scholarly journals SIMPLIFIED NUMERICAL FORM OF UNIVERSAL FINITE TYPE INVARIANT OF GAUSS WORDS

2013 ◽  
Vol 22 (08) ◽  
pp. 1350037
Author(s):  
TOMONORI FUKUNAGA ◽  
TAKAYUKI YAMAGUCHI ◽  
TAKAAKI YAMANOI
Keyword(s):  
Normal Form ◽  
Computer Program ◽  
Finite Type ◽  
Group Structure ◽  
Complete Classification ◽  
Finite Type Invariants ◽  
Partial Classification ◽  
Type Invariant

In this paper, we study the finite type invariants of Gauss words. In the Polyak algebra techniques, we reduce the determination of the group structure to transformation of a matrix into its Smith normal form and we give the simplified form of a universal finite type invariant by means of the isomorphism of this transformation. The advantage of this process is that we can implement it as a computer program. We obtain the universal finite type invariant of degrees 4, 5 and 6 explicitly. Moreover, as an application, we give the complete classification of Gauss words of rank 4 and the partial classification of Gauss words of rank 5 where the distinction of only one pair remains.

Progress in mineralogy

1977 ◽  
Vol 286 (1336) ◽  
pp. 235-237
Keyword(s):  
Earth Science ◽  
Complete Classification ◽  
Applied Physics ◽  
X Ray Diffraction ◽  
Full Effect ◽  
Spin Resonance ◽  
Great Progress ◽  
The Impact

Since the fiftieth anniversary of the Mineralogical Society in 1926, there has been an epoch of great progress, resulting from the impact of applied physics and from the generally widened horizons of Earth science. Description of the morphology of crystals by means of the goniometer, determination of the optics of minerals in transmitted light and of their chemistry by wet methods had already been carried to an advanced stage, but in the eight years up to 1934 the full effect of the application of X-ray diffraction to crystallography by von Laue, W. H. & W. L. Bragg, Jackson, Maugin, Pauling, W. H. Taylor, Warren, West and Wyckoff was felt, leading to a virtually complete classification of minerals on the basis of atomic structure (Bragg 1937). This has stood the test of time for all minerals save chrysotile, and has been fundamental to most other developments in mineralogy. Active fields in structure analysis today include the basis of ordering over octahedral and tetrahedral sites in silicates, and the factors controlling bond-lengths and angles; nuclear magnetic resonance (Bloch 1946; Purcell 1946) and electron spin resonance (Zavoisky 1945) are contributory techniques.

2- AND 3-VARIATIONS AND FINITE TYPE INVARIANTS OF DEGREE 2 AND 3

2013 ◽  
Vol 22 (08) ◽  
pp. 1350042 ◽  
Author(s):  
MIGIWA SAKURAI
Keyword(s):  
Finite Type ◽  
Lower Bounds ◽  
Virtual Knots ◽  
Finite Type Invariants ◽  
Type Invariant

Goussarov, Polyak and Viro defined a finite type invariant and a local move called an n-variation for virtual knots. In this paper, we give the differences of the values of the finite type invariants of degree 2 and 3 between two virtual knots which can be transformed into each other by a 2- and 3-variation, respectively. As a result, we obtain lower bounds of the distance between long virtual knots by 2-variations and the distance between virtual knots by 3-variations by using the values of the finite type invariants of degree 2 and 3, respectively.

FINITE TYPE INVARIANTS FOR SINGULAR SURFACE BRAIDS ASSOCIATED WITH SIMPLE 1-HANDLE SURGERIES

2004 ◽  
Vol 13 (01) ◽  
pp. 1-11
Author(s):  
MASAHIDE IWAKIRI
Keyword(s):  
Finite Type ◽  
Singular Surface ◽  
Finite Type Invariants ◽  
Type Invariant

S. Kamada introduced finite type invariants of knotted surfaces in 4-space associated with finger moves and 1-handle surgeries. In this paper, we define finite type invariants of surface braids associated with simple 1-handle surgeries and prove that a certain set of finite type invariants controls all finite type invariants. As a consequence, we see that every finite type invariant is not a complete invariant.

