DELTA LINK HOMOTOPY FOR TWO COMPONENT LINKS, II

2002 ◽  
Vol 11 (03) ◽  
pp. 353-362 ◽  
Author(s):  
YASUTAKA NAKANISHI ◽  
YOSHIYUKI OHYAMA
Keyword(s):  
Equivalence Relation ◽  
Necessary Condition ◽  
Oriented Link ◽  
Two Component ◽  
Numerical Invariants ◽  
Link Homotopy ◽  
Prime 2

In this note, we will study Δ link homotopy (or self Δ-equivalence), which is an equivalence relation of ordered and oriented link types. Previously, a necessary condition is given in the terms of Conway polynomials for two link types to be Δ link homotopic. A pair of numerical invariants δ1 and δ2 classifies all (ordered and oriented) prime 2-component link types with seven crossings or less up to Δ link homotopy. We will show here that for any pair of integers n1 and n2 there exists a 2-component link κ such that δ1(κ) = n1 and δ2(κ) = n2 provided that at least one of n1 and n2 is even.

scholarly journals SIGNATURES OF COLORED LINKS WITH APPLICATION TO REAL ALGEBRAIC CURVES

2005 ◽  
Vol 14 (07) ◽  
pp. 883-918 ◽  
Author(s):  
V. FLORENS
Keyword(s):  
Algebraic Curves ◽  
Betti Numbers ◽  
Alexander Polynomial ◽  
Necessary Condition ◽  
Oriented Link ◽  
Real Algebraic Curves

We construct the signature of a μ-colored oriented link, as a locally constant integer valued function with domain (S1 - {1})μ. It restricts to the Tristram–Levine's signature on the diagonal and the discontinuities can occur only at the zeros of the colored Alexander polynomial. Moreover, the signature and the related nullity verify the Murasugi–Tristram inequality. This gives a new necessary condition for a link to bound a smoothly and properly embedded surface in B4, with given Betti numbers. As an application, we achieve the classification of the complex orientations of maximal plane non-singular projective algebraic curves of degree 7, up to isotopy.

The self-smoothing number of knots and links

2018 ◽  
Vol 27 (12) ◽  
pp. 1850070
Author(s):  
Hideo Takioka
Keyword(s):  
The Self ◽  
Link Diagram ◽  
Oriented Link ◽  
Two Component ◽  
Crossing Point ◽  
Knots And Links

We call smoothing a self-crossing point of an oriented link diagram self-smoothing. By self-smoothing repeatedly, we obtain an oriented link diagram without self-crossing points. In this paper, we show that every knot has an oriented diagram which becomes a two-component oriented link diagram without self-crossing points by a single self-smoothing.

LINK HOMOTOPY AND QUASI SELF DELTA-EQUIVALENCE FOR LINKS

2000 ◽  
Vol 09 (05) ◽  
pp. 683-691 ◽  
Author(s):  
YASUTAKA NAKANISHI ◽  
TETSUO SHIBUYA
Keyword(s):  
Alexander Polynomial ◽  
Equivalence Relations ◽  
Necessary Condition ◽  
Link Homotopy

In this note, we will study on equivalence relations for links and we will give a necessary condition for equivalence in terms of the Alexander polynomial and show their differences.

scholarly journals On an integrable multi-component Camassa–Holm system arising from Möbius geometry

2021 ◽  
Vol 477 (2251) ◽  
pp. 20210164
Author(s):  
Jing Kang ◽  
Xiaochuan Liu ◽  
Changzheng Qu
Keyword(s):  
Integrable Systems ◽  
Invariant Curve ◽  
Necessary Condition ◽  
Miura Transformation ◽  
Möbius Geometry ◽  
Two Component ◽  
De Vries ◽  
Curve Flows

