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.

scholarly journals Movie moves for singular link cobordisms in 4-dimensional space

2016 ◽  
Vol 25 (02) ◽  
pp. 1650013
Author(s):  
Carmen Caprau
Keyword(s):  
Saddle Point ◽  
Cross Sections ◽  
Dimensional Space ◽  
The Other ◽  
Link Diagrams ◽  
Fixed Direction ◽  
Singular Link ◽  
Point Transformations

Two singular links are cobordant if one can be obtained from the other by singular link isotopy together with a combination of births or deaths of simple unknotted curves, and saddle point transformations. A movie description of a singular link cobordism in 4-space is a sequence of singular link diagrams obtained from a projection of the cobordism into 3-space by taking 2-dimensional cross-sections perpendicular to a fixed direction. We present a set of movie moves that are sufficient to connect any two movies of isotopic singular link cobordisms.

Elastic Scattering of Electrons and Positrons by Screened Nuclei

1966 ◽  
Vol 21 (9) ◽  
pp. 1321-1327 ◽  
Author(s):  
Elmar Zeitler ◽  
Haakon Olsen
Keyword(s):  
Elastic Scattering ◽  
Cross Section ◽  
Cross Sections ◽  
Relativistic Effects ◽  
The Other ◽  
Elastic Electron ◽  
Positron Scattering ◽  
Electrons And Positrons ◽  
Two Factors ◽  
The Cross

Results of calculations of cross sections for elastic electron and positron scattering are given in angular steps of 15 degrees for elements Z=6, 13, 29, 50, 82, and 92 and energies T=0.2, 0.4, 0.7, 1, 2, 4, and 10 MeV. The calculation is based on the separability of the cross section into two factors, one describing screening and the other, spin and relativistic effects. The first factor is obtained by the MOLIÈRE approximation 8. The second factor is taken from a paper by DOGGETT and SPENCER 5. Different screening potentials for Z=29 were applied.

Linking invariants of even virtual links

2017 ◽  
Vol 26 (12) ◽  
pp. 1750072 ◽  
Author(s):  
Haruko A. Miyazawa ◽  
Kodai Wada ◽  
Akira Yasuhara
Keyword(s):  
Lower Bound ◽  
The Other ◽  
Virtual Link ◽  
Link Diagram ◽  
Linking Number ◽  
Reidemeister Moves ◽  
Link Diagrams ◽  
Classical Link ◽  
Virtual Links ◽  
The Difference

A virtual link diagram is even if the virtual crossings divide each component into an even number of arcs. The set of even virtual link diagrams is closed under classical and virtual Reidemeister moves, and it contains the set of classical link diagrams. For an even virtual link diagram, we define a certain linking invariant which is similar to the linking number. In contrast to the usual linking number, our linking invariant is not preserved under the forbidden moves. In particular, for two fused isotopic even virtual link diagrams, the difference between the linking invariants of them gives a lower bound of the minimal number of forbidden moves needed to deform one into the other. Moreover, we give an example which shows that the lower bound is best possible.

THE ASSOCIATED PRODUCTION OF THE NEUTRAL TOP-PION WITH A PAIR OF THIRD FAMILY QUARKS AT THE HADRON COLLIDERS

2006 ◽  
Vol 21 (37) ◽  
pp. 2833-2843 ◽  
Author(s):  
XUELEI WANG ◽  
LILI YU ◽  
NAHONG SONG ◽  
WENNA XU
Keyword(s):  
Cross Sections ◽  
The Other ◽  
Tree Level ◽  
Hadron Colliders ◽  
Yukawa Couplings ◽  
The Third ◽  
Flavor Changing ◽  
Associated Production ◽  
Technicolor Model ◽  
The Cross

We study the associated production of the neutral top-pion [Formula: see text] with the third family quarks within the context of the topcolor-assisted technicolor model at the hadron colliders. The studies show that, at the Tevatron, the cross-sections of all these processes are too small to produce enough identified signals. But the cross-sections can be largely enhanced at the LHC. Specially for the processes [Formula: see text] and [Formula: see text], the cross-sections can reach the level of a few hundred fb even a few pb for the light neutral top-pion. With the high yearly luminosity 100 fb-1 at the LHC, over 104 signals can be produced via the above two processes. There exists an ideal flavor-changing mode to detect neutral top-pion, i.e. [Formula: see text], because the SM background of such production mode are very clean. Therefore, we can conclude that neutral top-pion should be observable at the LHC via the processes [Formula: see text] and [Formula: see text]. On the other hand, the statistics available at the LHC via these two processes might be enough to measure the Yukawa couplings [Formula: see text] and [Formula: see text]. Finally, it must be noted that the study of the process [Formula: see text] can give us a good chance to distinguish the TC2 model from the SM and MSSM because there does not exist such similar tree-level favor-changing process in these models.

VIRTUAL HOMOTOPY

2010 ◽  
Vol 19 (07) ◽  
pp. 935-960 ◽  
Author(s):  
H. A. DYE ◽  
LOUIS H. KAUFFMAN
Keyword(s):  
The Other ◽  
Virtual Link ◽  
Magnus Expansion ◽  
Reidemeister Moves ◽  
Link Diagrams ◽  
Virtual Links ◽  
Skein Relation

Two welded (respectively virtual) link diagrams are homotopic if one may be transformed into the other by a sequence of extended Reidemeister moves, classical Reidemeister moves, and self crossing changes. In this paper, we extend Milnor's μ and [Formula: see text] invariants to welded and virtual links. We conclude this paper with several examples, and compute the μ invariants using the Magnus expansion and Polyak's skein relation for the μ invariants.

