scholarly journals Band description of knots and Vassiliev invariants

2002 ◽  
Vol 133 (2) ◽  
pp. 325-343 ◽  
Author(s):  
KOUKI TANIYAMA ◽  
AKIRA YASUHARA
Keyword(s):  
Natural Number ◽  
Algebraic Condition ◽  
Vassiliev Invariants

In the 1990s, Habiro defined Ck-move of oriented links for each natural number k [5]. A Ck-move is a kind of local move of oriented links, and two oriented knots have the same Vassiliev invariants of order [les ] k−1 if and only if they are transformed into each other by Ck-moves. Thus he has succeeded in deducing a geometric conclusion from an algebraic condition. However, this theorem appears only in his recent paper [6], in which he develops his original clasper theory and obtains the theorem as a consequence of clasper theory. We note that the ‘if’ part of the theorem is also shown in [4], [9], [10] and [16], and in [13] Stanford gives another characterization of knots with the same Vassiliev invariants of order [les ] k−1.

scholarly journals SEIFERT SURFACES, COMMUTATORS AND VASSILIEV INVARIANTS

2007 ◽  
Vol 16 (10) ◽  
pp. 1295-1329
Author(s):  
E. KALFAGIANNI ◽  
XIAO-SONG LIN
Keyword(s):  
Fundamental Group ◽  
Lower Central Series ◽  
Central Series ◽  
Seifert Surface ◽  
Vassiliev Invariants ◽  
Seifert Surfaces

We show that the Vassiliev invariants of a knot K, are obstructions to finding a regular Seifert surface, S, whose complement looks "simple" (e.g. like the complement of a disc) to the lower central series of its fundamental group. We also conjecture a characterization of knots whose invariants of all orders vanish in terms of their Seifert surfaces.

scholarly journals Another characterization of congruence distributive varieties

2020 ◽  
Vol 57 (3) ◽  
pp. 284-289
Author(s):  
Paolo Lipparini
Keyword(s):  
Natural Number ◽  
Congruence Identity ◽  
Congruence Distributive

AbstractWe provide a Maltsev characterization of congruence distributive varieties by showing that a variety 𝓥 is congruence distributive if and only if the congruence identity … (k factors) holds in 𝓥, for some natural number k.

ON THE Cn-DISTANCE AND VASSILIEV INVARIANTS

2012 ◽  
Vol 21 (10) ◽  
pp. 1250097 ◽  
Author(s):  
SUMIKO HORIUCHI ◽  
YOSHIYUKI OHYAMA
Keyword(s):  
Natural Number ◽  
Natural Numbers ◽  
Vassiliev Invariants ◽  
Minimum Number

A local move called a Cn-move is closely related to Vassiliev invariants. A Cn-distance between two knots K and L, denoted by dCn(K, L), is the minimum number of times of Cn-moves needed to transform K into L. Let p and q be natural numbers with p > q ≥ 1. In this paper, we show that for any pair of knots K1 and K2 with dCn(K1, K2) = p and for any given natural number m, there exist infinitely many knots Jj(j = 1, 2, …) such that dCn(K1, Jj) = q and dCn(Jj, K2) = p - q, and they have the same Vassiliev invariants of order less than or equal to m. In the case of n = 1 or 2, the knots Jj(j = 1, 2, …) satisfy more conditions.

scholarly journals Finding Unavoidable Colorful Patterns in Multicolored Graphs

10.37236/8184 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Matt Bowen ◽  
Ander Lamaison ◽  
Alp Müyesser
Keyword(s):  
Natural Number ◽  
Complete Graph ◽  
Type Problem ◽  
Polish Spaces ◽  
Subgraph Isomorphic ◽  
Simple Characterization ◽  
Color Class ◽  
Ramsey Type

We provide multicolored and infinite generalizations for a Ramsey-type problem raised by Bollobás, concerning colorings of $K_n$ where each color is well-represented. Let $\chi$ be a coloring of the edges of a complete graph on $n$ vertices into $r$ colors. We call $\chi$ $\varepsilon$-balanced if all color classes have $\varepsilon$ fraction of the edges. Fix some graph $H$, together with an $r$-coloring of its edges. Consider the smallest natural number $R_\varepsilon^r(H)$ such that for all $n\geq R_\varepsilon^r(H)$, all $\varepsilon$-balanced colorings $\chi$ of $K_n$ contain a subgraph isomorphic to $H$ in its coloring. Bollobás conjectured a simple characterization of $H$ for which $R_\varepsilon^2(H)$ is finite, which was later proved by Cutler and Montágh. Here, we obtain a characterization for arbitrary values of $r$, as well as asymptotically tight bounds. We also discuss generalizations to graphs defined on perfect Polish spaces, where the corresponding notion of balancedness is each color class being non-meagre. 

