Triple points of unknotting discs and the Arf invariant of knots

1994 ◽  
Vol 116 (1) ◽  
pp. 119-129 ◽  
Author(s):  
Thomas Fiedler
Keyword(s):  
Triple Point ◽  
Critical Points ◽  
Normal Bundle ◽  
Morse Function ◽  
Euler Number ◽  
Double Points ◽  
Arf Invariant ◽  
Triple Points ◽  
Coordinate Functions

AbstractAn unknotting disc is the ‘trace’ in ℝ4 of a homotopy of a diagram of a knot in ℝ3, which shrinks the diagram to a point. In this paper we study unknotting discs which have as singularities only ordinary triple points. It turns out that the Arf invariant of the knot is determined by the number of triple points in which all three branches of the disc intersect pairwise with the same index. We call such a triple point coherent. This interpretation of the Arf invariant has a surprising consequence:Let S ⊂ ℝ4 be a taut immersed sphere which has as singularities only ordinary triple points. Then the number of coherent triple points in S is even. For example, it is easy to show that there is a taut immersed sphere S with Euler number six of the normal bundle and which has exactly three ordinary double points and no other singularities. So, our result implies that the three double points of S can not be pushed together to create an ordinary triple point without the appearance of new singularities.Here ‘taut’ means that the restriction of one of the coordinate functions on S has exactly two (non-degenerate) critical points, i.e. is a perfect Morse function.

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.

scholarly journals KNOTTED SURFACES AND EQUIVALENCIES OF THEIR DIAGRAMS WITHOUT TRIPLE POINTS

2012 ◽  
Vol 21 (03) ◽  
pp. 1250019 ◽  
Author(s):  
M. JABŁONOWSKI
Keyword(s):  
Triple Point ◽  
Finite Sequence ◽  
Closed Surface ◽  
Branch Points ◽  
Double Points ◽  
Generic Projection ◽  
Projection Image ◽  
Triple Points ◽  
Standard Projection ◽  
Three Space

The singularity set of a generic standard projection to the three space of a closed surface linked in four space, consists of at most three types: double points, triple points or branch points. We say that this generic projection image is p-diagram if it does not contain any triple point. Two p-diagrams of equivalent surface links are called p-equivalent if there exist a finite sequence of local moves, such that each of them is one of the four moves taken from the seven on the well known Roseman list, that connects only p-diagrams. It is natural to ask that whether any of two p-diagrams of equivalent surface links always p-equivalent? We introduce an invariant of p-equivalent diagrams and an example of linked surfaces that answers our question negatively.

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 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.

scholarly journals On singular sets of flat holomorphic mappings with isolated singularities

1978 ◽  
Vol 70 ◽  
pp. 47-80
Author(s):  
Hideo Omoto
Keyword(s):  
Algebraic Geometry ◽  
Singular Point ◽  
Critical Points ◽  
Euler Number ◽  
Holomorphic Mappings ◽  
Algebraic Varieties ◽  
Isolated Singularities ◽  
General Fiber ◽  
Singular Sets ◽  
Algebraic Maps

In [4] B. Iversen studied critical points of algebraic mappings, using algebraic-geometry methods. In particular when algebraic maps have only isolated singularities, he shows the following relation; Let V and S be compact connected non-singular algebraic varieties of dimcV = n, and dimc S = 1, respectively. Suppose f is an algebraic map of V onto S with isolated singularities. Then it follows thatwhere χ denotes the Euler number, μf(p) is the Milnor number of f at the singular point p, and F is the general fiber of f : V → S.

scholarly journals Some examples of minimally degenerate Morse functions

1987 ◽  
Vol 30 (2) ◽  
pp. 289-293 ◽  
Author(s):  
Frances Kirwan
Keyword(s):  
Riemannian Manifold ◽  
Critical Points ◽  
Morse Function ◽  
Compact Riemannian Manifold ◽  
Connected Components ◽  
Morse Inequalities ◽  
Poincaré Polynomial ◽  
Morse Functions ◽  
Image Position

Let X be a compact Riemannian manifold. If f:X→ℝ is a nondegenerate Morse function in the sense of Bott [2] then one has Morse inequalities which can be expressed in the formwhere Pt(X) is the Poincaré polynomial Σtidim Hi(X;ℚ of X ann {Cβ|β ∈B} are the connected components of the set of critical points for f For any polynomial Q(t)∈ℤ[t] we write Q(t)≧0 if all the coefficients of Q are nonnegative.

scholarly journals Umbilical Submanifolds and Morse Functions

1972 ◽  
Vol 48 ◽  
pp. 197-201 ◽  
Author(s):  
Katsumi Nomizu ◽  
Lucio Rodríguez
Keyword(s):  
Euclidean Space ◽  
Differentiable Function ◽  
Critical Points ◽  
Morse Function ◽  
Differentiable Manifold ◽  
Morse Functions

Let Mn be a differentiable manifold (of class C∞). By a Morse function on Mn we mean a differentiable function whose critical points are all non-degenerate. If f is an immersion of Mn into a Euclidean space Rm, we may obtain Morse functions on Mn in the following way.

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.

scholarly journals A Remark on the Heat Equation and Minimal Morse Functions on Tori and Spheres

2013 ◽  
Vol 9 (17) ◽  
pp. 11-20
Author(s):  
Carlos Cadavid ◽  
Juan Diego Vélez
Keyword(s):  
Riemannian Manifold ◽  
Heat Equation ◽  
Critical Points ◽  
Morse Function ◽  
Initial Condition ◽  
Morse Functions ◽  
Flat Tori

Let (M, g)be a compact, connected riemannian manifold that is homogeneous, i.e. each pair of pointsp, q∈M have isometric neighborhoods. This paper is a first step towards an understanding of the extent to which it is true that for each “generic” initial condition f0, the solution to∂f /∂t= ∆gf, f (·,0) =f0is such that for sufficiently larget, f(·, t) is a minimal Morse function, i.e., a Morse function whose total number of critical points is the minimal possible on M. In this paper we show that this is true for flat tori and round spheres in all dimensions.

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