arf invariant
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Author(s):  
Kengo Kawamura
Keyword(s):  

We introduce the notion of bicolored diagrams which are closely related to the region crossing changes. Moreover, we refine Cheng’s results on the region crossing changes and propose a certain way to calculate the Arf invariant of a proper link using a bicolored diagram.


2018 ◽  
Vol 69 (4) ◽  
pp. 1287-1302
Author(s):  
Nariya Kawazumi

Abstract We compute the mapping class group orbits in the homotopy set of framings of a compact connected oriented surface with non-empty boundary. In the case g≥2, the computation is some modification of Johnson’s results (D. Johnson, Spin structures and quadratic forms on surfaces, J. London Math. Soc. (2)22 (1980), 365–373; D. Johnson, An abelian quotient of the mapping class group ℐg, Math. Ann.249 (1980), 225–242) and certain arguments on the Arf invariant, while we need an extra invariant for the genus 1 case. In addition, we discuss how this invariant behaves in the relative case, which Randal-Williams (O. Randal-Williams, Homology of the moduli spaces and mapping class groups of framed, r-Spin and Pin surfaces, J. Topology7 (2014), 155–186) studied for g≥2.


2013 ◽  
Vol 155 (2) ◽  
pp. 257-269 ◽  
Author(s):  
ZHIYUN CHENG

AbstractWe prove that region crossing change on a link diagram is an unknotting operation if and only if the link is proper. This generalizes the related results in [10] and [2]. Furthermore by studying the relation between region crossing change and the Arf invariant, a new approach to the Arf invariant of proper links is given.


2012 ◽  
Vol 21 (09) ◽  
pp. 1250086 ◽  
Author(s):  
TAIZO KANENOBU

An oriented 2-component link is called band-trivializable, if it can be unknotted by a single band surgery. We consider whether a given 2-component link is band-trivializable or not. Then we can completely determine the band-trivializability for the prime links with up to 9 crossings. We use the signature, the Jones and Q polynomials, and the Arf invariant. Since a band-trivializable link has 4-ball genus zero, we also give a table for the 4-ball genus of the prime links with up to 9 crossings. Furthermore, we give an additional answer to the problem of whether a (2n + 1)-crossing 2-bridge knot is related to a (2, 2n) torus link or not by a band surgery for n = 3, 4, which comes from the study of a DNA site-specific recombination.


2011 ◽  
Vol 58 (2) ◽  
pp. 125-136
Author(s):  
Falko Lorenz ◽  
Peter Roquette

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