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 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.

scholarly journals Braid index bounds ropelength from below

2020 ◽  
Vol 29 (04) ◽  
pp. 2050019
Author(s):  
Yuanan Diao
Keyword(s):  
Lower Bound ◽  
General Formula ◽  
Crossing Number ◽  
Crossing Numbers ◽  
Braid Index ◽  
Alternating Links

For an unoriented link [Formula: see text], let [Formula: see text] be the ropelength of [Formula: see text]. It is known that in general [Formula: see text] is at least of the order [Formula: see text], and at most of the order [Formula: see text] where [Formula: see text] is the minimum crossing number of [Formula: see text]. Furthermore, it is known that there exist families of (infinitely many) links with the property [Formula: see text]. A long standing open conjecture states that if [Formula: see text] is alternating, then [Formula: see text] is at least of the order [Formula: see text]. In this paper, we show that the braid index of a link also gives a lower bound of its ropelength. More specifically, we show that there exists a constant [Formula: see text] such that [Formula: see text] for any [Formula: see text], where [Formula: see text] is the largest braid index among all braid indices corresponding to all possible orientation assignments of the components of [Formula: see text] (called the maximum braid index of [Formula: see text]). Consequently, [Formula: see text] for any link [Formula: see text] whose maximum braid index is proportional to its crossing number. In the case of alternating links, the maximum braid indices for many of them are proportional to their crossing numbers hence the above conjecture holds for these alternating links.

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.

Untangle problems, with knot theory

2020 ◽  
pp. 73-80
Author(s):  
Susan D'Agostino
Keyword(s):  
Knot Theory ◽  
Closed Loop ◽  
The Other ◽  
Mathematics Students ◽  
Crossing Numbers ◽  
Trefoil Knot ◽  
Figure Eight Knot

“Untangle problems, with knot theory” offers a basic introduction to the mathematical subfield of knot theory, including the classification of knots by crossing numbers. A mathematical knot is a closed loop that may or may not be tangled. Two knots are considered the same if one may be manipulated into the other using easy-to-understand techniques. Readers learn to identify knots by crossing numbers and encounter numerous hand-drawn sketches of knots, including the trivial knot, trefoil knot, figure-eight knot, and more. Mathematics students and enthusiasts are encouraged to employ knot theory methods for untangling problems in mathematics or life by asking whether they have encountered the problem before. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.

A REMARK ON THE DETERMINANT OF QUASI-ALTERNATING LINKS

2013 ◽  
Vol 22 (06) ◽  
pp. 1350031 ◽  
Author(s):  
K. QAZAQZEH ◽  
B. QUBLAN ◽  
A. JARADAT
Keyword(s):  
Crossing Number ◽  
Alternating Links ◽  
Alternating Link

We show that the crossing number of any link that is known to be quasi-alternating is less than or equal to its determinant. Based on this, we conjecture that the crossing number of any quasi-alternating link is less than or equal to its determinant. Thus if this conjecture is true, then it gives a new property of quasi-alternating links and easy obstruction to a link being quasi-alternating.

scholarly journals Generating links that are both quasi-alternating and almost alternating

2020 ◽  
pp. 2050090
Author(s):  
Hamid Abchir ◽  
Mohammed Sabak
Keyword(s):  
Infinite Family ◽  
Crossing Number ◽  
Alternating Links ◽  
Alternating Link

We construct an infinite family of links which are both almost alternating and quasi-alternating from a given either almost alternating diagram representing a quasi-alternating link, or connected and reduced alternating tangle diagram. To do that we use what we call a dealternator extension which consists in replacing the dealternator by a rational tangle extending it. We note that all non-alternating and quasi-alternating Montesinos links can be obtained in that way. We check that all the obtained quasi-alternating links satisfy Conjecture 3.1 of Qazaqzeh et al. (JKTR 22 (6), 2013), that is the crossing number of a quasi-alternating link is less than or equal to its determinant. We also prove that the converse of Theorem 3.3 of Qazaqzeh et al. (JKTR 24 (1), 2015) is false.

Some links with non-adequate minimal-crossing diagrams

1992 ◽  
Vol 111 (2) ◽  
pp. 283-289 ◽  
Author(s):  
Masao Hara ◽  
Makoto Yamamoto
Keyword(s):  
Crossing Number ◽  
Link Diagram ◽  
Polynomial Invariants ◽  
Crossing Numbers ◽  
Almost All ◽  
Alternating Link ◽  
Torus Links

To investigate invariants of links derived from their diagrams, the recent new polynomial invariants of links play important roles. Murasugi6, 7, Kauffman3 and Thistlethwaite 9 independently showed that the number of crossings in a proper connected alternating diagram of a link is the minimal-crossing number of the link and that the writhe of the diagram is invariant. Murasugi 8 also determined the minimal-crossing number of torus links. In 5, Lickorish and Thistlethwaite introduced the concept of an adequate link diagram and showed that the number of crossings in an adequate diagram of a semi-alternating link is the minimal-crossing number of the link. They also determined the minimal-crossing number of almost all Montesinos links. In this paper we show that for some links represented by plats and braids which are not adequate, the numbers of crossings in the diagrams are the minimal-crossing numbers of the links.

scholarly journals Characterization of quasi-alternating Montesinos links

2015 ◽  
Vol 24 (01) ◽  
pp. 1550002 ◽  
Author(s):  
K. Qazaqzeh ◽  
N. Chbili ◽  
B. Qublan
Keyword(s):  
Knot Theory ◽  
Infinite Family ◽  
Rational Tangles ◽  
Alternating Links ◽  
Alternating Link

Let L be a quasi-alternating link at a crossing c. We construct an infinite family of quasi-alternating links from L by replacing the crossing c by a product of rational tangles, each of which extends c. Consequently, we determine an infinite family of quasi-alternating Montesinos links. This family includes all classes of quasi-alternating Montesinos links that have been detected by Widmer [Quasi-alternating Montesinos links, J. Knot Theory Ramifications18(10) (2009) 1459–1469]. We conjecture that this family contains all non-alternating quasi-alternating Montesinos links.

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.

scholarly journals ENUMERATING THE PRIME ALTERNATING LINKS

2004 ◽  
Vol 13 (01) ◽  
pp. 151-173 ◽  
Author(s):  
STUART RANKIN ◽  
ORTHO FLINT
Keyword(s):  
Finite Sequence ◽  
Efficient Implementation ◽  
The Other ◽  
Large Set ◽  
Basic Scheme ◽  
Group Number ◽  
Alternating Links ◽  
Alternating Link

In [5], four knot operators were introduced and used to construct all prime alternating knots of a given crossing size. An efficient implementation of this construction was made possible by the notion of the master array of an alternating knot. The master array and an implementation of the construction appeared in [6]. The basic scheme (as described in [5]) is to apply two of the operators, D and ROTS, to the prime alternating knots of minimal crossing size n-1, which results in a large set of prime alternating knots of minimal crossing size n, and then the remaining two operators, T and OTS, are applied to these n crossing knots to complete the production of the set of prime alternating knots of minimal crossing size n. In this paper, we show how to obtain all prime alternating links of a given minimal crossing size. More precisely, we shall establish that given any two prime alternating links of minimal crossing size n, there is a finite sequence of T and OTS operations that transforms one of the links into the other. Consequently, one may select any prime alternating link of minimal crossing size n (which is then called the seed link), and repeatedly apply only the operators T and OTS to obtain all prime alternating links of minimal crossing size n from the chosen seed link. The process may be standardized by specifying the seed link to be (in the parlance of [5]) the unique link of n crossings with group number 1, the (n, 2) torus link.

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