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.

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.

scholarly journals RIBBONLENGTH OF TORUS KNOTS

2008 ◽  
Vol 17 (01) ◽  
pp. 13-23 ◽  
Author(s):  
BROOKE KENNEDY ◽  
THOMAS W. MATTMAN ◽  
ROBERTO RAYA ◽  
DAN TATING
Keyword(s):  
Crossing Number ◽  
Regular Polygons ◽  
Torus Knots ◽  
Torus Knot

Using Kauffman's model of flat knotted ribbons, we demonstrate how all regular polygons of at least seven sides can be realized by ribbon constructions of torus knots. We calculate length to width ratios for these constructions thereby bounding the Ribbonlength of the knots. In particular, we give evidence that the closed (respectively, truncation) Ribbonlength of a (q + 1,q) torus knot is (2q + 1) cot (π/(2q + 1)) (respectively, 2q cot (π/(2q + 1))). Using these calculations, we provide the bounds c1 ≤ 2/π and c2 ≥ 5/3 cot π/5 for the constants c1 and c2 that relate Ribbonlength R(K) and crossing number C(K) in a conjecture of Kusner: c1 C(K) ≤ R(K) ≤ c2 C(K).

Scientists and Broad Churchmen: an Early Victorian Intellectual Network

1964 ◽  
Vol 4 (1) ◽  
pp. 65-88 ◽  
Author(s):  
Walter F. Cannon
Keyword(s):  
Nineteenth Century ◽  
Indirect Method ◽  
Science And Religion ◽  
Matthew Arnold ◽  
Scientific Excellence ◽  
James Clerk Maxwell ◽  
Thomas Huxley ◽  
History Of ◽  
Lord Kelvin

The late Victorians popularized several ideas which have tended to obscure what was actually going on in intellectual matters in the early part of the nineteenth century. One of these is the notion that, whenever science and religion came into contact, some degree of scientific excellence was sacrificed, if only because the scientists themselves believed in the theological ideas. Another is the judgment that Dean Stanley, a “passive peaceable Protestant” always seeking compromise, was the typical Broad Churchman. And a third is the acceptance of Leslie Stephen's description of an arid “Cambridge rationalism” not only as enlightening (which it is) but also as complete.These and other similar misconceptions could be propagated because the later Victorian intellectual “aristocracy” or “self-reviewing circle,” as described so well by Noel Annan, was not continuous with that of the earlier period. Such physical descendants as did remain, notably Matthew Arnold and Leslie Stephen, played quite different roles in the new circle from those which their fathers had filled in the older, looser, grouping. The founders of the new aristocracy selected their mythic figures with an eye to current usefulness rather than with strict attention to the history of the earlier generation. This was to be expected. One could not expect Thomas Huxley to emphasize the great abilities of the geologist Adam Sedgwick when it was just such a reputation which supported “the old Adam” in his attack on Darwin's theories.In order to indicate the inadequacy of the three conceptions listed above, and others like them, it is the purpose of this article to use the indirect method of sketching the coming together of those men who were the mentors not only of Darwin but also of Stanley, of Tennyson, of Frederick Denison Maurice, of Lord Kelvin, and of James Clerk Maxwell.

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.

MANY CLASSICAL KNOT INVARIANTS ARE NOT VASSILIEV INVARIANTS

1994 ◽  
Vol 03 (01) ◽  
pp. 7-10 ◽  
Author(s):  
JOHN DEAN
Keyword(s):  
Crossing Number ◽  
Torus Knots ◽  
Knot Invariants ◽  
Vassiliev Invariants ◽  
Vassiliev Invariant ◽  
Bridge Number

We show that under twisting, a Vassiliev invariant of order n behaves like a polynomial of degree at most n. This greatly restricts the values that a Vassiliev invariant can take, for example, on the (2, m) torus knots. In particular, this implies that many classical numerical knot invariants such as the signature, genus, bridge number, crossing number, and unknotting number are not Vassiliev invariants.

