Knitting torus knots and links

2018 ◽  
pp. 119-137
Keyword(s):  
Torus Knots ◽  
Knots And Links

Incoherent nullification of torus knots and links

2019 ◽  
Vol 28 (05) ◽  
pp. 1950033
Author(s):  
Zac Bettersworth ◽  
Claus Ernst
Keyword(s):  
Upper Bound ◽  
Torus Knots ◽  
Number Formula ◽  
Knots And Links

In the paper, we study the incoherent nullification number [Formula: see text] of knots and links. We establish an upper bound on the incoherent nullification number of torus knots and links and conjecture that this upper bound is the actual incoherent nullification number of this family. Finally, we establish the actual incoherent nullification number of particular subfamilies of torus knots and links.

Nullification of Torus knots and links

2014 ◽  
Vol 23 (11) ◽  
pp. 1450058 ◽  
Author(s):  
Claus Ernst ◽  
Anthony Montemayor
Keyword(s):  
Torus Knots ◽  
The Family ◽  
Minimum Number ◽  
Knots And Links

It is known that a knot/link can be nullified, i.e. can be made into the trivial knot/link, by smoothing some crossings in a projection diagram of the knot/link. The minimum number of such crossings to be smoothed in order to nullify the knot/link is called the nullification number. In this paper we investigate the nullification numbers of a particular knot family, namely the family of torus knots and links.

scholarly journals Isogonal Weavings on the Sphere: Knots, Links, Polycatenanes

2020 ◽  
Author(s):  
Michael O'Keeffe ◽  
Michael Treacy
Keyword(s):  
Piecewise Linear ◽  
Infinite Family ◽  
Torus Knots ◽  
Vertex Transitive ◽  
Knots And Links ◽  
Complete Account

<p>We describe mathematical knots and links as piecewise linear – straight, non-intersecting sticks meeting at corners. Isogonal structures have all corners related by symmetry ("vertex" transitive). Corner- and stick-transitive structures are termed <i>regular</i>. We find no regular knots. Regular links are cubic or icosahedral and a complete account of these is given, including optimal (thickest-stick) embeddings. Stick 2-transitive isogonal structures are again cubic and icosahedral and also encompass the infinite family of torus knots and links. The major types of these structures are identified and reported with optimal embeddings. We note the relevance of this work to materials- and bio-chemistry.</p>

scholarly journals Finite n-quandles of torus and two-bridge links

2019 ◽  
Vol 28 (03) ◽  
pp. 1950028
Author(s):  
Alissa S. Crans ◽  
Blake Mellor ◽  
Patrick D. Shanahan ◽  
Jim Hoste
Keyword(s):  
Automorphism Groups ◽  
Cayley Graphs ◽  
Torus Knots ◽  
Knots And Links ◽  
Torus Links

We compute Cayley graphs and automorphism groups for all finite [Formula: see text]-quandles of two-bridge and torus knots and links, as well as torus links with an axis.

THE HOMFLY POLYNOMIAL FOR TORUS LINKS FROM CHERN–SIMONS GAUGE THEORY

1995 ◽  
Vol 10 (07) ◽  
pp. 1045-1089 ◽  
Author(s):  
J. M. F. LABASTIDA ◽  
M. MARIÑO
Keyword(s):  
Gauge Theory ◽  
Gauge Group ◽  
Homfly Polynomial ◽  
Fundamental Representation ◽  
Torus Knots ◽  
Polynomial Invariants ◽  
Chern Simons ◽  
Knots And Links ◽  
Torus Links

Polynomial invariants corresponding to the fundamental representation of the gauge group SU(N) are computed for arbitrary torus knots and links in the framework of Chern–Simons gauge theory making use of knot operators. As a result, a formula for the HOMFLY polynomial for arbitrary torus links is presented.

scholarly journals LEGENDRIAN KNOTS AND LINKS CLASSIFIED BY CLASSICAL INVARIANTS

2007 ◽  
Vol 09 (02) ◽  
pp. 135-162 ◽  
Author(s):  
FAN DING ◽  
HANSJÖRG GEIGES
Keyword(s):  
Rotation Number ◽  
Contact Structure ◽  
Analogous Result ◽  
Jet Space ◽  
Legendrian Knots ◽  
Torus Knots ◽  
Oriented Link ◽  
Linking Number ◽  
Tight Contact ◽  
Knots And Links

It is shown that Legendrian (respectively transverse) cable links in S3 with its standard tight contact structure, i.e. links consisting of an unknot and a cable of that unknot, are classified by their oriented link type and the classical invariants (Thurston–Bennequin invariant and rotation number in the Legendrian case, self-linking number in the transverse case). The analogous result is proved for torus knots in the 1-jet space J1(S1) with its standard tight contact structure.

