scholarly journals Triple-crossing number and moves on triple-crossing link diagrams

2019 ◽  
Vol 28 (11) ◽  
pp. 1940001 ◽  
Author(s):  
Colin Adams ◽  
Jim Hoste ◽  
Martin Palmer
Keyword(s):  
Triple Point ◽  
Crossing Number ◽  
Reidemeister Moves ◽  
Link Diagrams ◽  
Triple Points ◽  
Definition Of ◽  
Hopf Link

Every link in the 3-sphere has a projection to the plane where the only singularities are pairwise transverse triple points. The associated diagram, with height information at each triple point, is a triple-crossing diagram of the link. We give a set of diagrammatic moves on triple-crossing diagrams analogous to the Reidemeister moves on ordinary diagrams. The existence of [Formula: see text]-crossing diagrams for every [Formula: see text] greater than one allows the definition of the [Formula: see text]-crossing number. We prove that for any nontrivial, nonsplit link, other than the Hopf link, its triple-crossing number is strictly greater than its quintuple-crossing number.

Mosaic number of knots

2014 ◽  
Vol 23 (13) ◽  
pp. 1450069 ◽  
Author(s):  
Hwa Jeong Lee ◽  
Kyungpyo Hong ◽  
Ho Lee ◽  
Seungsang Oh
Keyword(s):  
Quantum System ◽  
Upper Bound ◽  
Crossing Number ◽  
System A ◽  
Definition Of ◽  
Hopf Link

Lomonaco and Kauffman developed knot mosaics to give a definition of a quantum knot system. This definition is intended to represent an actual physical quantum system. A knot n-mosaic is an n × n matrix of 11 kinds of specific mosaic tiles representing a knot or a link. The mosaic number m(K) of a knot K is the smallest integer n for which K is representable as a knot n-mosaic. In this paper, we establish an upper bound on the mosaic number of a knot or a link K in terms of the crossing number c(K). Let K be a nontrivial knot or a non-split link except the Hopf link. Then m(K) ≤ c(K) + 1. Moreover if K is prime and non-alternating except [Formula: see text] link, then m(K) ≤ c(K) - 1.

Thermodynamic Analysis of Isotope Effects on Triple Points and/or Melting Temperatures

1995 ◽  
Vol 50 (4-5) ◽  
pp. 337-346
Author(s):  
W. Alexander Van Hook
Keyword(s):  
Melting Point ◽  
Physical Properties ◽  
Thermodynamic Analysis ◽  
Triple Point ◽  
Isotope Effects ◽  
Inorganic Salts ◽  
Literature Information ◽  
Melting Temperatures ◽  
Triple Points

Àbstract Available literature information on triple point or melting point isotope effects (and related physical properties) is subjected to thermodynamic analysis and consistency checks. New values for the melting point isotope effects for C6H6/CgD6 and c-C6H12/c-C6D12 are reported. 6Li/7Li melting point isotope effects reported recently by Hidaka and Lunden (Z. Naturforsch. 49 a, 475 (1994)) for various inorganic salts are questioned

scholarly journals Triple-crossing projections, moves on knots and links and their minimal diagrams

2020 ◽  
Vol 29 (04) ◽  
pp. 2050015 ◽  
Author(s):  
Michał Jabłonowski ◽  
Łukasz Trojanowski
Keyword(s):  
Triple Point ◽  
Crossing Number ◽  
Systematic Method ◽  
Generating Set ◽  
New Type ◽  
Knots And Links ◽  
Minimal Generating Set

In this paper, we present a systematic method to generate prime knot and prime link minimal triple-point projections, and then classify all classical prime knots and prime links with triple-crossing number at most four. We also extend the table of known knots and links with triple-crossing number equal to five. By introducing a new type of diagrammatic move, we reduce the number of generating moves on triple-crossing diagrams, and derive a minimal generating set of moves connecting triple-crossing diagrams of the same knot.