Sir William Thomson and the Intercept Method

1972 ◽  
Vol 25 (1) ◽  
pp. 91-98
Author(s):  
Charles H. Cotter
Keyword(s):  
Nineteenth Century ◽  
Royal Society ◽  
New Method ◽  
William Thomson ◽  
Lord Kelvin ◽  
Short Method ◽  
Royal Society Of London

The year 1971 marked the first centenary of the publication of a paper on navigation which appeared in the Proceedings of the Royal Society of London in which the author, Sir William Thomson (later Lord Kelvin) described a new method of determining an astronomical position line. The method was impracticable and was not, therefore, adopted by practical seamen. Nevertheless, its design is ingenious and interesting, and an investigation of its principles adds lustre to the genius of its inventor—reputedly one of the most eminent philosophers of the nineteenth century. Although the method failed in the eyes of the mariners for whom it was intended, Thomson sparked off an interest in short-method tables which has persisted even to the present day.

scholarly journals Twisted Alexander polynomials with the adjoint action for some classes of knots

2014 ◽  
Vol 23 (10) ◽  
pp. 1450051 ◽  
Author(s):  
Anh T. Tran
Keyword(s):  
Knot Theory ◽  
Adjoint Action ◽  
Alexander Polynomial ◽  
Reidemeister Torsion ◽  
Torus Knots ◽  
Alexander Polynomials ◽  
Fibered Knots ◽  
Twisted Alexander Polynomial ◽  
The One ◽  
Better Than

We calculate the twisted Alexander polynomial with the adjoint action for torus knots and twist knots. As consequences of these calculations, we obtain the formula for the nonabelian Reidemeister torsion of torus knots in [J. Dubois, Nonabelian twisted Reidemeister torsion for fibered knots, Canad. Math. Bull.49(1) (2006) 55–71] and a formula for the nonabelian Reidemeister torsion of twist knots that is better than the one in [J. Dubois, V. Huynh and Y. Yamaguchi, Nonabelian Reidemeister torsion for twist knots, J. Knot Theory Ramifications18(3) (2009) 303–341].

ON CONJECTURES ABOUT POSITIVE BRAID KNOTS AND ALMOST ALTERNATING TORUS KNOTS

2010 ◽  
Vol 19 (11) ◽  
pp. 1471-1486
Author(s):  
MARKO STOŠIĆ
Keyword(s):  
Positive Integer ◽  
Homology Group ◽  
Knot Theory ◽  
Homology Theory ◽  
Khovanov Homology ◽  
Torus Knots ◽  
Link Homology

In this paper we resolve some conjectures concerning positive braid knots and almost alternating torus knots. Namely, we prove that the first Khovanov homology group of positive braid knot is trivial, as conjectured by Khovanov. Also, we generalize this result to show that the same is true in the case of Khovanov–Rozansky homology (sl(n) link homology) for any positive integer n. Moreover, by using the Khovanov homology theory, we prove the classical knot theory conjecture by Adams, that the only almost alternating torus knots are T3, 4 and T3, 5.

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 SIMPLEST HYPERBOLIC KNOTS

1999 ◽  
Vol 08 (03) ◽  
pp. 279-297 ◽  
Author(s):  
PATRICK J. CALLAHAN ◽  
JOHN C. DEAN ◽  
JEFFREY R. WEEKS
Keyword(s):  
Ad Hoc ◽  
Crossing Number ◽  
Hyperbolic Manifolds ◽  
Torus Knots ◽  
Initial Observation ◽  
Knot Complements ◽  
Standard Notion ◽  
Dehn Fillings

While the crossing number is the standard notion of complexity for knots, the number of ideal tetrahedra required to construct the complement provides a natural alternative. We determine which hyperbolic manifolds with 6 or fewer ideal tetrahedra are knot complements, and explicitly describe the corresponding knots in the 3-sphere. Thus, these 72 knots are the simplest knots according to this notion of complexity. Many of these knots have the structure of twisted torus knots. The initial observation that led to the project was the abundance of knot complements with small Seifert-fibered Dehn fillings among the census manifolds. Since many of these knots have rather large crossing number they do not appear in the knot tables. Our methods, while ad hoc, yield some detailed information about the knot complements as well as the manifolds that arise from exceptional surgeries on these knots.

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