scholarly journals ON CYCLES AND COVERINGS ASSOCIATED TO A KNOT

2013 ◽  
Vol 22 (13) ◽  
pp. 1350074
Author(s):  
LILYA LYUBICH ◽  
MIKHAIL LYUBICH
Keyword(s):  
Dynamical System ◽  
Abelian Group ◽  
Finite Type ◽  
Weak Topology ◽  
Knot Theory ◽  
Commutator Subgroup ◽  
Finite Abelian Group ◽  
Complete Classification ◽  
And Shifts

Let [Formula: see text] be a knot, G be the knot group, K be its commutator subgroup, and x be a distinguished meridian. Let Σ be a finite abelian group. The dynamical system introduced by Silver and Williams in [Augmented group systems and n-knots, Math. Ann.296 (1993) 585–593; Augmented group systems and shifts of finite type, Israel J. Math.95 (1996) 231–251] consisting of the set Hom (K, Σ) of all representations ρ : K → Σ endowed with the weak topology, together with the homeomorphism [Formula: see text] is finite, i.e. it consists of several cycles. In [Periodic orbits of a dynamical system related to a knot, J. Knot Theory Ramifications20(3) (2011) 411–426] we found the lengths of these cycles for Σ = ℤ/p,p is prime, in terms of the roots of the Alexander polynomial of the knot, mod p. In this paper we generalize this result to a general abelian group Σ. This gives a complete classification of depth 2 solvable coverings over [Formula: see text].

scholarly journals Petrographic structures and Hardy – Weinberg equilibrium

2020 ◽  
Vol 242 ◽  
pp. 133
Author(s):  
Yury VOYTEKHOVSKY ◽  
Alena ZAKHAROVA
Keyword(s):  
Quadratic Forms ◽  
Dimensional Space ◽  
Complete Classification ◽  
Algebraic Expression ◽  
Diagonal Form ◽  
Hardy Weinberg Equilibrium ◽  
Algebraic Forms ◽  
Canonical Diagonal Form

The article is devoted to the most narrative side of modern petrography – the definition, classification and nomenclature of petrographic structures. We suggest a mathematical formalism using the theory of quadratic forms (with a promising extension to algebraic forms of the third and fourth orders) and statistics of binary (ternary and quaternary, respectively) intergranular contacts in a polymineralic rock. It allows constructing a complete classification of petrographic structures with boundaries corresponding to Hardy – Weinberg equilibria. The algebraic expression of the petrographic structure is the canonical diagonal form of the symmetric probability matrix of binary intergranular contacts in the rock. Each petrographic structure is uniquely associated with a structural indicatrix – the central quadratic surface in n-dimensional space, where n is the number of minerals composing the rock. Structural indicatrix is an analogue of the conoscopic figure used for optical recognition of minerals. We show that the continuity of changes in the organization of rocks (i.e., the probabilities of various intergranular contacts) does not contradict a dramatic change in the structure of the rocks, neighboring within the classification. This solved the problem, which seemed insoluble to A.Harker and E.S.Fedorov. The technique was used to describe the granite structures of the Salminsky pluton (Karelia) and the Akzhailau massif (Kazakhstan) and is potentially applicable for the monotonous strata differentiation, section correlation, or wherever an unambiguous, reproducible determination of petrographic structures is needed. An important promising task of the method is to extract rocks' genetic information from the obtained data.