On fertility of knot shadows

2020 ◽  
Vol 29 (11) ◽  
pp. 2050080
Author(s):  
Ryo Hanaki
Keyword(s):  
Crossing Number ◽  
Number Formula ◽  
Definition Of

A knot [Formula: see text] is a parent of a knot [Formula: see text] if there exists a minimal crossing diagram [Formula: see text] of [Formula: see text] such that a subset of the crossings of [Formula: see text] can be changed to produce a diagram of [Formula: see text]. A knot [Formula: see text] with crossing number [Formula: see text] is fertile if for any prime knot [Formula: see text] with crossing number less than [Formula: see text], [Formula: see text] is a parent of [Formula: see text]. It is known that only [Formula: see text] are fertile for knots up to 10 crossings. However it is unknown whether there exist other fertile knots. A knot shadow is a diagram without over/under information at all crossings. In this paper, we introduce a definition of fertility for knot shadows. We show that if an alternating knot [Formula: see text] is fertile then the crossing number of [Formula: see text] is less than eight.

REIDEMEISTER MOVES FOR SURFACE ISOTOPIES AND THEIR INTERPRETATION AS MOVES TO MOVIES

1993 ◽  
Vol 02 (03) ◽  
pp. 251-284 ◽  
Author(s):  
J. SCOTT CARTER ◽  
MASAHICO SAITO
Keyword(s):  
Cross Sections ◽  
Height Function ◽  
The Other ◽  
Projection Direction ◽  
Reidemeister Moves ◽  
Link Diagrams ◽  
Fixed Direction ◽  
The Cross ◽  
Knot Diagrams

A movie description of a surface embedded in 4-space is a sequence of knot and link diagrams obtained from a projection of the surface to 3-space by taking 2-dimensional cross sections perpendicular to a fixed direction. In the cross sections, an immersed collection of curves appears, and these are lifted to knot diagrams by using the projection direction from 4-space. We give a set of 15 moves to movies (called movie moves) such that two movies represent isotopic surfaces if and only if there is a sequence of moves from this set that takes one to the other. This result generalizes the Roseman moves which are moves on projections where a height function has not been specified. The first 7 of the movie moves are height function parametrized versions of those given by Roseman. The remaining 8 are moves in which the topology of the projection remains unchanged.

Microstructural Evolution of Aluminum Interconnects During Post-Pattern Anneals: Correlation to Improved Em Lifetime

1994 ◽  
Vol 338 ◽  
Author(s):  
B. Miner ◽  
E.A. Atakov ◽  
A. Shepela ◽  
S. Bill
Keyword(s):  
Grain Boundary ◽  
Microstructural Evolution ◽  
Triple Point ◽  
Length Distribution ◽  
Microstructural Parameters ◽  
Statistical Measures ◽  
Before And After ◽  
Triple Points ◽  
Aluminum Interconnects ◽  
Cluster Length

ABSTRACTThe number of Al triple point junctions (Ntp) correlates inversely to electromigration lifetimes for partially bamboo interconnects that fail by grain boundary (GB) diffusion. This work emphasizes the evolution of statistical microstructural parameters, Ntp and cluster length distribution, during post-pattern anneals. In addition to statistical measures, the structure of specific clusters before and after anneal is compared from TEM images of the same area of the same sample.Each post-pattern anneal lowers Ntp and shortens the length of individual polycrystalline segments, but with diminishing returns for subsequent anneals. With a TiN capping layer, the statistical microstructural improvement is less but the longest clusters, those most probable as failure sites, lose triple points during anneal. The distribution of cluster lengths is characteristic for a process.

scholarly journals ON HOMOTOPIES WITH TRIPLE POINTS OF CLASSICAL KNOTS

2012 ◽  
Vol 21 (04) ◽  
pp. 1250038
Author(s):  
THOMAS FIEDLER ◽  
ARNAUD MORTIER
Keyword(s):  
Triple Point ◽  
Simple Proof ◽  
Intersection Index ◽  
Vassiliev Invariant ◽  
Triple Points ◽  
New Formula

We consider a knot homotopy as a cylinder in 4-space. An ordinary triple point p of the cylinder is called coherent if all three branches intersect at p pairwise with the same intersection index. A triple unknotting of a classical knot K is a homotopy which connects K with the trivial knot and which has as singularities only coherent triple points. We give a new formula for the first Vassiliev invariant v2(K) by using triple unknottings. As a corollary we obtain a very simple proof of the fact that passing a coherent triple point always changes the knot type. As another corollary we show that there are triple unknottings which are not homotopic as triple unknottings even if we allow more complicated singularities to appear in the homotopy of the triple homotopy.