TANGLE DECOMPOSITIONS OF DOUBLE TORUS KNOTS AND LINKS

1999 ◽  
Vol 08 (07) ◽  
pp. 931-939 ◽  
Author(s):  
MAKOTO OZAWA
Keyword(s):  
Torus Knots ◽  
Double Torus ◽  
Knots And Links

We characterize composite double torus knots and links, 2-string composite double torus knots.

scholarly journals STICK INDEX OF KNOTS AND LINKS IN THE CUBIC LATTICE

2012 ◽  
Vol 21 (05) ◽  
pp. 1250041 ◽  
Author(s):  
COLIN ADAMS ◽  
MICHELLE CHU ◽  
THOMAS CRAWFORD ◽  
STEPHANIE JENSEN ◽  
KYLER SIEGEL ◽  
...  
Keyword(s):  
Infinite Class ◽  
Torus Knots ◽  
3 Dimensional ◽  
Cubic Lattice ◽  
End To End ◽  
Knots And Links

The cubic lattice stick index of a knot type is the least number of sticks glued end-to-end that are necessary to construct the knot type in the 3-dimensional cubic lattice. We present the cubic lattice stick index of various knots and links, including all (p, p + 1)-torus knots, and show how composing and taking satellites can be used to obtain the cubic lattice stick index for a relatively large infinite class of knots. Additionally, we present several bounds relating cubic lattice stick index to other known invariants.

scholarly journals Isogonal weavings on the sphere: knots, links, polycatenanes

2020 ◽  
Vol 76 (5) ◽  
pp. 611-621
Author(s):  
Michael O'Keeffe ◽  
Michael M. J. Treacy
Keyword(s):  
Piecewise Linear ◽  
Infinite Family ◽  
Materials Chemistry ◽  
Torus Knots ◽  
Vertex Transitive ◽  
Knots And Links ◽  
Complete Account

Mathematical knots and links are described as piecewise linear – straight, non-intersecting sticks meeting at corners. Isogonal structures have all corners related by symmetry (`vertex'-transitive). Corner- and stick-transitive structures are termed regular. No regular knots are found. Regular links are cubic or icosahedral and a complete account of these (36 in number) is given, including optimal (thickest-stick) embeddings. Stick 2-transitive isogonal structures are again cubic and icosahedral and also encompass the infinite family of torus knots and links. The major types of these structures are identified and reported with optimal embeddings. The relevance of this work to materials chemistry and biochemistry is noted.

scholarly journals CLASSIFICATION OF TANGLE SOLUTIONS FOR INTEGRASES, A PROTEIN FAMILY THAT CHANGES DNA TOPOLOGY

2007 ◽  
Vol 16 (08) ◽  
pp. 969-995 ◽  
Author(s):  
DOROTHY BUCK ◽  
CYNTHIA VERJOVSKY MARCOTTE
Keyword(s):  
A Priori ◽  
Dna Topology ◽  
Dehn Surgery ◽  
Torus Knots ◽  
Circular Dna ◽  
Fourth Class ◽  
Two Systems ◽  
Knots And Links ◽  
Integrase Protein

A generic integrase protein, when acting on circular DNA, often changes the DNA topology by transforming unknotted circles into torus knots and links. Two systems of tangle equations — corresponding to two possible orientations of two DNA subsequences — arise when modelling this transformation. With no a priori assumptions on the constituent tangles, we utilize Dehn surgery arguments to completely classify the tangle solutions for each of the two systems. A key step is to combine work of our previous paper [10] with recent results of Kronheimer, Mrowka, Ozsváth and Szabó [39] and work of Ernst [23] to show a certain prime tangle must in fact be a Montesinos tangle. These tangle solutions are divided into three classes, common to both systems, plus a fourth class for one system that contains the sole Montesinos tangle. We discuss the possible biological implications of our classification, and of this novel solution.

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