, we consider which surgery on one component of a 2-bridge link yields a torus knot, a cable knot and a satellite knot in this paper. [ABSTRACT FROM AUTHOR]
- Published
- 2009
- Full Text
- View/download PDF
39. THE PROJECTION STICK INDEX OF KNOTS.
- Author
-
ADAMS, COLIN and SHAYLER, TODD
- Subjects
KNOT theory ,KNOT polynomials ,GRAPHICAL projection ,TORIC varieties ,MATHEMATICS - Abstract
The stick index of a knot K is defined to be the least number of line segments needed to construct a polygonal embedding of K. We define the projection stick index of K to be the least number of line segments in any projection of a polygonal embedding of K. In this paper, we establish bounds on the projection stick index for various torus knots. We then show that the stick index of a (p, 2p + 1)-torus knot is 4p, and the projection stick index is 2p + 1. This provides examples of knots such that the projection stick index is one greater than half the stick index. We show that for all other torus knots for which the stick index is known, the projection stick index is larger than this. We conjecture that a projection stick index of half the stick index is unattainable for any knot. [ABSTRACT FROM AUTHOR]
- Published
- 2009
- Full Text
- View/download PDF
40. BIQUANDLES AND THEIR APPLICATION TO VIRTUAL KNOTS AND LINKS.
- Author
-
FENN, ROGER
- Subjects
KNOT theory ,QUATERNIONS ,UNIVERSAL algebra ,EQUATIONS ,MATHEMATICAL analysis ,MATHEMATICS - Abstract
In this paper, based on a talk given at the Oberwolfach research centre in May 2008 I will describe how biquandles and their big brother, biracks, can be used to differentiate isotopy classes of virtual (and welded) knots and links. [ABSTRACT FROM AUTHOR]
- Published
- 2009
- Full Text
- View/download PDF
41. AN INVARIANT FOR SINGULAR KNOTS.
- Author
-
JUYUMAYA, J. and LAMBROPOULOU, S.
- Subjects
KNOT theory ,BRAID theory ,LOW-dimensional topology ,ALGEBRA ,MATHEMATICAL analysis ,MATHEMATICS - Abstract
In this paper we introduce a Jones-type invariant for singular knots, using a Markov trace on the Yokonuma–Hecke algebras Y
d,n (u) and the theory of singular braids. The Yokonuma–Hecke algebras have a natural topological interpretation in the context of framed knots. Yet, we show that there is a homomorphism of the singular braid monoid SBn into the algebra Yd,n (u). Surprisingly, the trace does not normalize directly to yield a singular link invariant, so a condition must be imposed on the trace variables. Assuming this condition, the invariant satisfies a skein relation involving singular crossings, which arises from a quadratic relation in the algebra Yd,n (u). [ABSTRACT FROM AUTHOR]- Published
- 2009
- Full Text
- View/download PDF
42. A PARTIAL ORDERING OF KNOTS AND LINKS THROUGH DIAGRAMMATIC UNKNOTTING.
- Author
-
DIAO, YUANAN, ERNST, CLAUS, and STASIAK, ANDRZEJ
- Subjects
MATHEMATICS ,GRAPHIC methods ,KNOTS & splices ,LINKS & link-motion ,JOINTS (Engineering) - Abstract
In this paper we define a partial ordering of knots and links using a special property derived from their minimal diagrams. A link $\mathcal{K}'$ is called a predecessor of a link $\mathcal{K}$ if $Cr(\mathcal{K}') < Cr(\mathcal{K})$ and a diagram of $\mathcal{K}'$ can be obtained from a minimal diagram D of $\mathcal{K}$ by a single crossing change. In such a case, we say that $\mathcal{K}' < \mathcal{K}$. We investigate the sets of links that can be obtained by single crossing changes over all minimal diagrams of a given link. We show that these sets are specific for different links and permit partial ordering of all links. Some interesting results are presented and many questions are raised. [ABSTRACT FROM AUTHOR]
- Published
- 2009
- Full Text
- View/download PDF
43. MINIMAL DEGREE SEQUENCE FOR TORUS KNOTS OF TYPE (p, q).
- Author
-
MADETI, PRABHAKAR and MISHRA, RAMA
- Subjects
MATHEMATICS ,MATHEMATICAL analysis ,RINGS of integers ,TORUS ,MANIFOLDS (Mathematics) - Abstract
In this paper we prove the following result: for coprime positive integers p and q with p < q, if r is the least positive integer such that 2p-1 and q + r are coprime, then the minimal degree sequence for a torus knot of type (p, q) is the triple (2p - 1, q + r, d) or (q + r, 2p - 1, d) where q + r + 1 ≤ d ≤ 2q - 1. [ABSTRACT FROM AUTHOR]
- Published
- 2009
- Full Text
- View/download PDF
44. SOLUTION OF THE HURWITZ PROBLEM FOR LAURENT POLYNOMIALS.
- Author
-
PAKOVICH, F.
