scholarly journals Symmetric criticality for tight knots

2014 ◽  
Vol 23 (02) ◽  
pp. 1450008 ◽  
Author(s):  
Jason Cantarella ◽  
Joseph H. G. Fu ◽  
Matt Mastin ◽  
Jennifer Ellis Royal
Keyword(s):  
Critical Configuration ◽  
Critical Configurations ◽  
Knots And Links

We prove a version of symmetric criticality for ropelength-critical knots. Our theorem implies that a knot or link with a symmetric representative has a ropelength-critical configuration with the same symmetry. We use this to construct new examples of ropelength-critical configurations for knots and links which are different from the ropelength minima for these knot and link types.

scholarly journals Algorithmic Aspects of a Chip-Firing Game

2001 ◽  
Vol 10 (6) ◽  
pp. 505-529 ◽  
Author(s):  
JAN VAN DEN HEUVEL
Keyword(s):  
Critical Configuration ◽  
Critical Configurations ◽  
Alternative Algorithm ◽  
Chip Firing

Algorithmic aspects of a chip-firing game on a graph introduced by Biggs are studied. This variant of the chip-firing game, called the dollar game, has the properties that every starting configuration leads to a so-called critical configuration. The set of critical configurations has many interesting properties. In this paper it is proved that the number of steps needed to reach a critical configuration is polynomial in the number of edges of the graph and the number of chips in the starting configuration, but not necessarily in the size of the input. An alternative algorithm is also described and analysed.

Knots and Links

2004 ◽  
Author(s):  
Peter R. Cromwell
Keyword(s):  
Knots And Links

scholarly journals Invariants of Random Knots and Links

2016 ◽  
Vol 56 (2) ◽  
pp. 274-314 ◽  
Author(s):  
Chaim Even-Zohar ◽  
Joel Hass ◽  
Nati Linial ◽  
Tahl Nowik
Keyword(s):  
Knots And Links ◽  
Random Knots

Stick numbers of Montesinos knots

2021 ◽  
pp. 2150013
Author(s):  
Hwa Jeong Lee ◽  
Sungjong No ◽  
Seungsang Oh
Keyword(s):  
Upper Bound ◽  
Crossing Number ◽  
Number Formula ◽  
Knots And Links

Negami found an upper bound on the stick number [Formula: see text] of a nontrivial knot [Formula: see text] in terms of the minimal crossing number [Formula: see text]: [Formula: see text]. Huh and Oh found an improved upper bound: [Formula: see text]. Huh, No and Oh proved that [Formula: see text] for a [Formula: see text]-bridge knot or link [Formula: see text] with at least six crossings. As a sequel to this study, we present an upper bound on the stick number of Montesinos knots and links. Let [Formula: see text] be a knot or link which admits a reduced Montesinos diagram with [Formula: see text] crossings. If each rational tangle in the diagram has five or more index of the related Conway notation, then [Formula: see text]. Furthermore, if [Formula: see text] is alternating, then we can additionally reduce the upper bound by [Formula: see text].

Piecewise-linear embeddings of knots and links with rotoinversion symmetry

2021 ◽  
Vol 77 (5) ◽  
Author(s):  
Michael O'Keeffe ◽  
Michael M. J. Treacy
Keyword(s):  
Piecewise Linear ◽  
Infinite Family ◽  
Knots And Links

This article describes the simplest members of an infinite family of knots and links that have achiral piecewise-linear embeddings in which linear segments (sticks) meet at corners. The structures described are all corner- and stick-2-transitive – the smallest possible for achiral knots.

scholarly journals Biquasiles and dual graph diagrams

2017 ◽  
Vol 26 (08) ◽  
pp. 1750048 ◽  
Author(s):  
Deanna Needell ◽  
Sam Nelson
Keyword(s):  
Dual Graph ◽  
Algebraic Structures ◽  
Combinatorial Structures ◽  
Reidemeister Moves ◽  
Knots And Links ◽  
The Way

We introduce dual graph diagrams representing oriented knots and links. We use these combinatorial structures to define corresponding algebraic structures which we call biquasiles whose axioms are motivated by dual graph Reidemeister moves, generalizing the Dehn presentation of the knot group analogously to the way quandles and biquandles generalize the Wirtinger presentation. We use these structures to define invariants of oriented knots and links and provide examples.

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.

Anomalies in Strongly Coupled Harmonic Quantum Brownian Motion

2013 ◽  
Vol 20 (01) ◽  
pp. 1350002 ◽  
Author(s):  
F. Giraldi ◽  
F. Petruccione
Keyword(s):  
Time Evolution ◽  
Power Law ◽  
Weak Coupling ◽  
Inverse Power ◽  
Strong Coupling Regime ◽  
Weak Coupling Regime ◽  
Spectral Densities ◽  
Inverse Power Law ◽  
Critical Configurations ◽  
Long Time

The exact dynamics of a quantum damped harmonic oscillator coupled to a reservoir of boson modes has been formally described in terms of the coupling function, both in weak and strong coupling regime. In this scenario, we provide a further description of the exact dynamics through integral transforms. We focus on a special class of spectral densities, sub-ohmic at low frequencies, and including integrable divergencies referred to as photonic band gaps. The Drude form of the spectral densities is recovered as upper limit. Starting from special distributions of coherent states as external reservoir, the exact time evolution, described through Fox H-functions, shows long time inverse power law decays, departing from the exponential-like relaxations obtained for the Drude model. Different from the weak coupling regime, in the sub-ohmic condition, undamped oscillations plus inverse power law relaxations appear in the long time evolution of the observables position and momentum. Under the same condition, the number of excitations shows trapping of the population of the excited levels and oscillations enveloped in inverse power law relaxations. Similarly to the weak coupling regime, critical configurations give arbitrarily slow relaxations useful for the control of the dynamics. If compared to the value obtained in weak coupling condition, for strong couplings the critical frequency is enhanced by a factor 4.

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