scholarly journals Lernaean knots and band surgery

2021 ◽  
Vol 33 (1) ◽  
pp. 23-46
Author(s):  
Yu. Belousov ◽  
M. Karev ◽  
A. Malyutin ◽  
A. Miller ◽  
E. Fominykh
Keyword(s):  
Knot Theory ◽  
Crossing Number ◽  
Connected Sum ◽  
Content Type

The paper is devoted to a line of the knot theory related to the conjecture on the additivity of the crossing number for knots under connected sum. A series of weak versions of this conjecture are proved. Many of these versions are formulated in terms of the band surgery graph also called the H ( 2 ) H(2) -Gordian graph.

THE ADDITIVITY OF CROSSING NUMBERS

2004 ◽  
Vol 13 (07) ◽  
pp. 857-866 ◽  
Author(s):  
YUANAN DIAO
Keyword(s):  
Knot Theory ◽  
Difficult Problem ◽  
Crossing Number ◽  
The Other ◽  
Connected Sum ◽  
Torus Knots ◽  
Crossing Numbers ◽  
Weaker Conditions ◽  
Alternating Links ◽  
Alternating Link

It has long been conjectured that the crossing numbers of links are additive under the connected sum of links. This is a difficult problem in knot theory and has been open for more than 100 years. In fact, many questions of much weaker conditions are still open. For instance, it is not known whether Cr(K1#K2)≥Cr(K1) or Cr(K1#K2)≥Cr(K2) holds in general, here K1#K2 is the connected sum of K1 and K2 and Cr(K) stands for the crossing number of the link K. However, for alternating links K1 and K2, Cr(K1#K2)=Cr(K1)+Cr(K2) does hold. On the other hand, if K1 is an alternating link and K2 is any link, then we have Cr(K1#K2)≥Cr(K1). In this paper, we show that there exists a wide class of links over which the crossing number is additive under the connected sum operation. This class is different from the class of all alternating links. It includes all torus knots and many alternating links. Furthermore, if K1 is a connected sum of any given number of links from this class and K2 is a non-trivial knot, we prove that Cr(K1#K2)≥Cr(K1)+3.

On the Question of Genericity of Hyperbolic Knots

2018 ◽  
Vol 2020 (21) ◽  
pp. 7792-7828
Author(s):  
Andrei V Malyutin
Keyword(s):  
Knot Theory ◽  
Crossing Number ◽  
Connected Sum ◽  
Satellite Knot

Abstract A well-known conjecture in knot theory says that the proportion of hyperbolic knots among all of the prime knots of $n$ or fewer crossings approaches $1$ as $n$ approaches infinity. In this article, it is proved that this conjecture contradicts several other plausible conjectures, including the 120-year-old conjecture on additivity of the crossing number of knots under connected sum and the conjecture that the crossing number of a satellite knot is not less than that of its companion.

ON THE ADDITIVITY OF CROSSING NUMBERS OF GRAPHS

2008 ◽  
Vol 17 (09) ◽  
pp. 1043-1050 ◽  
Author(s):  
JESÚS LEAÑOS ◽  
GELASIO SALAZAR
Keyword(s):  
Connected Graph ◽  
Knot Theory ◽  
Crossing Number ◽  
Connected Sum ◽  
Critical Graphs ◽  
Crossing Numbers ◽  
The Individual

We describe a relationship between the crossing number of a graph G with a 2-edge-cut C and the crossing numbers of the components of G-C. Let G be a connected graph with a 2-edge-cut C := [V1,V2]. Let u1u2, v1v2 be the edges of C, so that ui,vi ∈ Vi for i = 1,2, and let Gi := G[Vi] and G'i := Gi + uivi. We show that if either G1 or G2 is not connected, then cr (G) = cr (G1) + cr (G2), and that if they are both connected then cr (G) = cr (G'1) + cr (G'2). We use this to show how to decompose crossing-critical graphs with 2-edge-cuts into smaller, 3-edge-connected crossing-critical graphs. We also observe that this settles a question arising from knot theory, raised by Sawollek, by describing exactly under which conditions the crossing number of the connected sum of two graphs equals the sum of the crossing numbers of the individual graphs.

Totally knotted knots are prime

1982 ◽  
Vol 91 (3) ◽  
pp. 467-472
Author(s):  
J. C. Gomez-Larran¯aga
Keyword(s):  
Finite Number ◽  
Knot Theory ◽  
Piecewise Linear ◽  
Connected Sum ◽  
Higher Dimensional ◽  
The Family ◽  
Linear Category

Throughout, the word knot means a subspace of the 3-sphere S3 homeomorphic with the 1-sphere S1. Any knot can be expressed as a connected sum of a finite number of prime knots in a unique way (13), we consider the trivial knot a non-prime knot. (For higher dimensional knots, factorization and uniqueness have been studied in (1).) However given a knot it is difficult to determine if it is prime or not. We prove that totally knotted knots, see definition in §2, are prime in theorem 1, give a class of examples in theorem 2 and investigate how the last result can be applied to the conjecture that the family Y of unknotting number one knots are prime. (See problem 2 in (5).) At the end, prime tangles as defined by W. B. R. Lickerish are used to prove that in a certain family of knots, related somewhat to Y, there is just one non-prime knot: the square knot. The paper should be interpreted as being in the piecewise linear category. Standard definitions of 3-manifolds and knot theory may be found in (6) and (11) respectively.