ON A LOCAL MOVE FOR VIRTUAL KNOTS AND LINKS

2009 ◽  
Vol 18 (11) ◽  
pp. 1577-1596 ◽  
Author(s):  
TOSHIYUKI OIKAWA
Keyword(s):  
Sufficient Conditions ◽  
Virtual Link ◽  
Link Invariant ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Linking Number ◽  
Reidemeister Moves ◽  
Link Diagrams ◽  
Necessary And Sufficient ◽  
Knots And Links

We define a local move called a CF-move on virtual link diagrams, and show that any virtual knot can be deformed into a trivial knot by using generalized Reidemeister moves and CF-moves. Moreover, we define a new virtual link invariant n(L) for a virtual 2-component link L whose virtual linking number is an integer. Then we give necessary and sufficient conditions for two virtual 2-component links to be deformed into each other by using generalized Reidemeister moves and CF-moves in terms of a virtual linking number and n(L).

scholarly journals A KHOVANOV TYPE INVARIANT DERIVED FROM AN UNORIENTED HQFT FOR LINKS IN THICKENED SURFACES

2013 ◽  
Vol 24 (10) ◽  
pp. 1350078 ◽  
Author(s):  
KEIJI TAGAMI
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Equivalence Class ◽  
Chain Complex ◽  
Quantum Field ◽  
Link Homology ◽  
Stable Equivalence ◽  
Reidemeister Moves ◽  
Link Diagrams ◽  
Topological Quantum

Two link diagrams on compact surfaces are strongly equivalent if they are related by Reidemeister moves and orientation preserving homeomorphisms of the surfaces. They are stably equivalent if they are related by the two previous operations and adding or removing handles. Turaev and Turner constructed a link homology for each stable equivalence class by applying an unoriented topological quantum field theory (TQFT) to a geometric chain complex similar to Bar-Natan's one. In this paper, by using an unoriented homotopy quantum field theory (HQFT), we construct a link homology for each strong equivalence class. Moreover, our homology yields an invariant of links in the oriented I-bundle of a compact surface.

scholarly journals Topological acoustic triple point

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Sungjoon Park ◽  
Yoonseok Hwang ◽  
Hong Chul Choi ◽  
Bohm-Jung Yang
Keyword(s):  
Triple Point ◽  
Euler Number ◽  
Longitudinal Mode ◽  
High Symmetry ◽  
Time Inversion ◽  
Goldstone Theorem ◽  
Nodal Structure ◽  
Transverse Modes ◽  
Degeneracy Point ◽  
Triple Points

AbstractAcoustic phonon is a classic example of triple degeneracy point in band structure. This triple point always appears in phonon spectrum because of the Nambu–Goldstone theorem. Here, we show that this triple point can carry a topological charge $${\mathfrak{q}}$$ q that is a property of three-band systems with space-time-inversion symmetry. The charge $${\mathfrak{q}}$$ q can equivalently be characterized by the skyrmion number of the longitudinal mode, or by the Euler number of the transverse modes. We call triple points with nontrivial $${\mathfrak{q}}$$ q the topological acoustic triple point (TATP). TATP can also appear at high-symmetry momenta in phonon and spinless electron spectrums when Oh or Th groups protect it. The charge $${\mathfrak{q}}$$ q constrains the nodal structure and wavefunction texture around TATP, and can induce anomalous thermal transport of phonons and orbital Hall effect of electrons. Gapless points protected by the Nambu–Goldstone theorem form a new platform to study the topology of band degeneracies.

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