In this paper, we mainly study the geometric background, integrability and peaked solutions of a ( 1 + n ) -component Camassa–Holm (CH) system and some related multi-component integrable systems. Firstly, we show this system arises from the invariant curve flows in the Möbius geometry and serves as the dual integrable counterpart of a geometrical ( 1 + n ) -component Korteweg–de Vries system in the sense of tri-Hamiltonian duality. Moreover, we obtain an integrable two-component modified CH system using a generalized Miura transformation. Finally, we provide a necessary condition, under which the dual integrable systems can inherit the Bäcklund correspondence from the original ones.

scholarly journals INSTANTON FLOER HOMOLOGY FOR TWO-COMPONENT LINKS

2012 ◽  
Vol 21 (05) ◽  
pp. 1250054 ◽  
Author(s):  
ERIC HARPER ◽  
NIKOLAI SAVELIEV
Keyword(s):  
Euler Characteristic ◽  
Floer Homology ◽  
Integral Homology ◽  
Oriented Link ◽  
Linking Number ◽  
Two Component

For any oriented link of two components in an integral homology 3-sphere, we define an instanton Floer homology whose Euler characteristic is twice the linking number between the components of the link. We show that, for two-component links in the 3-sphere, this Floer homology does not vanish unless the link is split. We also relate our Floer homology to the Kronheimer–Mrowka instanton Floer homology for links.

scholarly journals Self C2-equivalence of two-component links and invariants of link maps

2018 ◽  
Vol 27 (13) ◽  
pp. 1842012
Author(s):  
Sergey A. Melikhov
Keyword(s):  
Two Component ◽  
String Links ◽  
Link Homotopy

We use Kirk’s invariant of link maps [Formula: see text] and its variations due to Koschorke and Kirk–Livingston to deduce results about classical links. Namely, we give a new proof of the Nakanishi–Ohyama classification of two-component links in [Formula: see text] up to [Formula: see text]-link homotopy. We also prove its version for string links, which is due (in a slightly different form) to Fleming–Yasuhara. The proofs do not use Clasper Theory.

Boundary links are self delta-equivalent to trivial links

2007 ◽  
Vol 143 (2) ◽  
pp. 449-458 ◽  
Author(s):  
TETSUO SHIBUYA ◽  
AKIRA YASUHARA
Keyword(s):  
Equivalence Relation ◽  
Link Homotopy ◽  
Trivial Link

AbstractSelf Δ-equivalence is an equivalence relation for links, which is stronger than link-homotopy defined by J. W. Milnor. It was shown that any boundary link is link-homotopic to a trivial link by L. Cervantes and R. A. Fenn and by D. Dimovski independently. In this paper we will show that any boundary link is self Δ-equivalent to a trivial link.

scholarly journals Delta link homotopy for two component links, III

2003 ◽  
Vol 55 (3) ◽  
pp. 641-654 ◽  
Author(s):  
Yasutaka NAKANISHI ◽  
Yoshiyuki OHYAMA
Keyword(s):  
Two Component ◽  
Link Homotopy

SELF Ck-MOVE, QUASI SELF Ck-MOVE AND THE CONWAY POTENTIAL FUNCTION FOR LINKS

2004 ◽  
Vol 13 (07) ◽  
pp. 877-893
Author(s):  
TETSUO SHIBUYA ◽  
AKIRA YASUHARA
Keyword(s):  
Potential Function ◽  
Necessary Conditions ◽  
Alexander Polynomial ◽  
Necessary Condition ◽  
Conway Polynomial ◽  
Link Homotopy

Nakanishi and Shibuya gave a relation between link homotopy and quasi self delta-equivalence. And they also gave a necessary condition for two links to be self delta-equivalent by using the multivariable Alexander polynomial. Link homotopy and quasi self delta-equivalence are also called self C1-equivalence and quasi self C2-equivalence respectively. In this paper, we generalize their results. In Sec. 1, we give a relation between self Ck-equivalence and quasi self Ck+1-equivalence. In Secs. 2 and 3, we give necessary conditions for two links to be self Ck-equivalent by using the multivariable Conway potential function and the Conway polynomial respectively.

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