scholarly journals COMPLEXITY OF LINKS IN 3-MANIFOLDS

2009 ◽  
Vol 18 (10) ◽  
pp. 1439-1458 ◽  
Author(s):  
EKATERINA PERVOVA ◽  
CARLO PETRONIO
Keyword(s):  
Lower Bounds ◽  
Crossing Number ◽  
Upper And Lower Bounds ◽  
Recent Theory ◽  
Asymptotic Estimates ◽  
Connected Sum ◽  
Hyperbolic Volume ◽  
Do So

We introduce a complexity c(X) ∈ ℕ for pairs X = (M,L), where M is a closed orientable 3-manifold and L ⊂ M is a link. The definition employs simple spines, but for well-behaved X 's , we show that c(X) equals the minimal number of tetrahedra in a triangulation of M containing L in its 1-skeleton. Slightly adapting Matveev's recent theory of roots for graphs, we carefully analyze the behaviour of c under connected sum away from and along the link. We show in particular that c is almost always additive, describing in detail the circumstances under which it is not. To do so we introduce a certain (0, 2)-root for a pair X, we show that it is well-defined, and we prove that X has the same complexity as its (0, 2)-root. We then consider, for links in the 3-sphere, the relations of c with the crossing number and with the hyperbolic volume of the exterior, establishing various upper and lower bounds. We also specialize our analysis to certain infinite families of links, providing rather accurate asymptotic estimates.

scholarly journals On measures of nonplanarity of cubic graphs

2018 ◽  
Vol 11 (2) ◽  
Author(s):  
Leonid Plachta
Keyword(s):  
Upper Bounds ◽  
Crossing Number ◽  
Cubic Graphs ◽  
Minimum Order ◽  
Edge Deletion ◽  
Connected Sum ◽  
Minimal Genus ◽  
Two Measures

We study two measures of nonplanarity of cubic graphs G, the genus γ (G), and the edge deletion number ed(G). For cubic graphs of small orders these parameters are compared with another measure of nonplanarity, the rectilinear crossing number (G). We introduce operations of connected sum, specified for cubic graphs G, and show that under certain conditions the parameters γ(G) and ed(G) are additive (subadditive) with respect to them.The minimal genus graphs (i.e. the cubic graphs of minimum order with given value of genus γ) and the minimal edge deletion graphs (i.e. cubic graphs of minimum order with given value of edge deletion number ed) are introduced and studied. We provide upper bounds for the order of minimal genus and minimal edge deletion graphs.

HAMILTONIAN CYCLES AND ROPE LENGTHS OF CONWAY ALGEBRAIC KNOTS

2006 ◽  
Vol 15 (01) ◽  
pp. 121-142 ◽  
Author(s):  
YUANAN DIAO ◽  
CLAUS ERNST
Keyword(s):  
Real Number ◽  
Knot Theory ◽  
Crossing Number ◽  
Hamiltonian Cycles ◽  
Open Question

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 c1, 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).

The Gordian Knot

2020 ◽  
pp. 235-247
Author(s):  
Nicholas Mee
Keyword(s):  
Nineteenth Century ◽  
Knot Theory ◽  
Crossing Number ◽  
Important Class ◽  
Torus Knots ◽  
Lord Kelvin

Chapter 22 includes a brief survey of knots and their uses. The nineteenth-century physicist Lord Kelvin suggested that atoms might be knots in the aether. This idea led to the development of knot theory as a branch of mathematics. Knots are classified by their crossing number. As the crossing number increases, the number of prime knots rises rapidly. This chapter explains an important class of knots known as torus knots that can be produced by winding a string around a torus. Knots that are formed of more than one component are known as links.

Natural classification of knots

2006 ◽  
Vol 463 (2078) ◽  
pp. 569-582 ◽  
Author(s):  
Alessandro Flammini ◽  
Andrzej Stasiak
Keyword(s):  
Knot Theory ◽  
Three Dimensional ◽  
Crossing Number ◽  
Topological Domains ◽  
Natural Classification ◽  
Quantum Numbers ◽  
Simple Quantum ◽  
Knot Space ◽  
Writhe Number

The principal objective of the knot theory is to provide a simple way of classifying and ordering all the knot types. Here, we propose a natural classification of knots based on their intrinsic position in the knot space that is defined by the set of knots to which a given knot can be converted by individual intersegmental passages. In addition, we characterize various knots using a set of simple quantum numbers that can be determined upon inspection of minimal crossing diagram of a knot. These numbers include: crossing number; average three-dimensional writhe; number of topological domains; and the average relaxation value.

Erratum: Connected sum of representations of knot groups

2016 ◽  
Vol 25 (05) ◽  
pp. 1692001 ◽  
Author(s):  
Jinseok Cho
Keyword(s):  
Knot Theory ◽  
Connected Sum

We fix the errors in the paper ‘Connected sum of representations of knot groups’ [J. Cho, Connected sum of representations of knot groups, J. Knot Theory Ramifications 24(3) (2015) 18, 1550020].

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