INTEGRAL REGION CHOICE PROBLEMS ON LINK DIAGRAMS

概要

and a knot is a link with one component. A link in the 3-space is presented as the natural
projection image on the 2-plane R2 where the singular points are transverse double points
with over/under information. This presentation is called a link diagram or a diagram of
the link. A diagram of a link in the 3-sphere S3 = R3 ∪ {∞} is given on the 2-sphere
S2 = R2 ∪ {∞} similarly. For each link diagram, a connected component of the complement
of the projection image on R2 or S2 is called a region.
In [10], Shimizu defined a region crossing change at a region for a diagram to be the
crossing change at all the crossings on the boundary of the region as an unknotting operation
for a knot diagram, which was proposed by Kengo Kishimoto. For example in Fig.1, the left
diagram is changed to the right diagram, choosing the region marked with ∗ as illustrated on
the middle and changing the three crossings on the boundary of the marked region. In [3, 4],
Cheng and Gao gave a necessary and sufficient condition that a region crossing change is an
unknotting operation on a link diagram.
It is known that a region crossing change can be interpreted as follows. We call a diagram
ignored over/under information a projection. Let each crossing of the given projection be
equipped with a score 0 or 1 modulo 2. We choose a region of the projection. Then the
scores of all the crossings on its boundary are increased by 1 modulo 2. For example, the
region crossing change illustrated on Fig.1 is interpreted as Fig.2. Shimizu showed that the
scores of all the crossings on any knot diagram become 0 by some choices of regions. Cheng
and Gao induced a Z2 -homomorphism from region crossing changes on link diagrams. In
[6, 7], Hashizume studied structures of their Z2 -homomorphism.
As an extension of a region crossing change to an integral range, Ahara and Suzuki pro2020 Mathematics Subject Classification. 57K10. ...

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