- Subjects
POLYNOMIALS ,ALGEBRA ,APPROXIMATION theory ,BERNOULLI polynomials ,RANDOM polynomials ,MATHEMATICS ,MATHEMATICAL analysis - Abstract
We investigate the following existence problem for rational functions: for a given collection Π of partitions of a number n to define whether there exists a rational function f of degree n for which Π is the branch datum. An important particular case when the answer is known is the one when the collection Π contains a partition consisting of a single element (in this case, the corresponding rational function is equivalent to a polynomial). In this paper, we provide a solution in the case when Π contains a partition consisting of two elements. [ABSTRACT FROM AUTHOR]
- Published
- 2009
- Full Text
- View/download PDF
45. ALEXANDER QUANDLES OF ORDER 16.
- Author
-
MURILLO, GABRIEL and NELSON, SAM
- Subjects
ISOMORPHISM (Mathematics) ,KNOT theory ,LOW-dimensional topology ,ALGEBRAIC topology ,MATHEMATICS - Abstract
Isomorphism classes of Alexander quandles of order 16 are determined, and classes of connected quandles are identified. This paper extends the list of distinct connected finite Alexander quandles. [ABSTRACT FROM AUTHOR]
- Published
- 2008
- Full Text
- View/download PDF
46. NEW INVARIANTS OF SIMPLE KNOTS.
- Author
-
KEARTON, C. and WILSON, S. M. J.
- Subjects
NUMBER theory ,ALGEBRA ,KNOT theory ,LOW-dimensional topology ,MATHEMATICS - Abstract
Our longterm plan is to classify knot modules and pairings by utilizing the power of computational number theory. The first step in this is to define invariants for which any given value arises from only finitely many modules: this is the purpose of the present paper. [ABSTRACT FROM AUTHOR]
- Published
- 2008
- Full Text
- View/download PDF
47. DIRECTED GRAPHS AND KRONECKER INVARIANTS OF PAIRS OF MATRICES.
- Author
-
TOWBER, JACOB
- Subjects
MATRICES (Mathematics) ,ALGEBRA ,MATHEMATICS ,GRAPH theory ,EIGENVALUES - Abstract
Call two pairs (M,N) and (M′,N′) of m × n matrices over a field K, simultaneously K-equivalent if there exist square invertible matrices S,T over K, with M′ = SMT and N′ = SNT. Kronecker [2] has given a complete set of invariants for simultaneous equivalence of pairs of matrices. Associate in the natural way to a finite directed graph Γ, with v vertices and e edges, an ordered pair (M,N) of e × v matrices of zeros and ones. It is natural to try to compute the Kronecker invariants of such a pair (M,N), particularly since they clearly furnish isomorphism-invariants of Γ. Let us call two graphs "linearly equivalent" when their two corresponding pairs are simultaneously equivalent. There have existed, since 1890, highly effective algorithms for computing the Kronecker invariants of pairs of matrices of the same size over a given field [1,2,5,6] and in particular for those arising in the manner just described from finite directed graphs. The purpose of the present paper, is to compute directly these Kronecker invariants of finite directed graphs, from elementary combinatorial properties of the graphs. A pleasant surprise is that these new invariants are purely rational — indeed, integral, in the sense that the computation needed to decide if two directed graphs are linearly equivalent only involves counting vertices in various finite graphs constructed from each of the given graphs — and does not involve finding the irreducible factorization of a polynomial over K (in apparent contrast both to the familiar invariant-computations of graphs furnished by the eigenvalues of the connection matrix, and to the isomorphism problem for general pairs of matrices). [ABSTRACT FROM AUTHOR]
- Published
- 2008
- Full Text
- View/download PDF
48. COMBINATORIC AND DIAGRAMMATIC STUDY IN KNOT THEORY.
- Author
-
CHUN-CHUNG HSIEH
- Subjects
KNOT theory ,QUANTUM field theory ,PERTURBATION theory ,COMBINATORICS ,MATHEMATICS - Abstract
Motivated by Massey–Milnor linking and Chern–Simon–Witten perturbative quantum field theory, we developed some combinatorial and diagrammatic study in this paper, aiming at knot theory in the combinatoric aspect. [ABSTRACT FROM AUTHOR]
- Published
- 2007
- Full Text
- View/download PDF
49. CHIRALITY OF ALTERNATING KNOTS IN S × I.
- Author
-
FLEMING, THOMAS
- Subjects
SPATIAL analysis (Statistics) ,STATISTICAL correlation ,SPATIAL systems ,KNOT theory ,LOW-dimensional topology ,MATHEMATICS - Abstract
In this paper, we will explore the properties of knots that lie in a surface cross an interval. We show that any alternating knot nontrivially embedded in S × I is chiral. This demonstrates that knots in S × I behave differently from knots in the three sphere, where alternating knots (like the figure eight) can be amphichiral. We use a generalized Kauffman bracket polynomial, as well as geometric and covering space techniques. [ABSTRACT FROM AUTHOR]
- Published
- 2006
- Full Text
- View/download PDF
50. THE ALEXANDER POLYNOMIAL OF (1,1)-KNOTS.
- Author
-
CATTABRIGA, A.
- Subjects
KNOT theory ,LOW-dimensional topology ,POLYNOMIALS ,ALGEBRAIC topology ,MANIFOLDS (Mathematics) ,MATHEMATICS - Abstract
In this paper we investigate the Alexander polynomial of (1,1)-knots, which are knots lying in a 3-manifold with genus one at most, admitting a particular decomposition. More precisely, we study the connections between the Alexander polynomial and a polynomial associated to a cyclic presentation of the fundamental group of an n-fold strongly-cyclic covering branched over the knot K, which we call the n-cyclic polynomial of K. In this way, we generalize to all (1,1)-knots, with the only exception of those lying in S
2 ×S1 , a result obtained by Minkus for 2-bridge knots and extended by the author and M. Mulazzani to the case of (1,1)-knots in S3 . As corollaries some properties of the Alexander polynomial of knots in S3 are extended to the case of (1,1)-knots in lens spaces. [ABSTRACT FROM AUTHOR]- Published
- 2006
- Full Text
- View/download PDF
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