The Centre of Shear of Aerofoil Sections

1953 ◽  
Vol 57 (508) ◽  
pp. 235-237 ◽  
Author(s):  
John A. Jacobs
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Lateral Surface ◽  
The Other ◽  
Principal Axes ◽  
Unstrained State ◽  
Uniform Cross Section ◽  
Single Force ◽  
The Cross ◽  
Fixed End

Consider a cantilever beam of uniform cross section whose generators are parallel to the z-axis and whose lateral surface is free from surface tractions. The line of centroids of the cross sections in the unstrained state is taken as the z-axis, and the x- and y-axes are the principal axes of the cross section at the centroid of the fixed end z = 0.The other end of the beam (z = l) is subject to forces which reduce to a single force with components (Wx, Wv, 0), transverse to the z-axis, acting through the load point L of this end section (see Fig. 1). The co-ordinates of L are taken as (p, q, l).

scholarly journals Addition to the methodology of research into permanent teeth hardness

2010 ◽  
Vol 62 (3) ◽  
pp. 739-746 ◽  
Author(s):  
Sanja Gnjato
Keyword(s):  
Standard Deviation ◽  
Cross Section ◽  
Cross Sections ◽  
Arithmetic Mean ◽  
The Other ◽  
Permanent Teeth ◽  
Hardness Profile ◽  
Average Hardness ◽  
The Cross ◽  
Statistical Indicators

This paper examines permanent teeth hardness (microhardness) using the Vickers method. An original methodology was developed and adopted for preparing the experimental material, i.e. the cross sections into four characteristic locations on the tooth: corona dentis, cervix dentis, pars medialis radicis dentis and apex radicis dentis. A new 'hardness profile' was introduced, which connects hardness and location along the cross section of the tooth. Hardness was measured 664 times on experimental cross sections with total a average hardness of 73.17 HV according to Vickers and a standard deviation of 55.68 HV. The derived descriptive statistical indicators of tooth hardness were calculated for equivalent cross sections, individual teeth, teeth groups and tooth localizations. Two algorithms were developed for determining the rank of tooth hardness - one for estimating the rank of arithmetic mean of the hardness of the cross sections of the teeth, and the other for estimating the rank of hardness for individual teeth. .

scholarly journals THE MILNOR $\bar{\mu}$ INVARIANTS AND NANOPHRASES

2013 ◽  
Vol 22 (02) ◽  
pp. 1250142 ◽  
Author(s):  
YUKA KOTORII
Keyword(s):  
The Other ◽  
Generalize Formula ◽  
Reidemeister Moves ◽  
Link Diagrams ◽  
Virtual Links ◽  
Link Homotopy

Two link diagrams are link homotopic if one can be transformed into the other by a sequence of Reidemeister moves and self-crossing changes. Milnor introduced invariants under link homotopy called [Formula: see text]. Nanophrases, introduced by Turaev, generalize links. In this paper, we extend the notion of link homotopy to nanophrases. We also generalize [Formula: see text] to the set of those nanophrases that correspond to virtual links.

Ineffective sets and the region crossing change operation

2020 ◽  
Vol 29 (03) ◽  
pp. 2050010
Author(s):  
Miles Clikeman ◽  
Rachel Morris ◽  
Heather M. Russell
Keyword(s):  
Upper Bounds ◽  
The Other ◽  
Link Diagram ◽  
Complete Collection ◽  
Link Diagrams ◽  
Change Operation ◽  
Knot Diagrams

Region crossing change (RCC) is an operation on link diagrams in which all crossings incident to a selected region are changed. Two diagrams are called RCC-equivalent if one can be transformed to the other via a sequence of RCCs. RCC is an unknotting operation but not an unlinking operation. A set of regions of a diagram is called ineffective if RCCs at every region in that set have no net effect on the crossings of the diagram. The main result of this paper is a construction of the complete collection of ineffective sets for any link diagram. This involves a combination of linear algebraic and diagrammatic techniques including a generalization of checkerboard shading called tricoloring. Using this construction of ineffective sets, we provide sharp upper bounds on the maximum number of RCCs needed to transform between RCC-equivalent knot diagrams and reduced 2- and 3-component link diagrams with fixed underlying projections.

scholarly journals VIRTUAL KNOT DIAGRAMS AND THE WITTEN–RESHETIKHIN–TURAEV INVARIANT

2005 ◽  
Vol 14 (08) ◽  
pp. 1045-1075 ◽  
Author(s):  
H. A. DYE ◽  
LOUIS H. KAUFFMAN
Keyword(s):  
Fundamental Group ◽  
Virtual Link ◽  
Virtual Knot ◽  
Reidemeister Moves ◽  
Link Diagrams ◽  
Classical Link ◽  
Kirby Calculus ◽  
Knot Diagrams

The Witten–Reshetikhin–Turaev invariant of classical link diagrams is generalized to virtual link diagrams. This invariant is unchanged by the framed Reidemeister moves and the Kirby calculus. As a result, it is also an invariant of the 3-manifolds represented by the classical link diagrams. This generalization is used to demonstrate that there are virtual knot diagrams with a non-trivial Witten–Reshetikhin–Turaev invariant and trivial 3-manifold fundamental group.

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