THE TOPOLOGICAL IHX RELATION

2001 ◽  
Vol 10 (02) ◽  
pp. 309-329
Author(s):  
NATHAN HABEGGER
Keyword(s):  
Finite Type ◽  
Finite Type Invariants

We give an exposition of the classification of finite type invariants of homology 3-spheres. A new conceptually-based proof of the topological IHX relation, needed to show the well-definedness of diagrams to manifolds, is given

CLASPER-MOVES AMONG RIBBON 2-KNOTS CHARACTERIZING THEIR FINITE TYPE INVARIANTS

2006 ◽  
Vol 15 (09) ◽  
pp. 1163-1199 ◽  
Author(s):  
TADAYUKI WATANABE
Keyword(s):  
Finite Type ◽  
Finite Type Invariants ◽  
Topological Interpretation ◽  
Type Invariant

Habiro found in his thesis a topological interpretation of finite type invariants of knots in terms of local moves called Habiro's Ck-moves. Ck-moves are defined by using his claspers. In this paper we define "oriented" claspers and RCk-moves among ribbon 2-knots as modifications of Habiro's notions to give a similar interpretation of Habiro–Kanenobu–Shima's finite type invariants of ribbon 2-knots. It works also for ribbon 1-knots. Furthermore, by using oriented claspers for ribbon 1-knots, we can prove Habiro–Shima's conjecture in the case of ℚ-valued invariants, saying that ℚ-valued Habiro–Kanenobu–Shima finite type invariant and ℚ-valued Vassiliev–Goussarov finite type invariant are the same thing.

scholarly journals Brunnian links, claspers and Goussarov–Vassiliev finite type invariants

2007 ◽  
Vol 142 (3) ◽  
pp. 459-468 ◽  
Author(s):  
KAZUO HABIRO
Keyword(s):  
Finite Type ◽  
Finite Type Invariants ◽  
Type Invariant ◽  
Brunnian Links

AbstractGoussarov and the author independently proved that two knots in S3 have the same values of finite type invariants of degree <n if and only if they are Cn-equivalent, which means that they are equivalent up to modification by a kind of geometric commutator of class n. This property does not generalize to links with more than one component.In this paper, we study the case of Brunnian links, which are links whose proper sublinks are trivial. We prove that if n ≥ 1, then an (n+1)-component Brunnian link L is Cn-equivalent to an unlink. We also prove that if n ≥ 2, then L can not be distinguished from an unlink by any Goussarov–Vassiliev finite type invariant of degree <2n.

scholarly journals FINITE TYPE INVARIANTS AND MILNOR INVARIANTS FOR BRUNNIAN LINKS

2008 ◽  
Vol 19 (06) ◽  
pp. 747-766 ◽  
Author(s):  
KAZUO HABIRO ◽  
JEAN-BAPTISTE MEILHAN
Keyword(s):  
Quadratic Form ◽  
Finite Type ◽  
Finite Type Invariants ◽  
Type Invariant ◽  
Milnor Invariants ◽  
Link Homotopy ◽  
Homotopy Invariants ◽  
Brunnian Links

A link L in the 3-sphere is called Brunnian if every proper sublink of L is trivial. In a previous paper, Habiro proved that the restriction to Brunnian links of any Goussarov–Vassiliev finite type invariant of (n + 1)-component links of degree < 2n is trivial. The purpose of this paper is to study the first nontrivial case. We show that the restriction of an invariant of degree 2n to (n + 1)-component Brunnian links can be expressed as a quadratic form on the Milnor link-homotopy invariants of length n + 1.

scholarly journals FINITE TYPE LINK HOMOTOPY INVARIANTS II: Milnor's $\bar\mu$-Invariants

2000 ◽  
Vol 09 (06) ◽  
pp. 735-758 ◽  
Author(s):  
BLAKE MELLOR
Keyword(s):  
Finite Type ◽  
Higher Order ◽  
Finite Type Invariants ◽  
Homotopy Invariant ◽  
Type Invariant ◽  
Link Homotopy ◽  
Homotopy Invariants

We define a notion of finite type invariants for links with a fixed linking matrix. We show that Milnor's link homotopy invariant [Formula: see text] is a finite type invariant, of type 1, in this sense. We also generalize this approach to Milnor's higher order [Formula: see text] invariants and show that they are also, in a sense, of finite type. Finally, we compare our approach to another approach for defining finite type invariants within linking classes.

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