A characterization of lambda-terms transforming numerals

2016 ◽  
Vol 26 ◽  
Author(s):  
PAWEŁ PARYS
Keyword(s):  
Natural Number ◽  
Linear Transformation ◽  
Fixed Number ◽  
The Other ◽  
Natural Numbers ◽  
Data Types ◽  
Fixed Type ◽  
Intersection Types ◽  
The One

AbstractIt is well known that simply typed λ-terms can be used to represent numbers, as well as some other data types. We show that λ-terms of each fixed (but possibly very complicated) type can be described by a finite piece of information (a set of appropriately defined intersection types) and by a vector of natural numbers. On the one hand, the description is compositional: having only the finite piece of information for two closed λ-terms M and N, we can determine its counterpart for MN, and a linear transformation that applied to the vectors of numbers for M and N gives us the vector for MN. On the other hand, when a λ-term represents a natural number, then this number is approximated by a number in the vector corresponding to this λ-term. As a consequence, we prove that in a λ-term of a fixed type, we can store only a fixed number of natural numbers, in such a way that they can be extracted using λ-terms. More precisely, while representing k numbers in a closed λ-term of some type, we only require that there are k closed λ-terms M1,. . .,Mk such that Mi takes as argument the λ-term representing the k-tuple, and returns the i-th number in the tuple (we do not require that, using λ-calculus, one can construct the representation of the k-tuple out of the k numbers in the tuple). Moreover, the same result holds when we allow that the numbers can be extracted approximately, up to some error (even when we only want to know whether a set is bounded or not). All the results remain true when we allow the Y combinator (recursion) in our λ-terms, as well as uninterpreted constants.

APPLICATION OF BRAIDING SEQUENCES, I: ON THE CHARACTERIZATION OF VASSILIEV AND POLYNOMIAL LINK INVARIANTS

2010 ◽  
Vol 12 (05) ◽  
pp. 681-726 ◽  
Author(s):  
A. STOIMENOW
Keyword(s):  
Finite Number ◽  
Polynomial Growth ◽  
Embedded Graphs ◽  
Vassiliev Invariants ◽  
Link Invariants ◽  
Polynomial Coefficients ◽  
Linearly Independent

We apply the concept of braiding sequences to extend the polynomial growth result for Vassiliev invariants to links, tangles and embedded graphs. It implies the non-existence of Vassiliev invariants that depend on any finite number of link polynomial coefficients, and allows to define two norms on the space of Vassiliev invariants. Then we show that (apart from well-known relations) the coefficients of the link polynomials are linearly independent.

scholarly journals Permutations that preserve asymptotically null sets and statistical convergence

Filomat ◽  
2020 ◽  
Vol 34 (14) ◽  
pp. 4821-4827
Author(s):  
Jeff Connor
Keyword(s):  
Natural Number ◽  
Statistical Convergence ◽  
Asymptotic Density ◽  
Arbitrary Natural Number ◽  
Convergent Sequences

The main result of this article is a characterization of the permutations ?: N ? N that map a set with zero asymptotic density into a set with zero asymptotic density; a permutation has this property if and only if the lower asymptotic density of Cp tends to 1 as p ? ? where p is an arbitrary natural number and Cp = {l : ?-1(l)? lp}. We then show that a permutation has this property if and only if it maps statistically convergent sequences into statistically convergent sequences.

WEB DIAGRAMS AND REALIZATION OF VASSILIEV INVARIANTS BY KNOTS

2000 ◽  
Vol 09 (05) ◽  
pp. 693-701 ◽  
Author(s):  
YOSHIYUKI OHYAMA
Keyword(s):  
Natural Number ◽  
Vassiliev Invariants

Recently it has been proved by K. Taniyama, S. Yamada and the author that for any natural number n and any knot K, there are infinitely many unknotting number one knots, all of whose Vassiliev invariants of order less than or equal to n coincide with those of K. In this paper we give another proof of this result by using web diagrams.

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