Region crossing change, bicolored diagram and Arf invariant

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.

The mapping class group orbits in the framings of compact surfaces

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.

When is region crossing change an unknotting operation?

We 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.

BAND SURGERY ON KNOTS AND LINKS, II

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.